PARALLELOGRAM
, in Geometry, is a quadrilateral right-lined sigure, whose opposite sides are parallel to each other.
A Parallelogram may be conceived as generated by the motion of a right line, along a plane, always parallel to itself.
Parallelograms have several particular denominations, and are of several species, according to certain particular circumstances, as follow:
When the angles of the Parallelogram are right ones, it is called a Rectangle.—When the angles are right, and all its sides equal, it is a square.—When the sides are equal, but the angles oblique ones, the figure is a Rhombus or Lozenge. And when both the sides and angles are unequal, it is a Rhomboides.
Every other quadrilateral whose opposite sides are neither parallel nor equal, is called a Trapezium.
Properties of the Parallelogram.—1. In every Parallelogram ABDC, the diagonal divides the figure into two equal triangles, ABD, ACD. Also the opposite angles and sides are equal, viz, the side AB = CD, and AC = BD, also the angle A = [angle] D, and the [angle] B = [angle] C. And the sum of any two succeeding angles, or next the same side, is equal to two right angles, or 180 degrees, as [angle] A + [angle] C = [angle] C + [angle] D = [angle] D + [angle] B = [angle] B + [angle] A = two right-angles.
2. All Parallelograms, as ABDC and abDC, are equal, that are on the same base CD, and between th<*> same parallels Ab, CD; or that have either the same or equal bases and altitudes; and each is double a triangle of the same or equal base and altitude.
3. The areas of Parallelograms are to one another in the compound ratio of their bases and altitudes. If their bases be equal, the areas are as their altitudes; and if the altitudes be equal, the arcas are as the bases. And when the angles of the one Parallelogram are equal to those of another, the areas are as the rectangles of the sides about the equal angles.
4. In every Parallelogram, the sum of the squares of the two diagonals, is equal to the sum of the squares of all the four sides of the figure, viz, . Also the two diagonals bisect each other; so that AE = ED, and BE = EC.
5. To find the Area of a Parallelogram.—Multiply any one side, as a base, by the height, or perpendicular let fall upon it from the opposite side. Or, multiply any two adjacent sides together, and the product by the sine of their contained angle, the radius being 1 : viz, The area is [angle] C.
Complement of a Parallelogram. See COMPLEMENT.
Centre of Gravity of a Parallelogram. See CENTRE of Gravity, and Centrobaric Method.
Parallelogram, or Parallelism, or PENTAGRAPH, also denotes a machine used for the ready and exact reduction or copying of designs, schemes, plans, prints, &c, in any proportion. See Pentagraph.
Parallelogram of the Hyperbola, is the Parallelogram formed by the two asymptotes of an hyperbola, and the parallels to them, drawn from any point of the curve. This term was first used by Huygens, at the end of his Dissertatio de Causa Gravitatis. This Parallelogram, so formed, is of an invariable magnitude in the same hyperbola; and the rectangle of its sides is equal to the power of the hyperbola.
This Parallelogram is also the modulus of the logarithmic system; and if it be taken as unity or 1, the hyperbolic sectors and segments will correspond to Napier's or the natural logarithms; for which reason these have been called the hyperbolic logarithms. If the Parallelogram be taken = .43429448190 &c, these sectors and segments will represent Briggs's logarithms; in which case the two asymptotes of the hyperbola make between them an angle of 25° 44′ 25″1/2.
Newtonian or Analytic Parallelogram, a term us<*>d for an invention of Sir Isaac Newton, to <*>ind the first term of an infinite converging series. It is sometimes called the Method of the Parallelogram and Ruler; because a ruler or right line is also used in-it.
This Analytical Parallelogram is formed by dividing any geometrical Parallelogram into equal small squares or Parallelograms, by lines drawn horizontally and per| pendicularly through the equal divisions of the sides of the Parallelogram. The small cells, thus formed, are filled with the dimensions o<*> powers of the species x and y, and their products.
For instance, the powers of y, as y° or 1, y, y2, y3, y4, &c, being placed in the lowest horizontal range of cells; and the powers of x, as x° = 1, x, x2, x3, &c, in the vertical column to the left; or vice versa; these powers and their products will stand as in this figure:
Now when any literal equation is proposed, involving various powers of the two unknown quantities x and y, to find the value of one of these in an infinite series of the powers of the other; mark such of the cells as correspond to all its terms, or that contain the same powers and products of x and y; then let a ruler be applied to two, or perhaps more, of the Parallelograms so marked, of which let one be the lowest in the left hand column at AB, the other touching the ruler towards the right hand; and let all the rest, not touching the ruler, lie above it. Then select those terms of the equation which are represented by the cells that touch the ruler, and from them find the first term or quantity to be put in the quotient.
Of the application of this rule, Newton has given several examples in his Method of Fluxions and Infinite Series, p. 9 and 10, but without demonstration; which has been supplied by others. See Colson's Comment on that treatise, p. 192 & seq. Also Newton's Letter to Oldenburg, Oct. 24, 1676. Maclaurin's Algebra, p. 251. And especially Cramer's Analyses des Lignes Courbes, p. 148.—This author observes, that this invention, which is the true foundation of the method of series, was but imperfectly understood, and not valued as it deserved, for a long time. He thinks it however more convenient in practice to use the Analytical Triangle of the abbé de Gua, which takes in no more than the diagonal cells lying between A and C, and those which lie between them and B.
Parallelogram Protractor, a mathematical instrument, consisting of a semicircle of brass, with four ru- lers in form of a Parallelogram, made to move to any angle. One of these rulers is an index, which shews on the semicircle the quantity of any inward and outward angle.
