b2 | (x2+}ax+p)2={ apx+p2+2px3

where the terms may be continued I may be found in terms of the

2q

. Having thus found the values of P, q, and r, the four values of a, in the proposed equation, are also determined from the assumed equation

(x2+ax+p)2 -(qx+r)2=0, or
x2+ {ax+ p = ±(qx+r); whence
x2+(a−q) x = r—p, or
x2+({a+q) x = —r—p,
by taking the ambiguous sign by
which (qx+r) is effected both +
and; whence the four required

roots are

2. Euler's Method.

pqr

p3-fp2+gp—h=0;
whence the three quantities p, q,
and r, become known, and conse-
quently the roots of the proposed
equation, x1-ax2-br—c=0,
being as follows; viz.
When b is positive,

He assumes a general biquadra. Consequently the cubic equation

tic under the form

x-ax2-bx-c=0,

and supposes its root to be

x = √ p + √ q + √r;

which, squared, becomes
x2 = p + q +r+2(√ pq + √ pr+
√gr);

or, making p+q+r=f,
it becomes
x2-f=2(√pq+ √pr + √qr;
squaring this, we have

( 4 (pq+pr + qr)
-x+ —2ƒx2+ƒ2 = +8(√p gr+√q2pr
+√r2pg).
Making pq+pr+gr=g, and put-
ting the latter part under the form
8√prq(√p+√q+√r),

and substituting at the same time
Pqrh, we obtain
x2=2ƒs2+ƒ2=4g+8x√h

will be

or 4-2fx-8/h.x+(ƒ2-4g)=0; 4th. so that 2f = a, or ƒ = {a

++

+ +

5

-= 1

2

5

=

2

3

-6

For the construction of biquadratic equations, see Construction. BISSEXTILE, 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.

The day thus added is also called bissextile; Cæsar 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 calendus 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.

Thus, 18204 leaves 0, therefore 1820 is bissextile; and 1823 4 leaves 3, therefore 1823 is the third 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, cæteris 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. length, breadth, and thickness.

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 introduced 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 BODY, in Physics, is a solid, exJulian account is bissextile or leap-tended, palpable substance; of ityear, is a common year in the self merely passive, and indifferent Gregorian account, unless the num- either to motion or rest; but capaber of centuries can be divided by ble of any sort of motion, and all four without a remainder. Thus, figures and forms. 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.

The bissextile, or number of years after it, is the remainder, upon dividing the date by 4.

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 any 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 fluid.

A solid Body is that in which the attractive power of the particles of which it is composed exceed their repulsive power, and, cousequently, they are not readily moved one among another; and, therefore, the body will retain any figure that is given to it.

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 considers, 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 fluid Body is that in which A keen contest was for some the attractive and repulsive pow-time carried on among the contiers of the particles are in exact nental mathematicians, respecting equilibrio, and therefore yields the solution of this problem, which to the slightest impression. was first accurately given by Jaines Bernoulli. It is the same as the cycloid.

Fluid bodies are also distinguished into nonelastic and elastic, or fluids properly so called, and ariform fluids or gases.

BRANCH of a Curve, in Geometry, is a term used to denote cerRegular BODIES, or Platonic Bo-tain parts of a curve, which are DIES, are those which have all their sides, angles, and planes, similar and equal, of which there are only five, viz.

1. Tetraedron, con- (4 equilateral tained under

triangles. 2. Hexaedron, 6 squares. 3. Octaedron, 8 triangles. 4. Dodecaedron,.. 12 pentagons. 5. Icosaedron, • • • 20 triangles.

infinitely extended without return. ing 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

x2 b2

y= +
a

will have an infinite parabolic branch, which will have the common parabola for its asymptote, of 22 which the equation is y = For

BOW 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 being supposed infinite, the last to his celebrated problem of the term vanishes, and the equation "Curve of swiftest Descent," name- becomes simply y = ly, 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 pos

sible.

At first view of this problem, it would be imagined that a right

a' which is the equation of the common parabola.

Hyperbolic BRANCHES, are those which have a right line, or an hy. perbola of a superior or inferior degree for their asymptote. For example, the curve whose equa

its asymptote will be the infinite ordinate passing through its origin. It may also have for its asymptote the common hyperbola.

BRIDGE, a work of carpentry or masonry, built over a river or canal, for the convenience of passing from one side to the other.

BURNING Glass, a convex lens which transmits the rays of light, but in their passage refracts or inclines them towards a common point in the axis called the focus; and by thus combining together in a single point the power of all the rays transmitted through the glass, a very great degree of heat is accumulated in that point, which will fuse bodies that are infusible in the greatest culinary heat that can be produced.

BURNING Mirrors, or Specula, are concave reflecting surfaces, commonly of metal, which reflect the rays of light falling upon them, but at the same time incline them towards a determined point or focus, where their accumulated effect operates in the most power ful manner, burning and dissipating the hardest and most infusibie bodies.

Convex lenses were very imperfectly understood by the anci ents; but they seem to have had burning mirrors in greater perfection than the moderns, at least if we may credit the relations of several eminent historians, who assert that Archimedes, by means of such mirrors, burned and destroyed the Roman fleet, which, under Marcellus, was employed at the siege of Syracuse; and that Pro clus in the same way destroyed the navy of Vitellius, at the siege of Byzantium.

"When the fleet of Marcellus was within bow-shot," says Tzetzes, chil. 2. hist. 35, "the old man (Archimedes) brought an hexago nal mirror which he had previ ously prepared, at a proper distance from which he also placed other smaller mirrors, of the same kind, that moved in all directions

on hinges, which, when placed in the sun's rays, directed them upon the Roman fleet, whereby it was reduced to ashes."

Of the moderns, the most remarkable burning glasses are those of Magine, of 20 inches diameter; of Sepatala, of Milan, near 42 inches diameter, and which burnt at the distance of 15 feet; of Settala, of Vilette, of Tschirmhausen, of Butfon, of Trudaine, and of Parker.

Mr. Parker, of Fleet-street, London, succeeded in the construction of a lens of flint glass, 3 feet in diameter, which, when fixed in its frame, exposed a surface of 32 inches; the distance of the focus 6 feet 9 inches, and its diameter 1 inch. The rays from this large lens were received and transmitted through a smaller one of 13 inches diameter, its focul length 29 inches, and diameter of its focus inch; so that this second lens increased the power of the former, as 82 to 3, or rather more than to 1.

From a variety of experiments made with these lenses, the following are selected to serve as a specimen of their powers: Substances fused, and Time of Fusion. with their Weight Gold,. · • pure. Silver, ditto. Copper, ditto Platina, ditto. Nickell.

Wt. in Time
Grs. in Sec.

• 20. • 4

20.

3

33. · • 20

• 10. • · 3

. 16.

· · 3