42 On the Drawing of Figures of Crystals. 
18, Suppose for example the right rectangular prism has been 
projected, (fig 9.) and it is required to place on its angles the plane 
QP, whose parametric ratio is 2: 1:1. Since 2 refers to the vertical 
axis, we lay off on the lateral edge (e) twice as many parts of this 
edge as of each of the terminal edges (€ and €.) Consequently, 
by taking a point in the edge e distant from a, $ the length of e, and 
a point in each é and é, } their respective lengths, and then join- 
ing these points, the conditions will be complied with, and the 
plane 2P will be constructed. If the plane to be introduced were 
4P2 the parametric ratio of which is 4 2: 1, (in which 4 refers to 
the vertical axis and 2 to the longer horizontal,) we should in the 
same manner mark off 4 parts of e, 2 of € and 1 of €; if the plane 
were 4P2, (in which 2 refers to the shorter horizontal axis,) 2 parts 
of é should be laid off, and L of €. By connecting the points thus 
determined, the plane 4P2 or 4P2 would be delineated. If the 
plane were 2Pa (2: :1), which represents a plane on the 
longer terminal edge, 2 parts of e should be laid off, and 1 of &; 
from the determined points in e and é, lines should be drawn to the 
opposite edges parallel with the edge é, and by connecting the ex- 
tremities of the lines thus drawn, the desired representation of a 
plane 2Pa would be completed. The same should be repeated 
on all the similar edges. This will suffice to illustrate the manner 
of substituting the edges for the axes, and also the method of de- 
lineating single planes. 
19. The manner of determining the intersections of planes, we 
may illustrate by an example. Suppose it were required to place 
the planes P, 2P, 4P2 and 2P2 ona right rectangular prism. Two 
rectangular prisms should first be accurately projected by the method 
which has been explained. One, of a size which may be considered 
convenient for a representation of the crystal, drawn with light pen- 
cil marks; the other of larger dimensions, for the purpose of deter- 
mining the direction of the intersections ; these intersections when 
determined are to be transferred to the smaller figure. On fig 9, 
we may first lay down the plane P, by drawing lines connecting the 
centers of the three edges about the angle. These lines are neces- 
sarily parallel to the diagonals of the three faces; the triangle mno 
is therefore the plane P. By connecting the points m, b, n, the plane 
2P is constructed ; for the plane mbn cuts off 2 parts of e to 1 of each 
é and €, as the expression 2P requires. To lay off 4P2 (4: 11 2-) 
Let the whole edge ab represent 4; then an (4 of &) will equal 2 
