134 Mr Levy on the Modes of Notation 
rhomboid, the sign of which is b n , and will become 
cos J ( [b n . b n ) __ 1 
cos J ( b n : b n ) ~ U 
By making in the same formula x = 1, y — o and z — — n, it 
will correspond to the case of a dodecaedron produced by n 
rows in breadth on the inferior edges of the rhomboid, the sign 
of which is d n , and will become 
cos \ (d n . d n ) 
cos | ( d n : d n ) 
Lastly, By supposing x — — 1,^ = 1, and z = n , it will 
correspond to the case of a dodecaedron produced by n rows in 
breadth on the lateral angles of the primitive, the sign of which 
is e n , and will become 
cos \ (e n . e n ) __ n — 1 
eos ^ ( en * en) ^ 
These three formulae will immediately give the law of decre- 
ment by the simple subtraction of two logarithms, when two of 
the incidences of the faces of the dodecaedron will be known. 
The first shews than when n — 2, the angle ( b n . b n ) = 
{bn : bn), that is to say, that a decrement by two rows on the 
superior edges will produce dodecaedrons with isosceles tiiangu- 
lar planes. 
The second ormula makes the two angles {d n . d n ), ( d n : dn) 
equal, only when n = 1, in which case the result of the decre- 
ment is the lateral planes of a six-sided prism. 
The third formula shews that when n — 3 the angle {e n . e n ) — 
(en : en), that is to say that a decrement by three rows on the 
lateral angles of a rhomboid will produce dodecaedrons with isos- 
celes triangular planes. 
Returning now to the general case, the origin of hypothetical 
primitive forms, and the reasons for which a dodecaedron re- 
sulting from an intermediary decrement upon the angles of the 
primitive rhomboid, is, and has always been found to result of a 
very simple decrement on the edges or angles of the hypotheti- 
cal primitive form, may readily be discovered. For, it is ob- 
vious from the four preceding formulas, that if the dodecaedron, 
