BIQ
Bill
438
Used in the late dispatches of the gallant and much-lamentedGeneral Sir John Moore .
151 PAR T!ENT is a number that divides another into twoequal parts, without a remainder. Thus 2 is a bipartient to 4,and 5 a bipartient to 10.
BIPARTITE, [from bintis and partinr, Lat.] having two cor-respondent parts ; divided into two.
Bipartite Leaf, in botany, a leaf divided into two seg-ments.
BIPATENT, \])ipateas, Lat.] open on both sides.
BIPED, [bipe. r, Lat.] an animal with two feet.
BIPEXNAl'ED, [from binns and penna, Lat.] having twowings.
BlPENNIS, a two-edged axe, used anciently by the Amazonsin light; as well as by the seamen, to cut asunder the cordage ofthe enemy’s vessels.
HI l’ETALOUS, [of bis, Lat. and tvi7aXo»,Gr.] a flower consistingof two leaves.
Ill PUNCTUAL, [from bis and punetum,'] having two points.
BIQUADRATE, or BIQUADRATIC, in algebra, tin;fourth power, arising from the multiplication of a square numberor quantity by itself.
BIQUADRATIC Equation, is that which rises to four di-mensions, or in which the unknown quantity rises to the fourthpower ; as x* + or 3 -f- bx' 2 -f- cx -|- d = 0. The first resolution ofa biquadratic equation was given in Cardan’s Algebra, 'chap. 3<),being the invention of his pupil and friend Lewis Ferrari, aboutthe year 1540. This is effected by means of a cubic equation,•and is indeed a method of depressing the biquadratic equation toa cubic, which Cardan demonstrates, and applies in a great va-riety of examples. Des Cartes has given another solution of bi-quadratic equations, in the 3d book of his Geometry. lie re-solved the given biquadratic equation into two quadratics, bymeans of a cubic, equation, in this manner: First, let the 2d termor 3d power be taken away out of the equation after which it willstand thus,
a 1 -f- px- -|- qx f-r = 0. Find y in this cubic equation. +/>•*)
j / 6 + 2 P'f = o;
— 4 r)
and then the values of x inthese two quadratics.
** — Jr + x>f + !/> + — = 0,-y
^ + yx + i>f + ip— — == o,
2 y
and these values of x will be the roots of the given biquadraticequation.
Ex. Let the equation be .r 4 — 17.i 2 — 20.r — 6 = 0.
Hence p = — 17 ,q — — 20, and r = — 6 ;and the cubic equation is y e ‘ — 34<y‘ -j- 313i/-—400 = 0,the root of which is y 2 =16, or y = 4;hence the two qua- \ x -— 4,r — 3 = 0dratics are $ x- -f- 4x -|- 2 = 0,
the four roots of which are 2 ± \ /, 7 and — 2 ^ ^/2.
The celebrated Leonard Euler gave, in the 6th volume of thePetersburg!! Ancient Commentaries, for 1738, an ingenious andgeneral method of resolving equations of all degrees, by means ofthe equation of the next lower degree, and among them of thebiquadratic equation by means of the cubic ; and lliis last was alsogiven more at large in his treatise of Algebra , translated from theGerman into French in 1774, in two vols. Svo. The method isthis: Let .t 4 — ax- — bx — c = 0, be the given biquadratic equa-tion, wanting the second term. Take _/’=-'«, g = T V 4 ' 2 +and /i=£/> 2 ; with which values of J\ g, A, Yorm the cubicequation z 3 — Jz- fl-g; — h = 0. Find the three roots of this cu-bic equation, and let them be called p, tj, r. Then shall the fourroots of the proposed biquadratic be these following, viz.
When ~b is positive.1st. '[> + V 7 + \ / ' r
2d. ^//i -j- s /q — y/r
3d. \//> \/y r
4th. s fp — v /y — s / r
When \b is negative:s /p + s /q — y/r,
VP — \'<1 + V r <
— V P + \/? + \ /r >
— y/p — \ /f ] — v r -
For the construction of biquadratic equations, see Des Cartes’sGeometry, with the Commentaries of Schooten and others; Ba-ker’s Geometrical Key; Slusius’s Mesolabium ; 1’Hospital's Co-nic. Sections; Wolsius’s Elcmenta Mathescos, &c.
Biquadratic Parabola, a curve of the third order, havingtwo infinite legs, and expressed by one or other of these threeequations, viz.
a 3 x — y*
a-x ~ y* — u-y 2
a \r = i/ 4 — » -)- /i .y 3 — aby 2 ,
where x = the absciss, y = the ordinate, a and b = givenquantities. But the equation of this curve is exhibited in a gene-ral manner for the several kinds of this curve, thus:
ay = 2 * + bx 3 + cx 2 + dx -f- e;
where x is the absciss, and —y or T y is the ordinate, also a, b,c, d, c, are constant quantities; the beginning ot the absci.,s beingat any point in the indefinite line to which the ordinate-: are r< -ferred. But if the beginning of the absciss be where this lineintersects the curve, then the nature of the curve will be definedby a more simple equation. When the curve has no serpentinepart, the equation is still more simple. The biquadratic parabolais one of the parabolas of the higher kind, which are sometimescalled paraboloids.
lUQUALAR, among the Algerines, a cook of the divan.
BIQUINTILE, an aspect of the planeis, when they are 144degrees distant from each other. It is thus called, because theyare distant from one another by twice the fifth part of 360 de-grees.
Bill, or BE11R, a town of Asiatic Turkey, in Diarbeck. Lon.38. 6. E. Lat. 36. 10. N.
BIRCH, in botany. See Betui.a.
Birch, 'Thomas,' D. D. an eminent historical and biogra-phical writer, was born in London in 1705. His parents, whowere (makers, intended him for trade, but the love of learningprevailed, and he was permitted to pursue his inclination on con-dition that he should provide for himself, lie was usher in threeschools kept by quakers, which sect, however, he quitted, andin 1730 was ordained a minister of the church of England ; soonafter which he obtained the living of Citing in Essex. In 1735 hewas elected a fellow of the royal and antiquarian societies. Hewas presented to several church preferments in succession, thelast of which was the rectory of Dcpden in Essex, which he heldwith the united rectories of St. Margaret Patten and St. Gabriel,Eenchurch-street. In 1752 he was elected, one of the secretariesof the Royal Society, and the year following the archbishop of Can-terbury conferred on him the degree of D. D. He was also ap-pointed one of the trustees of the British Museum . He was killedby a fall from his horse in 1766. Dr. Birch had a considerableshare in the General Dictionary, historical and critical, 10 vols.folio; and published the lives ot' Mr. Boyle, archbishop I illotson,Henry, prince of Wales, and other works of alike kind. He alsowrote an Enquiry into the share which Charles I. had in the'Transactions of the Earl of Glamorgan, 8vo. 1747. A Historyof (he Royal Society , 4 vols. 4to. 1756, &c. Memoirs of theReign of Queen Elizabeth, 2 vols. 4to. &c. I le left his books,MSS. and 500/. to the British museum, the money to go towardsincreasing the stipend of the two assistant librarians.
BIRD, in zoology. See Zoologa', Ornithology , andAvks.
Bird, Beam, or Petty-Chaps. See Motacilla.
Bird, Black. SecTunnus.
Bird, Blue. See Motacilla.
Bird, Call, or Decoy Bird, a bird trained up to call andallure oilier birds into the fowlers nets. See Bird-catching.
Bird, Canary. See Fringiu.a.
Birds, in heraldry, according to their several kinds, representeither the contemplative or active life. '1 hey are reckoned theemblems of liberty, expedition, readiness, swillness, and fear.They are more honourable bearings than fishes, because they par-ticipate more of air and fire, the two noblest and highest ele-ments, than of earth and water.
Birds, Method of preserving. Various methods.have beenattempted, for preserving birds from putrefaction, so as to retain
their