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PARALLELOPIPED, Properties of the. All parallelopipeds, prisms, and cylinders, &c., whose bases and heights are equal, are themselves equal.

Every upright prism is equal to a rectangular parallelo piped of equal base and altitude.

A diagonal plane divides the parallelopiped into two equal prisms : a rectangular prism, therefore, is half a parallelo piped upon the same base, and of the same altitude.

All parallelopipeds, prisms, cylinders, &c., are in a ratio compounded of their bases and altitudes ; wherefore, if their bases be equal, they are in proportion to their altitudes; and conversely.

All parallelopipeds, cylinders, &c., are in a triplicate ratio of the homologous sides ; and also of their altitudes.

Equal parallelopipeds, prisms, cones. cylinders, &c., are in the reciprocal ratio of their bases and altitudes.

Rectangular parallelopipeds, contained under the corres ponding lines of three ranks of proportionals, are themselves proportionals.

To measure the surface and solidity of a parallelopiped.— Find the areas of the parallelograms r L M K, L M o N, and omae (See PARALLELOGRAM); add these into One SUM, multiply that sum by 2, and the product will be the surface of the parallelopiped.

" lf, then, the base, I L M K. be multiplied by the altitude, M o, the product will be the solidity.

Suppose L = 36, = 15, M 0 = 12 ; then LIKM= 36 x 15 = 540; ',mos= 36X 12=432; MOKP=15 X 12 = 180. The sum of which is = 1152, which multi plied by 2, gives the superficics equal to 2304. And 540 X 12 gives the solidity equal to 6480. Or the solid content of a parallelopiped may be obtained by multiplying the area of the base by the altitude of the parallelopiped. Thus, if the two dimensions of the base be 16 and 12 inches, and the height of the solid 10 inches; then the area of the base being 192. the content of the solid will be 1920 cubical inches.

The parallelopiped with oblique angles is a figure very common to many kinds of especially of the softer sort.

PA Il.11.LELDGIZAM, (from the Greek a at patina, a figure) in geometry, a quadrilateral right-lined figure, whose opposite sides are parallel and equal to each A parallel gram is generated by the. equable motion of a right line alway s parallel to itself.

When the parallelogram has all its four angles right, and only its opposite sides equal, it is a or an oblong.

When the angles are all right, and the sides are all equal, it is called a M pare, w hich some make a species of parallelo gram, others not.

If all the sides are equal, and the angles unequal, it is called a rhomhas, lozenge.

It' both the sides and angles be unequal, it is called a rhaattcide..

E%ery other quadrilateral, whose opposite sides are neither parallel nor opal, is called a trapezium. See each of these articles.

In every species of parallelogram, a diagonal divides it into two equal parts; the angles diagonally opposite are equal ; the opposite angles of the same side are, equal to two right an:_f. es; and every two sides are. together,

greater than the ding nal. Every quadrilateral, whose oppo site sides are eyed, is a prallelogram.

Two parallelograms on the same, or on an equal base, and of the same height, or between the same parallels, are equal. I lemee two triangles on the same base, and a the saute height are also equal ; as are all parallelograms or triangles whatever, whose bases and altitudes are equal :um mg themselves.

lenee, also, every triangle is half a parallelogram upon the same or an equal base, and of the saute altitude, or the same parallels. !knee, also, it triangle is equal to a ffarallehtmam having the same base and half the alti tude, or h ill' the base and the same altitude.

Parallelograms, therefore, are. in a given ratio. com pounded of their bases and altitudes. Miter the altitudes be equal, they are as the bases; and conversely.

In similar parallelograms and trian!des, tfie altitudes are ptoportional to the sides; :old the bases are cut thereby. 1 knee similar parallelograms and triangles are in a duplicate ratio of their homologous sides, also oh their altitudes, and the segments of their bases; they :ire, tlieretlfre, as the squares of the sides, altitudes, and hut yms semnents of the bases.

hi every parallelogram, the soul of the squares of the two diagonals is equal to the sum of the squares or the four sides: and the two dimjonals bisect each other.

This proposition Al. de Lagny takes to be one of the most important in all geometry ; he even ranks it with the cele brated forty-seventh of Euclid, and with that of the simili tude of triangles; and adds, that the whole first book of Euelid is only a ['articular case of it. Fur if the parallelo gram be rectangular, it follows, that the two diagonals are equal ; and, of consequence, the square of a dia gonal, or, which comes to the same thing, the square of the 6. potlannse of a right angle is equal to the squares of the If the parallelom.am be not reet:nigular. and, of conse quence, the two diagonals be nut equal, which is the most general case, the proposition becomes of vast extent ; it may serve, t'or instance, in the whole theory of compound motions, &c.

There are three ways of demonstrating. this proposition ; the first by trigonometryy which requires twenty-one opera tions; the second geometrical and analytical, whieli requires fifteen. Rut M. De 11.m.Tn• gives a more concise one, in the Menlo; rex de 1' :lead., which only requires seven.

BARA1.1.1.11.0oRANI, Set PENTAGRA1'11.

1'A I;.\ (front the Greek Trapa, through, and 1tt..7-prw, to nnasu•e) it) cultic sect nuts, a constant right line in each of the three sections; called also tons rectum. In the parabola. the rectangle of the parameter and an abscissa are equal to the square of the correspondent semi-ordinate. See AttAll").A.

111 all ellipsis and hyperbola, the parameter is a third pro portii mai to a conjugate and transverse axis. See ELLI•sis 111"PER1301.A.