84. The vanishing line and parallel of the vertex to any plane will be those also of all planes whatever which are parallel to the first ; and the radial and vanishing point of an original line will bo those also of all lines parallel to the first, whether they lie in one plane, or in dif. ferent ones.
85. The perspective projections, or images, of any number of original parallel lines, will be either parallel lines, parallel to the originals (73), or will be lines passing through the respective intersecting points of (771, and through their common vanishing point; and the points in which the indefinite images of original lines, not parallel, cut one another, will be those of the points in which the originals cut one another.
86. It. has been shown that the image of a line is parallel to that line's director ; if, therefore, two or more lines have a common station point, and consequently a common director, or if the station points of two or more linen lie in one director, the images of those lines in either cane will be parallel lines; and in these cases only can original lines, not parallel, have parallel images.
87. The ratios which exist between the definitive images and the original segments of lines are easily deducible, either geometrically or analytically ; but as these theorems do not lead to rules of frequent practical utility, we forbear, with one or two exceptions, entering into them. Let n' a represent a finite portion of an original line, bisected by the point B', then the ray. v v a', v B, and the radial v r of B" a, will be harnionical lines ; the definite image of the original line will consequently be harmonically divided by the iraagee 6, of B' and 0, and by the vanishing point r. Conversely, if any segment of an indefinite image of a line he bisected by a point, the segment of the original line between its station point and the original of the image farthest from that station point will be harmonically divided by the originals of the other two points. If the point which bluets a finite line • B be the station point of that line, the image a 6 of A D will be bisected by the vanishing point P.
88. If an original finite line • B be parallel to the plane of the picture, its image a b will be to A B in the ratio of the distance of the picture (70) to the perpendicular distance of the plane, parallel to the picture in which • B lies from the vertex; and if • n be divided by a point D In any ratio, the image a b will be divided by d, that of D in the same ratio.
89. If an original plane figure be parallel to the plane of the picture, the image of that figure will be similar to the original ; its periphery will be to that of the original, In the ratio of the distance of the picture I to the perpendicular distance of the plane of the original figure from the vertex; and the area of the image will be to that of the original, as the squares of 'these lines. If, therefore, the distance of the picture be equal to that of the original parallel plane from the vertex, the Image of an original figure in that plane will be equal, as well an similar, to the original : this may occur if the original plane coincide with that of the picture, or if the vertex he at equal distances from both, and lie between them ; or if the vertex be infinitely and therefore equally distant from both on the same aide.* 90. If an original plane, or planes, be parallel to the plane of the picture, their vanishing plane will coincide with the vertical plane : no such planes can therefore have Any vanishing line.
91. If an original plane, or planer, be perpendicular to the plane of the picture, their vanishing plane will pass through the distance of the picture (70); consequently the vanishing line of such plane, or planes, will pass through the centre of the picture.
142. If an original plane pass through the vertex, its vanishing plane will coincide with it; the intersecting and vanishing lines will there. fore coincide in one, as will also the etatiou lines, and parallel of the vertex ; and the images of all lines and plane figures, in such an original plane, will coincide in one line, that in which the plane itself cuts the plane of the picture.
93. The vanishing planes of two original planes will form the same dihedral angle that the original planes form with each other, and the line in which the vanishing planes intersect will pass through the vertex and be parallel to that in which the original planes Intersect each other ; it will, therefore, be the radial of this latter-named inter section. The intersection of the two vanishing planes, or this radial, will out the plane of the picture in the vanishing point of the inter. sections of the original planes, which vanishing point will obviously be the intersection of the two vanishing lines determined by the two vanishing planes.