189 
of Edinburgh^ Session 1863 - 64 . 
It is also shown that no angle of one can be equal to that of any 
other muarif system, with the exception of 180° which belongs to 
all systems. 
The general character of a muarif angle is, that its cosine is 
rational; the value of its sine may or may not involve the square 
root of an unsquare number ; if the sine be rational, the angle be- 
longs to the tetragonal system ; but if otherwise, the irreducible surd 
involved in the expression for the sine becomes the modulus or 
mastar of the system, and all angles having the same irreducible 
surd in the values of their sines, the cosines being rational, belong 
to the same system. 
The modulus of the common or tetragonal system is thus VI, 
that of the trigonal system is V3 ; while the modulus of any other 
system is the irreducible surd in the common expression for the area 
of any trigon belonging to it. 
The sixth and last section of the paper contains a few miscellane- 
ous propositions. The first group of problems are cases of this 
general one, “ To construct a trigon, of which the three sides and 
the lines dividing one or more of the angles into equal parts may 
be all commensurable.” 
When only one angle is proposed to be divided, or when two 
angles are to he divided, we can assume these as the proper multi- 
ples of muarif angles of any system whatever; but when the three 
angles are to he divided, we find ourselves restricted in the choice 
of the system. 
Thus, if we wish that the lines bisecting each of the angles be 
rational, we must use the common or the tetragonal system, because 
the half of 180° belongs to it. While, if we wish that the lines 
trisecting the three angles be all rational, we must take the trigonal 
system, because the third part of 180° is among its angles. 
And it is remarkable that we cannot construct a trigon oi‘ which 
the sides and the lines dividing its angles into any other number 
of equal parts than two and three, may be all rational, because rro 
aliquot part of the half revolution except tire half and the third can 
belong to any muarif system. 
P.S. — Since the paper was read, a treatise by Professor Gill of 
New York, on the “ Application of the Angular Analysis to the 
