The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

64 NON-EUCLIDEAN GEOMETRY [c. III.
an acute angle /3, the two sides enclosing the acute angle
being c and 1, and the other two a and mn'.
If we interchange c and 1, and m' and a, we obtain the same
quadrilateral. It follows that, given the right-angled triangle
a, b, c, (X, /x), there exists another right-angled triangle whose
elements are al, b,, cl, (X, /,l), where
al=m', bl=b, cl-l, e =yl, =I l = —
Thus, starting with the right-angled triangle
a, b, c, (X, /),..........................1)
we obtain a second right-angled triangle whose elements are
m, b, 1, Y 5 - ).......................(2)
If we now take the sides and opposite angles of this triangle
in the reverse order, i.e. write it as the triangle
b, m', 1, (2 -c, T),
we obtain another right-angled triangle with the elements
c', m'I, a (X r - ).................... (3)
Writing this as
i', c', a', (-, X)
we obtain another with the elements
1 c, IC, - a, ( - )...............(4)
From this we obtain in its turn, a, m, 7 2 - )...................... )
Again, from this we have
b, a, c, (/, X ),.........................(6)
the last being the original triangle.