IN TRUE PERSPECTIVE. 
489 
figures ; for while B moves downwards in the direction of 
B B^', A is depressed in that of A A^^, and D in that of 
D D^^. The limits of this change are, the coincidence of 
B with B", of A with A^^, and of D with D"^ which makes 
the projection of the angle BAC^BDCrrO, that of 
the angle ACD = ABD=: 180°. It is evident that the 
ratio of A A" to A^ A" must be equal to that of B B" to 
B^ B^^5 and of D D^^ to D^^, because we may imagine 
the square A B C D to be part of a rectangular triangle 
revolving round one of its sides ; a triangle produced in the 
above figure by lengthening M N till it is intersected by 
B A and B D. But the ratio of A^^ . ^iv ^ii j^^jj^g 
equal to that of A^^^ A : A^^^ A^^, it will also be equivalent 
to that of A^^ C : A^^ D^^3 for the equality and similarity of 
the triangles B A"^ A, A A" C, C D" D, and D D^" B. 
The ratio of A"C to A"D", and that of B^B" to A"D", 
depend therefore entirely upon the choice of the position 
in which the hexahedron is to be represented ; but the ra- 
tio of A^^A^ to A^ A^^ is a consequence of the two suppo- 
sitions. If, for instance, we suppose A^^C = |A^^D^^, and 
^iv I A^^ D^^, the projection of the square upon a 
plane perpendicular to the visual ray will be that repre- 
sented Fig. 6., denoted A^ B^ C. This is the position 
adopted in M. Mohs' Works, for the figures of crystals 
belonging to the pyramidal and tessular systems. 
The projection of the square A B C D having thus been 
obtained, upon a plane perpendicular to the visual ray, it 
is required to find the length of those lines which in the 
projection appear vertical, and represent the lateral edges 
of the hexahedron. 
Suppose Fig. 7. to be a vertical section, in the plane of 
the visual ray. The line B B^^ in this figure will be the 
projection of the face B A C D in Fig. 6. If this line be 
turned round the point B" (which is the projection of the 
