in a restrained sense, is used by mathematicians for a determinate quantity to which a variable one continually approaches; in which sense, the circle may be said to be the limit of its circumscribed and inscribed polygons. In algebra, the term limits is applied to two quantities, one of which is greater, and the other less, than another quantity; and in this sense it is used, in speaking of the limits of equations, whereby their solution is much facilitated. See Algebra.

LINING, the art of painting in water-colours, in contradiction to painting, which is done in oil-colours. See Painting.

Limning is by far more ancient than painting in oil; this last being first invented by John Van Eyck, a Flemish painter, in 1410.

In liming, all colours are proper enough, except the white, made of lime, which is only used in fresco. The azure and ultramarine must always be mixed with size or gum; but there are always applied two layers of hot size, before the size colours are laid on: the colours are all ground in water, each by itself; and, as they are required in working, are diluted with size water.

When the piece is finished, they go over it with the white of an egg, well beaten; and then with varnish, if required.

To limn or draw a face in colours: having all the materials in readiness, lay the prepared colour on the card even and thin, free from hairs and spots, over the place where the picture is to be. The ground being laid, and the party placed in a due position, begin the work; which is to be done at three sittings. At the first, you are only to dead-colour the face, which will require about two hours. At the second sitting, go over the work more curiously, adding its particular graces or deformities. At the third sitting, finish the whole; carefully remarking whatever may conduce to render the piece perfect, as the cast of the eyes, moles, scars, gestures, and the like.