187 
of Edinburgh^ Session 1863 - 64 . 
The solution is also extended to this more general problem : 
having given any polygon inscribed in a circle, to find another ap- 
proximating to it, and having its sides commensurable with the 
diameter, and its area with the square of the diameter. 
Afterwards, it is shown how to construct a trigon having its sides 
and the lines bisecting its angles all rational. 
In the third part of the paper the construction of polygons 
having their sides, and also the ordinates of their corners integer, 
is discussed. 
The only regular polygons which can be used to cover surface 
are the trigon^ the tetragon^ and the hexagon. Of these the regular 
tetragon or square is the one in common use for the measurement 
of surface ; but, viewing the matter abstractly, we may as well 
measure surface by triangular inches as by square inches. Since 
the regular hexagon contains exactly six regular trigons, it follows 
that, as far as the doctrine of commensurables is concerned, there 
are only two possible systems of surface measurement, — viz.. That 
with the square, and that with the equilateral trigon as the super- 
ficial unit. 
The fourth section of the paper is occupied in discussing the tri- 
gonal system of measurement. Just as the right-angled trigon is 
the guide to the theory of tetragonal commensurables ; the trigon 
having an angle of 120° is the guide to the theory of trigonal com- 
mensurables. The leading property of such a trigon is, that the 
square of the subtense exceeds the squares of the two containing 
sides by their rectangle ; but this is an enunciation in tetragonal 
language : stated appropriately it is this, that the equilateral trigon 
constructed on the subtense, is equivalent to those on the two sides 
together with the original trigon ; for in this system, the surface of 
an equilateral trigon represent the second power of a number, and 
that of a trigon of 120° (or 60°) the product of two numbers. The 
arithmetical representative of such a trigon is 
-f _ c. 
When the trigon is in its lowest terms, that is, when a, h and c 
have no common divisor, c cannot be divisible by 2, by 3, by 5, 
by 11, or in general by any prime number which is not of the form 
6n + 1, and conversely, it is shown that every prime number of the 
