Article 22
Gudermanian Angle

If a circle be used instead of the ellipse of Art. 20, the gudermanian of the hyperbolic sectorial measure will be equal to the radian measure of the angle of the corresponding circular sector (see eq. (6), and Art. 3, Prob. 2). This angle will be called the gudermanian angle; but the gudermanian function $v$, as above defined, is merely a number, or ratio; and this number is equal to the radian measure of the gudermanian angle $\theta$, which is itself usually tabulated in degree measure; thus

Prob. 55.

Show that the gudermanian angle of $u$ may be constructed as follows:

Take the principal radius $OA$ of an equilateral hyperbola, as the initial line, and $OP$ as the terminal line, of the sector whose measure is $u$; from $M$, the foot of the ordinate of $P$, draw $MT$ tangent to the circle whose diameter is the transverse axis; then $AOT$ is the angle required.²

Prob. 56.

Show that the angle $\theta$ never exceeds $90^{\circ }$.

Prob. 57.

The bisector of angle $AOT$ bisects the sector $AOP$ (see Prob. 13, Art. 9, and Prob. 53, Art. 21), and the line $AP$. (See Prob. 1, Art. 3.)

Prob. 58.

This bisector is parallel to $TP$, and the points $T$, $P$ are in line with the point diametrically opposite to $A$.

Prob. 59.

The tangent at $P$ passes through the foot of the ordinate of $T$, and intersects $TM$ on the tangent at $A$.

Prob. 60.

The angle $APM$ is half the gudermanian angle.