DIAGONAL, in geometry, a right line drawn across a quadrilateral figure, from one angle to another, by some called the diameter, and by others the diameter of the figure. Thus A C, (Plate IV. Mis eel. fig. 7.) is called a diagonal.
It is demonstrable, 1. That every dia gonal divides a parallelogram into two equal parts. 2. That two diagonals drawn in any parallelogram bisect each other. 3. A G, passing through the mid dle point of the diagonal of a parallelelo gram, divides the figure into two equal parts. 4..The diagonal of a square is in. commensurable with one of its sides. 5. That the sum of the squares of the two diagonals of every parallelogram is equal to the sum of the squares of the four sides. This proposition is of great use in the theory of compound motions ; for, in an oblique angled parallelogram, the greater diagonal being the subtense of an obtuse, and the lesser of an acute angle, which is the complement of Ate former, if the obtuse angle be conceiz. ed to grow till it be infinitely great regard to the acute one, the great diago-:. nal becomes the sum of the the two sides, , and the lesser one nothing. Now two contiguous sides ofa being known, together with the angle they in.
elude, it is easy to find one of the diago nals in numbers, and then the foregoing proposition gives the other. This second diagonal is the line that would be de scribed by a body, impelled at the same time by two forces, which should have the fame ratio to each other as the contiguous sides have, and act in those two directions ; and the body would de scribe this diagonal in the same time as it would have described either of the contiguous sides in, if only impelled by the force corresponding thereto. 6. In any trapezium, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals, together with four times the square of the dis tance between the middle points of the diagonals. 7. In any trapezium, the sum of the squares of the two diagonals is double the sum of the squares of two lines bisecting the two pairs of opposite sides. 8. In any quadrilateral inscribed in a circle, the rectangle of the two diagonals is equal to the sum of the two rectangles under the two pairs of oppo site sides.