STE EUTOMY, (from cepeoc, solid, and relent, section,) the science and art of cutting solids tinder certain specified conditions.
Stereotomy may be regarded as a branch of stereography, which is the science of solids in general. Mr. Hamilton has intitled his complete body of perspective, Stereography, which perhaps would have been more properly called Stereolomy, as the perspective representation of every object in nature is the section of a pyramid or wilco( rays. But as it has not been the object of writers on perspective to show the rules for finding the sections of solids in general, under certain specified conditions of the cutting plane, nor of finding any other sections besides those of cones and pyramids, it is the express intention of this article to explain the general prin ciples of the science for any given law, by which the surface of the solid may be constituted of straight lines, or that the surface may agree with the common section of two planes disposed in given positions. And as nothing of the kind has yet appeared, perhaps this attempt may be the more accept able, particularly as in its principles the whole art of dial ing is included, and the mechanical arts of masonry and car pentry. The art of stone-cutting, the squaring and cutting of timbers, and the formation of hand-rails, depend entirely upon the sections of solids.
Properties of Planes and Solids demonstrated in the eleventh book of Euclid's Elements, 'useful in Stereotomy.
Proposition 1.—One part of a straight line cannot be in a plane, and another part above it.
Proposition 11.—Two straight lines which cut each other are ir. one plane, and three straight lines which meet each other are in one plane.
Proposition 111.—If two planes cut each other, their com mon section is a straight line.
Proposition IV.—If a straight line stand at right angles to each of two straight lines in the point of their intersection, it shall also be at right angles to the plane which passes through them.
Proposition V.—If three straight lines meet all in one point, and a straight line stands at right angles to each of them in that point, these three first straight lines are in one and the same plane.
Proposition Vi.—If two straight lines be at right angles to the same plane, they shall be parallel to each other.
Proposition VII.—If two straight lines be parallel, the straight line drawn from any point in one to any point in the other, is in the same plane with the parallels.
Proposition VIII.—If two straight lines be parallel, and one of them at right angles to a plane, the other shall alsb be at right angles to the same plane.
Proposition IX.—Two straight lines which aro each of them parallel to the same straight line, and not in the same plane with it, are parallel to each other.
Proposition two straight lines meeting each other be parallel to two others that also meet, but are not in the same plane with the first two, both couples will contain equal angles.
Proposition X1.—PROBLEM.—To draw a straight line perpendicular to a plane, from a given point in space above the plane.
Draw any straight line in the plane, and from the given point above the plane draw a second straight line at right angles to the first ; from the point where the perpendicular meets the first line, draw a third straight line in the plane, at right angles to the first ; and, lastly, from the given point in space draw a fourth line at right angles to the third ; and the fourth straight line, thus drawn, will be perpendicular to the plane.
Proposition X11.—PROBLEM.—To erect a straight line at right angles to a given plane front a given point in the plane.
From any given point above the plane draw a straight line perpendicular to the plane, and through the given point in the plane draw a second line parallel to the first ; which second line will be the perpendicular required.
Proposition the same point in a given plane there cannot be two straight lines at right angles to the plane, upon the seine side of it ; and there can be but one perpen dicular to a plane from a point above.