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1.2.3 plus and minus alternately in the latter case. ls we raise (1 + 1) to any power n, we shall have the co-efficients only ; hence the sum of all the coefficients of a binomial are equal to the same power of 2, as that to which the binomial is raised. Also, the sum of the positive co-efficients is equal to the sum of the negative ones, which therefore destroy each other. If also we consider the latter part of the series, it will be found that the co-efficients from either extreme are the same, increasing from each end to the centre term, when the number of terms is odd, or to the two centre terms when the number is even. The number of terms is 1 greater than the exponent of the power. It is odd for even powers and even for odd powers. The index of the first, or leading quantity, is the same as that of the power, and in the succeeding terms it decreases always by 1 ; while that of the second part increases by I, whereby the sun of the indices is always the same in each term. As to the co-efficients, the first is always unity, and the second the same as the index of the power and for the rest multiply the co efficient of each preceding term by

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at pleasure, the law of series being obvious. When the quantity to be expanded comes under the form of a fraction, the denominator must be placed under a negative index, and brought up into the numerator. BIQUADRATE, or Biquadr Atic Power, in Algebra, is the square squared, or fourth power of a quantity; thus, 16 is the biquadrate of 2. B I QUADRATIC Root, is the fourth root of any proposed quantity; thus 2 is the biquadratic root of 16. The biquadratic root of a number is found by extracting the square root of it, and then the square root of that root. BIQUADRATIC Equation, is an equation in which the unknown quantity rises to the fourth power; as a 4-H aa:3 + bao-H cr–H d = 0 in which a, b, c, and d, may be any numbers whatever, positive or negative, or any of them equal 0. A biquadratic equation is the highest order of equation that admits of a general solution; all higher ones being soluble only in particular cases. The following are two of the methods of solution.

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Hence, as the value b is nega

tive, the four roots are the following:

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For the construction of biquadratic equations, see Construction. BISS EXTILE, or Leap-Year, in Chronology, a year consisting of 366 days, happening once every four years, by reason of the addition of a day in the month of February to recover the six hours which the earth occupies in its annual course, beyond the 365 days ordinarily allowed for it. he day thus added is also called bissextile; Caesar having ap pointed it to be introduced by reckoning the 24th of February twice ; and as this day, in the old account, was the same as the sixth of the calends of March, which had been long celebrated among the Romans on account of the expulsion of Tarquin, it was called “bis sextus calendas Martii ;” and from hence we have derived the name bissextile. By the statute 21 Hen. III. to prevent misunderstandings, the intercalary day, and that next before it, are to be accounted as one day. The astronomers concerned in reforming the calendar, by order of Pope Gregory XIII. in 1582, observing that the bissextile in four years added forty-four minutes more than the sun spent in returning to the same point of the zodiac, and computing that these supernumerary minutes in 133 years would form a day; to prevent any

changes being thus insensibly in

troduced into the seasons, directed that in the course of 400 years there should be three bissextiles retrenched; so that every centisimal year, which according to the Julian account is bissextile or leapyear, is a common year, in the Gregorian account, unless the number of centuries can be divided by four without a remainder. Thus, 1600 and 2000 are bissextile; but 1700, 1800, and 1900, are common. But, with the exceptions of the above even centuries, any year which exactly divides by four is leap-year; and when there is any remainder, it indicates the number of years since leap-year. he bissextile, or number of years after it, is the remainder, upon gividing the date by 4.

Thus, 1820-4 leaves 0, therefore 1820 is bissextile; and 1823 - 4 leaves 3, therefore 1823 is the Loird after bissextile. BLACK, an epithet applied to any thing opaque and porous, which imbibes the greater part of the light that falls on it, reflects little or none, and therefore exhibits no colour. Bodies of a black colour are found more inflammable, because the rays of light falling on them are not reflected outwards, but enter the body, and are often reflected and refracted within it, till they are stifled and lost. They are also found lighter, carteris paribus, than white bodies, being more porous. BLUE, one of the seven primitive colours of the rays of light, into which they are divided when refracted through a glass prism. The blue colour of the sky is a remarkable phenomenon, which has been variously accounted for by different philosophers. Newton observes, that all the vapours when they begin to condense and coalesce into natural particles, become first of such a bigness as to reflect the azure rays, before they can constitute clouds of other colours. Bouguer ascribes this blueness of the sky to the constitution of the air itself, being of such a nature that the fainter coloured rays are incapable of making their way through any very considerable portion of it. BODY, or Solid, in Geometry, has three dimensions, viz. 1.É. breadth, and thickness. BODY, in Physics, is a solid, extended, palpable substance; of itself merely passive, and indifferent either to motion or rest; but capable of any sort of motion, and all figures and forms. Bodies are either hard, soft, or elastic. A hard Body is that whose parts do not yield to any stroke or percussion, but which retains its figure unaltered. A soft Body is that whose parts yield to the stroke or impression, without restoring themselves again. An elastic Body is that whose parts yield to *} stroke, but im

mediately restore themselves again, and the body retains the same figure as at first. . We know not, however, of any bodies that are perfectly hard, soft, or elastic; but all possess these properties in a greater or less degree. Bodies are also either solid or jluid. A solid Body is that in which the attractive power of the particles of which it is composed exceed their repulsive power, and, consequently, they are not readily moved one among another; and, therefore, the body will retain any figure that is given to it. A fluid Body is that in which the attractive and repulsive powers of the particles are in exact equilibrio, and therefore yields to the slightest impression. Fluid bodies are also distinguished into monelastic and elastic, or fluids properly- so called, and “s. fluids or gases. egular Bodies, or Platonic BoD1Es, are those which have all their sides, angles, and planes, similar and equal, of which there are only five, viz. 1. Tetraedron, con- ;" equilateral tained under triangles. 2. Hexaedron, - - - 6 squares. 3. Octaedron, - - - 8 triangles. 4. Dodecaedron, . . 12 pentagons. 5. Icosaedron, ... • - 20 triangles. B O W, Compass, for drawing arches of very large circles; it consists of a beam of wood or brass, with three long screws that govern or bend a lath of wood of steel to any arch. The term is also sometimes used to denote very small compasses employed in describing archs, too small to be accurately drawn by the common compasses. BRACHYSTOCHRONE, is the name which John Bernoulli gave to his celebrated problem of the “Curve of swiftest Descent,” namely, to find the curve along which a body would descend from a given point A, to another given point B, both in the same vertical plane, in the shortest time possible. At first view of this problem, it would be imagined that a right

line, as it is the shortest path from

one point to another, must likewise be the line of swiftest descent; but the attentive geometer will not hastily assert this, when he cousiders, that in a concave curve, described from one point to another, the moving body descends at first in a direction more approaching to a perpendicular, and consequently acquires a greater velocity than down an inclined plane; which greater velocity is to be set against the length of the path, which may cause the body to arrive at the point B sooner through the curve than down the plane. A keen contest was for some time carried on among the contimental mathematicians, respecting the solution of this problem, which was first accurately given by Jaimes Bernoulli. It is the same as the cycloid. BRANCH of a Curve, in Geometry, is a term used to denote certain parts of a curve, which are infinitely extended without returning upon themselves; being called also infinite branches : such are the legs of the parabola and hyperbola. The infinite branches of curves, are either of the parabolic or hyperbolic kind. Parabolic BRANches are those which may have for an asymptote a parabola of a superior or inferior order: thus, for example, the curve of which the equation is a2 ... b

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