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

60, 61, 62] THE MEASUREMENT OF ANGLES 105
equation o = II (a), connecting the segment and the corresponding angle of parallelism, has had only a geometrical significance.
In it oa has stood for a certain definite acute angle, which has
the property that the perpendicular to one of its bounding lines,
at a distance a from the angular point, is parallel to the other
bounding line.
When it comes to assigning numerical values to angles, the
choice of one number is sufficient, if, in addition, the angle
zero is denoted by O. In the Non-Euclidean Trigonometry
we shall assign the number 2 to the right angle. All other
angles will have the numerical values proper to them on this
scale.
In the rest of this work, when we use the equation o = II (a),
both o and a will be numbers, the one the measure of the angle
on this scale, the other the measure of the segment on one of
the scales agreed upon below (~ 55), in which the unit segment
is the distance apart of two concentric Limiting-Curves, when
1
the ratio of the arcs cut off by two of their axes is e or ek.
It should perhaps be remarked that in dealing with the
trigonometrical formulae in the previous sections the measure
of the segment, and not the segment itself, is what we have
meant to denote by the letters in the different equations.
~ 62. The Trigonometrical Functions of the Angle.
The Trigonometrical Functions
sin c, cos c, tan a, etc.,
are defined by the equations:
ia - ia ia - ia
e -e e +e
sin oc = C cos a =
2i 2
sin o 1
tan c = --, coto= ta
Cos O tan o
1 1
see c==, cosec c= —
Cos oc sin oc
The fundamental equation of the Hyperbolic Trigonometry
is tanh a = cos oc,
when c = II(a).