CURVATURE, of a line, is the peculiar manner of its bending or flexure, by which it becomes a curve of such and such pe culiar properties. Any two arches of curve lines touch each other, when the same right line is the tangent of both at the same point ; but when they are ap plied upon each other, in this manner, they never perfectly coincide, unless they are similar arches of equal and similar figures and the curvature of lines admit of indefinite variety. Because the curva ture is uniform in a given circle, and may be varied at pleasure in them, by enlarg ing or diminishing their diameters, the curvature of circles serves for measuring that of other lines.
Of all the circles that touch a curve in any given point, that is said to have the same curvature with it, which touches it so closely, that no circle can be drawn through the point of contact between them. And this circle is called the circle of curvature ; its centre, the centre of curvature ; and its semidiameter, the ray • of curvature belonging to the point of contact. As in all figures, rectilinear , ones excepted, the position of the tan gent is continually varying, so the curva ture is continually varying in all curvi linear figures, the circle only excepted. As the curve is separated from its tangent by its curvature, so it is separated from the circle of curvature in consequence of the increase or decrease of its curvature ; and as its curvature is greater or less, ac cording as it is more or less inflected from the tangent, so the variation of cur vature is greater or less, according as it is more or less separated from the circle of curvature.
When any two curve lines touch each other in such a manner that no circle can pass between them, they must have the same curvature ; for the circle that touches the one so closely that no circle can pass between them must touch the other in the same manner., And it can be made appear, that circles may touch curve lines in this manner ; that there may be indefinite degrees of more or less intimate contact between the curve and the circle of curvature ; and that a conic section may be described, that shall have the same curvature with a given line at a I given point, and the same variation of a curvature, or a contact of the same kind with the circle of curvature. The rays of curvature of similar arches, in similar figures, are in the same ratio as any ho mologous lines'of these figures, and the variation of curvature is the same. See Guava.