OVAL

, an oblong curvilinear figure, having two unequal diameters, and bounded by a curve line returning into itself. Or a figure contained by a single curve line, imperfectly round, its length being greater than its breadth, like an egg: whence its name.

The proper Oval, or egg-shape, is an irregular figure, being narrower at one end than the other; in which it differs from the ellipse, which is the mathematical Oval, and is equally broad at both ends.—The common people confound the two together: but geometricians call the Oval a False Ellipse.

The method of describing an Oval chiefly used among artificers, is by a cord or string, as FHf, whose length is equal to the greater diameter of the intended Oval, and which is fastened by its extremes to two points or pins, F and f, planted in its longer diameter; then, holding it always stretched out as at H, with a pin or pencil carried round the inside, the Oval is described: which will be so much the longer and narrower as the two fixed points are farther apart. This Oval so described is the true mathematical ellipse, the points F and f being the two foci.

Another popular way to describe an Oval of a given length and breadth, is thus: Set the given length and breadth, AB and CD, to bisect each other perpendicularly at E; with the centre C, and radius AE, describe an arc to cross AB in F and G; then with these centres, F and G, and radii AF and BG, describe two little arcs HI and KL for the smaller ends of the Oval; and lastly, with the centres C and D, and radius CD, describe the arcs HK and IL, for the flatter or longer sides of the Oval.— Sometimes other points, instead of C and D, are to be taken by trial, as centres in the line CD, produced if necessary, so as to make the two last arcs join best with the two former ones.

Oval denotes also certain roundish figures, of various and pleasant shapes, among curve lines of the higher kinds. These figures are expressed by equations of all dimensions above the 2d, and more especially the even dimensions, as the 4th, 6th, &c. Of this kind is the equation , which denotes the Oval B, in shape of the section of a pear through the middle, and is easily described by means of poínts. For, if| <*> circle be described whose diameter AC is = a, and AD be perpendicular and equal to AC; then taking any point P in AC, joining DP, and drawing PN parallel to AD, and NO parallel to AC; and lastly taking PM = NO, the point M will be one point of the Oval sought.

In like manner the equation expresses several very pretty Ovals, among which the following 12 are some of the most remarkable. For when the equation has four real unequal roots, the given equation will denote the three following species, in fig. 1, 2, 3:

When the two less roots are equal, the three species will be expressed as in fig. 4, 5, 6, thus:

When the two less roots become imaginary, it will denote the three species as exhibited in fig. 7, 8, 9:

When the two middle roots are equal, the species will be as appears in fig. 10: when two roots are equal, and two more so, the species will be as in fig. 11: and when the two middle roots become imaginary, the species will be as appears in fig. 12: