See also:

• BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term)

BINOMIAL (from the See also:

• LAT

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

• TERM

term)  , in See also:

• MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

mathematics, a word first introduced by See also:

• ROBERT
• ROBERT (1275—1343)
• ROBERT, HUBERT (1753-1808)
• ROBERT, LOUIS LEOPOLD (1794-1835)

Robert See also:

• RECORDE, ROBERT (c. 1510-1558)

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 See also:

• BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term)

binomial theorem is a celebrated theorem, originally due to See also:

• SIR

Sir See also:

• ISAAC (Hebrew for " he laughs," on explanatory references to the name, see ABRAHAM)

Isaac See also:

• NEWTON
• NEWTON, ALFRED (1829–1907)
• NEWTON, JOHN (1725-1807)
• NEWTON, JOHN (1823-1895)
• NEWTON, SIR CHARLES THOMAS (1816-1894)
• NEWTON, SIR ISAAC (1642-1727)

Newton, by which any See also:

• POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power of a binomial can be expressed as a See also:

• SERIES (a Latin word from serere, to join)

series . In its See also:

• MODERN

modern See also:

• FORM (Lat. forma)

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 See also:

• ARTICLE (from Lat. articulus, a joint)

article See also:

• ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

ALGEBRA for the See also:

• PROOF (in M. Eng. preove, proeve, preve, &°c., from O. Fr . prueve, proeve, &c., mod. preuve, Late. Lat. proba, probate, to prove, to test the goodness of anything, probus, good)

proof and applications of this theorem; here we shall only treat of the See also:

• HISTORY

history of its See also:

• DISCOVERY

discovery . The See also:

• ORIGINAL

original form of the theorem was first given in a See also:

• LETTER (through Fr. lettre from Lat. littera or litera, letter of the alphabet; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

letter, dated the 13th of See also:

• JUNE

June 1676, from Sir Isaac Newton to See also:

• HENRY
• HENRY (1129-1195)
• HENRY (c. 1108-1139)
• HENRY (c. 1174–1216)
• HENRY (Fr. Henri; Span. Enrique; Ger. Heinrich; Mid. H. Ger. Heinrich and Heimrich; O.H.G. Haimi- or Heimirih, i.e. " prince, or chief of the house," from O.H.G. heim, the Eng. home, and rih, Goth. reiks; compare Lat. rex " king "—" rich," therefore " mig
• HENRY, EDWARD LAMSON (1841– )
• HENRY, JAMES (1798-1876)
• HENRY, JOSEPH (1797-1878)
• HENRY, MATTHEW (1662-1714)
• HENRY, PATRICK (1736–1799)
• HENRY, PRINCE OF BATTENBERG (1858-1896)
• HENRY, ROBERT (1718-1790)
• HENRY, VICTOR (1850– )
• HENRY, WILLIAM (1795-1836)

Henry See also:

• OLDENBURG

Oldenburg for communication to Wilhelm G . See also:

• LEIBNITZ (LEIBNIZ), GOTTFRIED WILHELM

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 See also:

• ROOT (late O.E. rot, adopted from Scand., cf. Norw. and Swed. rot, Dan. rod; the true O.E. word was wyrt, plant, represented in Ger. Wurz or Wurzel; the ultimate root is the same in both words, and is seen in Lat. radix)
• ROOT, ELIHU (1845– )

root is required, p the first See also:

• TERM

term of that quantity, and q the quotient of the See also:

• REST (O. Eng. rest, reste, bed, cognate with other Teutonic forms, e.g. Ger. Rast, Riiste, rest, and probably Gothic Rasta, league, i.e. resting or stopping place)

rest divided by p, me the power, which may be a See also:

• POSITIVE (or PORTABLE) ORGAN

positive or negative integer or a fraction, and a, b, c, &c., the several terms in See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

order, e.g. a = p'", b= aq, c = 2n nbq, and so on . In a second letter, dated the 24th of See also:

• OCTOBER

October 1676, to Olden-See also:

• BURG

burg, Newton gave the See also:

• TRAIN (M. Eng. trayn or trayne, derived through Fr. from Late Lat. trahinare, to drag, draw, Lat. trahere, cf. trail, trace, ultimately from the same source)

train of reasoning by which he devised the theorem . " In the beginning of my mathematical studies, when I was perusing the See also:

• WORKS

works of the celebrated Dr See also:

• WALLIS, JOHN (1616-1703)

Wallis, and considering the series by the See also:

• INTERPOLATION (from Lat. interpolare, to alter, or insert something fresh, connected with pollee, a polish)

interpolation of which he exhibits the See also:

• AREA

area of the circle and See also:

• HYPERBOLA

hyperbola (for instance, in this series of curves whose See also:

• COMMON

common See also:

• BASE

base or See also:

• AXIS (Lat. for " axle ")

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 See also:

• POWERS, HIRAM (1805-1873)

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 See also:

• FOURTH

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 See also:

• CON

con- tinued multiplication of the terms of this seriesm I o x'--'L---1 xm3-2 ... , &c . . . . This See also:

• RULE

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 See also:

• JOHN
• JOHN (1167–1216)
• JOHN (1290-c. 1320)
• JOHN (1296-1346)
• JOHN (1371–1419)
• JOHN (1468-1532)
• JOHN (1801-1873)
• JOHN (Heb. llni')
• JOHN (ZAPOLYA) (1487-1540)
• JOHN, 4TH MARQUESS OF TWEEDDALE (c. 1695-1762)
• JOHN, DON (1545-1578)
• JOHN, DON (1629–1679)
• JOHN, GOSPEL OF ST
• JOHN, or HAYS (1513–1571)
• JOHN, THE APOSTLE
• JOHN, THE EPISTLES OF

John Wallis's investigations in the method of interpolation . Newton gave no proof, and it was in the Ars Conjectandi (r 713) that See also:

• JAMES
• JAMES (Gr. 'IlrKw,l3or, the Heb. Ya`akob or Jacob)
• JAMES (JAMES FRANCIS EDWARD STUART) (1688-1766)
• JAMES, 2ND EARL OF DOUGLAS AND MAR(c. 1358–1388)
• JAMES, DAVID (1839-1893)
• JAMES, EPISTLE OF
• JAMES, GEORGE PAYNE RAINSFOP
• JAMES, HENRY (1843— )
• JAMES, JOHN ANGELL (1785-1859)
• JAMES, THOMAS (c. 1573–1629)
• JAMES, WILLIAM (1842–1910)
• JAMES, WILLIAM (d. 1827)

James See also:

• BERNOULLI

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 See also:

• EULER, LEONHARD (1707-1783)

Euler's proof for negative and fractional values being faulty, and was finally given by Niels Heinrik See also:

• ABEL (better ABELL), THOMAS (d. 1540)
• ABEL (Hebrew for breath)
• ABEL, KARL FRIEDRICH (1725-1787)
• ABEL, NIELS HENRIK (1802-1829)
• ABEL, SIR FREDERICK AUGUSTUS, BART

Abel . The multi- (or poly-) nomial theorem has for its See also:

• OBJECT

object the expansion of any power of a multinomial and was discussed in 1697 by See also:

• ABRAHAM, or ABRAM (Hebrew for " father is high ")

Abraham See also:

• DEMOIVRE, ABRAHAM (1667-1754)

Demoivre (see COMBINATORIAL See also:

• ANALYSIS (Gr. avci and Meta, to break up into parts)

ANALYSIS) . gondence of Scientific Men of the 17th See also:

• CENTURY (from Lat. centuria, a division of a hundred men)

Century (1841); M . Cantor, eschichte der Mathematik (1894-1901) . BINTURONG (Arctictis binturong), the single See also:

• SPECIES

species of the viverrine genus Arctictis, ranging from See also:

• NEPAL, NEPAUL

Nepal through the See also:

• MALAY

Malay See also:

• PENINSULA
• PENINSULA (Lat. paeninsula, from paene, almost, and insula, an island)

Peninsula to See also:

• SUMATRA

Sumatra and See also:

• JAVA

Java . This See also:

• ANIMAL
• ANIMAL (Lat. animalis, from anima, breath, soul)

animal, also called the See also:

• BEAR
• BEAR, BLACK
• BEAR, BROWN
• BEAR, GRIZZLY
• BEAR, ISABELLINE
• BEAR, WHITE

bear-See also:

• CAT
• CAT, CIVET
• CAT, HOUSE

cat, is allied to the See also:

• PALM (Lat. palma, Gr. iraX6.jn )
• PALM, JOHANN PHILIPP (1768-1806)

palm-civets, or paradoxures, but differs from the rest of the See also:

• FAMILY

family (Viverridae) by its tufted ears and See also:

• LONG, GEORGE (1800-1879)
• LONG, JOHN DAVIS (1838– )

long, bushy, prehensile tail, which is thick at the root and almost equals in length the See also:

• HEAD (in 0. Eng. heafad; the word is common to Teutonic languages; cf. Dutch hoofd, Ger. Haupt, generally taken to be in origin connected with Lat. caput, Gr. KerbOvi7)
• HEAD, SIR EDMUND WALKER, BART
• HEAD, SIR FRANCIS BOND, BART

head and See also:

• BODY

body together (from 28 to 33 inches) . The See also:

• FUR (connected with O. Fr. forre, a sheath or case; so " an outer covering ")

fur is long and coarse, of a dull See also:

• BLACK
• BLACK, ADAM (1784-1874)
• BLACK, JEREMIAH SULLIVAN (1810–1883)
• BLACK, WILLIAM (1841-1898)

black See also:

• HUE

hue with a See also:

• GREY, CHARLES GREY, 2ND EARL (1764-1845)
• GREY, HENRY GREY, 3RD EARL (1802-1894)
• GREY, LADY JANE (1537-1554)
• GREY, SIR EDWARD
• GREY, SIR GEORGE (1812–1898)

grey See also:

• WASH, THE

wash on the head and fore-limbs . In habits the binturong is nocturnal and arboreal, inhabiting forests, and living on small vertebrates, See also:

• WORMS

worms, See also:

• INSECTS

insects and fruits . It is said to be naturally fierce, but when taken See also:

• YOUNG
• YOUNG, A
• YOUNG, ARTHUR (1741-1820)
• YOUNG, BRIGHAM (1801-1877)
• YOUNG, CHARLES MAYNE (1777–1856)
• YOUNG, EDWARD (1683–1765)
• YOUNG, JAMES (1811-1883)
• YOUNG, THOMAS (1773-1829)

young is easily tamed and becomes See also:

• GENTLE (through the Fr. gentil, from Lat. gentilis, belonging to the same gens, or family)

gentle and playful .