188 Proceedings of the Royal Society 
form 6?^ + 1, and every product of such primes, may represent the 
subtense of 120°, the other sides being rational. 
It is worthy of remark, as a notable relation between the proper- 
ties of numbers and those of surface, that 4 and 6 are the only 
moduli which separate prime numbers into two classes ; all primes, 
with the exception of 2, being of one or other of forms 1^-1 and 
47i-t-l ; while, excepting 2 and 3, all belong either to the form 
6^^=l or to 6^4-1. Four squares may lie round a point, and the 
form 4n-t-l, includes all the hypotenuses of right-angled trigons ; 
six equilateral trigons lie round a point, and the form Gn-pl con- 
tains all the subtenses of 120° or of 60°. 
A list is given of trigons of 120°, in the lowest terms, of which 
the subtense does not exceed 1000, accompanied by the value of 
the smaller angle. 
This leads to the recognition of muarif angles of the trigonal 
system, possessing properties analogous to those of the common or 
tetragonal system. 
Thus, if at the extremities of any base trigonal muarif angles be 
made, all the segments into which the sides of these angles cut each 
other are commensurable with the base, and all the areas with its 
equilateral trigon. 
And similarly, if at the point of contact of a straight line and 
circle, trigonal muarif angles be made, if the extremities of the 
chords be joined, and if tangents be applied at those extremities, 
all the segments so formed are commensurable with the side, and 
all the areas with the area of the circumscribed regular trigon ; and 
it may be remarked that in the trigonal system, the inscribed trigon 
is commensurable with the circumscribed, whereas the inscribed 
square is incommensurable with the circumscribed. 
The existence of these two distinct, yet analogous systems of 
muarif angles, naturally suggests the inquiry, whether there may 
not be other systems as well. 
The fifth section of the paper treats of muarif systems in general ; 
it shows that if a trigon be constructed with sides, proportional to 
any three integers whatever, its angles belong to a system of muarif 
angles possessing properties analogous to those of the two preced- 
ing systems, the assumed trigon, or any other one of the system, 
becoming the unit of surface. 
