186 Froceedings of the Boyal Society 
This part of the subject is completed by a table showing every 
form of right-angled trigon, having its sides expressed in integer 
numbers, with the hypotenuse under 1000. 
In the second part of the paper, the properties of the angles of 
such right-angled trigons are investigated : to these angles the 
name muarif is given, and their values are entered opposite each 
of the trigons in the above-mentioned table. Muarif angles are 
defined to be those which have their sines and cosines rational ; and 
it is shown that the sines and cosines of the sum, or difference of 
two muarif angles, are also rational, this property being analogous 
to the arithmetical proposition, that the product of the sum of one 
pair of squares by the sum of another pair, is also the sum of two 
squares. 
This property of muarif angles is then applied to the demonstra- 
tion of various theorems, and to the solution of several problems. 
In the first place, it is shown that if a trigon be constructed with 
two of its angles muarif, the three sides, the three altitudes, the 
radius of the circumscribing circle, and the radii of the four circles 
of contact, are all commensurable, while the area also is commen- 
surable with their squares. 
This proposition is then extended thus : that if at the ends of 
any line assumed as a base, muarif angles (in any number) be 
made, the sides of these extended indefinitely intercept segments, 
which are all commensurable with the base, and include areas 
which are all commensurable with the square of the base. 
And, farther, that the same property is extended to the sides of 
all muarif angles made at any of the intersections of the above- 
mentioned lines. 
Also, it is shown that if a straight line be drawn to touch a 
circle, and if at the point of contact any number of muarif angles 
be made, if the extremities of the chords thus formed be joined, 
and if tangents be applied at those extremities, all the lines being 
continued indefinitely, then all the intercepted distances are com- 
mensurable with the diameter, and all the areas with the square of 
the diameter. 
It is then shown how to construct a muarif angle which may ap- 
proximate with any required degree of precision to a given angle, and 
thence how to find a rational trigon approximating to a given shape. 
