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to the six residuals or apotomes there treated of also, and of whichmention is made under the article Apotomf.. These apotomes,however, become binomials by only changing the sign of the lat-ter term from minus to plus, which therefore are as below ;

First 3 + 5, 4th 4 -f- \/ 3,

2 d 18 + 4, 5th \/ 6 + 2,

3d \/ 24 -[- 18, 6 th a,/ 6 -j— aJ 2.

Binomial Root, in algebra, a root composed ot only twoparts connected with the signs plus or minus.

Binomial, to Extract the Sciuare Root of a, as ofa + ^/b, or c -f- b. Various rules have been given for thispurpose. The first is that of Lucas De Burgo, in his_ “ Summade Arith.” &c. which is this : When one part, as a, is rational,divide it into two parts such, that their product may be equal to]th of the number under the radical b ; then shall tlie sum of theroots of those parts be the root of the binomial sought: or theirdifference is the root when the quantity is residual. That is, if

c + e = a, and ce = D - , then is .y'' c + y'' e — a/ a b theroot sought. As if the binomial be 23 448 ; then the parts

of 23 are 16 and 7, and their product is 112, which is ilh of 44S ;therefore the sum of their roots A -\- \/ 7 is the root sought of23 + a,/ 44S. De Burgo gives also another rule for the sameextractions, which is this: The given binomial being, for exam-ple, \/ 0 + ^/ b, its root will be { a/ c + 4\/ c —b +v' i V c — 4V c — b.

Binomials, to Extract the Cubic and other higherRoots of. This is useful in resolving cubic and higher equa-tions, and was introduced with the resolution of those equationsby Tartalea and Cardan. The rules for such extractions are in agreat measure tentative. The following rule given by Sir IsaacNewton applies to all roots of binomials whatsoever. Of thegiven quantity a ± b, let a be the greater term, and c the index ofthe root to be extracted. Seek the least number n, whose power7tc can be divided by ua — bb without a remainder, and let the

quotient be < 7 ; compute c \/ a-\-b.^/q in the nearest integernumber, which call r; divide q by its greatest rational divi-sor, calling the quotient s; and let the nearest integer number

r +11.

above- r be t : so shall

2s

t-V + fly/ t~S'~ - 71

V?

be the root sought, if

the root can be extracted. And this rule is demonstrated favs’Gravesande in his Commentary on Newton’s Arithmetic.

Binomial r I heorem, in algebra, the celebrated theorem in-vented by Sir Isaac Newton tor raising a binomial to any power,or for extracting any root of it by an approximating infiniteseries. Sir Isaac Newton , after he had discovered this rule forgenerating the coefficients of the powers of 1 + x, when theindexes of those powers were whole numbers, conjectured thatit might be possibly true likewise when they were fractions. lietherefore resolved to try whether it was or not, by applying it tosuch indexes in a few easy instances, and particularly to the in-dexes \ and j, which, if the rule held good in the case of frac-tional ind exes, would enable him to find serieses equal to thevalues of 1 -f- x ]4 and 1 i or the square root and cube rootof the bino mial 1 -f- x, and when he had in this manner obtaineda series for 1 + . 1 ] 4 or the square root of 1 -j- x, he multiplied thesaid series into itself, and found that the product was 1 -|-x; inlike manner he obtained a series which he suspected to be equalto k+TU, and by multiplying the said series twice into itself, theproduct lie found to be 1 +x ; hence he concluded that the for-mer series was really equal to the square root of 1 + x, and thelatter to its cube root, and from these and a few more such trialshe .concluded universally, that the rule was true in all cases, whe-ther the index m stood for a whole number, or a fraction of any

kind, as 4 , 4 , \, or, in general, —. This discovery of New-

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toll’s appears first to have been given to the world, but without ademonstration, by Dr. Wallis in his Algebra , chap. 85, in thevear 1(385, though it was inserted in Sir Isaac ’s first letter to Mr.Oklenlmrgh, the secretary to the Royal Society , dated Julv 13,voi.i,— no. 28.

1676; and the said letter was shewn to Mr. Leibnitz , and proba-bly to other mathematicians of that time, yet it remained for someyears without a demonstration, either in the case of integral.powersor roots. At last it was demonstrated, in the case of integralpowers, by means of the properties of figurative numbers,by Mr. James Bernoulli , in die 3d chapter of his treatise deArte Coujectandi, published in 1713, about eight years afterhis death, but probably written in liie latter years of the preced-ing century. \arious other demonstrations have been given ofthis very important theorem by more modern mathematicians,some of which are by means of Fluxions, and others more legallyfrom the pure principles of algebra only ; among the best of thiskind is that given by Dr. Hutton in his Mathematical Tracts,,vol. 1 , where a full account is given of what others have doneupon this subject. A demonstration of it is also given from alge-braical principles, by Dr. A. Robertson, in the Philosophical Trans-actions for 18013. A complete demonstration was pointed out byM. Lagrange, which is correctly exhibited in Mr. Manning’s Al-gebra, vols. i. and ii, dee Algebra , § 30, where an algebraicdemonstration is given, and the theorem applied to raising powersand extracting roots. The binomial theorem is engraven on themonument of Sir Isaac Newton , as one of his greatest discoveries.

BINOMINAL, [from bums and nomen, a name,] composedof two parts, or members.

BINOMIUS, [from bis and nomcn, name,] in middle-agewriters, denotes a person with two names - .

BINTAN, an island in the East Indies, S. of Malacca.

BIOCOLYTiE, [from /S^, violence, and to hin-

der,] in the Greek empire, an order of officers appointed toprevent the violence frequently committed by the soldiers. Theywere suppressed by the emperor Justinian .

BIOGRAPHER, [/3i^ and ys-tfiu,] a writer of lives; a relu-ter, not of the history of nations, but of the actions of particularpersons.

BIOGRAPHY, a species of history which records the livesand characters of remarkable persons. This is at once the mostentertaining and instructive kind of history. It admits of all thepainting and passion of romance ; but witfi this capital difference,that our passions are more keenly interested, because the charac-ters and incidents are not only agreeable to nature, but strictly true.Eew books are more proper to be put into the hands of youngpeople, than those which contain well-written and authentic re-cords of great and good men. We regret that the extensive na-ture of our work, and the limits to which we are necessarily con-fined, prevent us from entering more largely into this departmentof our plan, and compel us, however reluctant, to pass by somenames we should wish to insert. “Non omnia possumns oivmes.”We recommend to our readers the follow ing among others as pe* -culiarly interesting biographical works: Plutarclds Lives, byLanghorne ; Johnson’s Lives of the l’oets; Doddridge ’s Life of -Colonel Gardiner; Orton’s Life of Doddridge ; II ay ley’s Livesof Milton and Cow’per ; Lord Teignmonth f s Life of- Sir WilliamJones; Betham’s Lives of Eminent Women; and Boswell’s Lifeof Johnson.

BION, a bucolic poet, a native of Smyrna, lived at the sametime with Ptolemy Philadelphus, whose reign reached from the4th year of the 123d Olympiad to the 2 d year of the 133d. Hewas an incomparable poet, if we may believe the lamentations ofhis disciple Moschus. His few pieces which are left do not con-tradict this testimony. The best editions of Bioii and Mosclmsare Schwebelius, 8 vo. Veuel. 17-56-; Aerkin’s, 8 vo. Oxon. 1748;and Wakefield’s, 4to. 1795, beautifully printed by Hensley.

BIORNBURG, a town of Sweden, near the Gulf of Bothnia .

BIOT1IANATI, [from 0 <«, violence, and yreva«r, death,] insome medical writers, denotes those who die a violent death.The word is also written, and with more propriety, biutlmnati,sometimes binolhanti. In a more particular sense,, it denotesthose who kill themselves.

Biothanati, [from life, and 5«vre«.-, death, alluding lothe belief of a future life after death,] was a name of reproachgiven by the Heathens to the primitive Christians, for their con-stancy and forwardness to lay down their lives in martyrdom.

BlOVAC, BIHOVAC, or BIVOUAC, [Kr. from iscy teach,a double guard, German,] a guard at night performed by the wholearmv; winch either at a siege, or lying before an enemy, every even-inn draws out from its tents or huts, and continues all’ll ght in arms.

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