43. If a liue be conceived to move always perpendicular to a co ordinate plane, and peas through an original curve, its intersection with the plane will be the projection of the curve ' • this projection being the section of the cylindroidal surface formed by the generating line. If the curve be any other than a circle or an ellipse, its pro jection can only be practically described by finding the projections of a sufficient number of points of the original, from some known pro perty of the curve; or from the mode of its generation; and the required projection must be drawn by hand through the points thus determined. It is obvious that the projection of any plane curve which is parallel to the co-ordinate plane must be equal and similar to the original. But if the original curve be a circle, or an ellipse, the projecting line during its motion will generate a right or oblique cylin drical surface, the section of which by the co-ordinate plane must be either a circle or an ellipse.
49. Whatever may be the oblique position of an original circle with respect to the co-ordinate plane, there must be one diameter which is parallel to that plane ; now the projection of this diameter being equal to the original, must be greater than those of all the other diameters of the original circle, which are all necessarily oblique to the plane : and since the projection of every diameter must be a diameter of the projected curve, the projection of this parallel one must be the major axis of the ellipse. This diameter of the original circle parallel to the co-ordinate plane is that which is parallel to the trace of the plane. The conjugate axle of the ellipse will be the projection of that diameter of the original circle which is perpendicular to the former, or to the trace of the plane.
50. The projection of a sphere on a co-ordinate plane must be a circle of the same radius as the sphere, this circle being the projection of that great circle of the solid which is parallel to the co-ordinate plane.
51. In the applications of practical geometry to the arts, the object is either to delineate the forme to which materials are to be reduced, or to guide workmen in making and putting together machinery; or, in the construction of edifices of every description.
52. Owing to the symmetry of the machines or edifices, the forms most commonly required to be delineated are reducible to series of rectangular geometrical solids, the planes of which are either parallel or perpendicular to the horizon. The plans, elevations, sections, pro files, &c., furnished to the workman by the draughtsman, are the projections on rectangular co-ordinate planes, assumed to be parallel to the planes of the machines or edifices, made to a reduced scale ; the plan being such a projection, made on a horizontal plane, and the elevation on a eertical plane. When the building or engine is
supposed to be laid open, by being cut by a plane, so as to show its interior, the projection made on this supposition is termed a section, or profile.
53. It is obvious, from these assumptions, that the various plane rectilinear, or mixed, figures which are produced by the intersections or boundaries of the various surfaces of the original objects, are repro seated on the draughtsman's plans, ace., by figures similar to the original forms ; and that those plane surfaces of the original object which are vertical to the horizon appear only as right lines on a plan, bounding the figures which are the representatives of original planes parallel to the horizon ; and, conversely, these last-mentioned surfaces are rupreaented by lines in the elevations, while the vertical plane figures of the original are projected into similar plane figures on these elevations or protilea.
54. llence two, at lout, and commonly three, such projections, on rectangular co-ordinate planes, are requisite to convey an idea of the forms of an original object ; but since these forms of the original are represented of their true dimensions and proportions, such projections are sufficient, and indispensable, as guides to the mechanics who are to construct or put together the edifice or machine.
55. The principles of projection enable us, as far as regards the rectangular parallelopiped, the solid of most frequent occurrence, to combine the two purposes for which such projections are employed ; that is, to convey, by one image or figure, an accurate idea of the relative position of the parallel and vertical planes of an original object, reducible to this form, and at the same time to preserve one constant and correct proportion between the magnitudes of the original and of its representative* 56. It has been shown (26, 27) that the projections of definite right lines, inclined in eapial angles to the co-onlioate plane, will be in a constant proportion to the originals ; if, therefore, the three plane right angles forming a solid angle of a rectangular parallelopided be inclined in equal dihedral angles to the co-ordinate plaice, all lines parallel to the three edges of that solid angle will be projected into lines bearing one constant retie to the originals, and forming with each other equal angles, which are the projections of the right ones formed by the original lines.