WHAT ARE HYPERBOLIC SCALES?

The hyperbolic scales display one or more of the three main hyperbolic functions:

 sinh x = (ex - e-x) /2 The hyperbolic sine of x, pronounced "shine" or "sinsh". cosh x = (ex + e-x) /2 The hyperbolic cosine of x, pronounced "cosh". tanh x = (ex - e-x) / (ex + e-x) The hyperbolic tangent of x, pronounced "than".

The Hyperbolic functions are related to the usual trigonometrical functions by means of imaginary circular angles thus:

sinh x = -i sin(ix)
cosh x = cos(ix)
tanh x = -i tan(ix)

where i is the imaginary unit with the property that i2 = -1.

More detailed information on the use of hyperbolic functions on slide rules can be found in:

Hyperbolic Functions on Slide Rules and Civil Engineering Applications by Pierre vander Meulen from the Slide Rule Gazette Autumn 2001.

and

The Highest Class Electrical Engineer's 'Universal' Duplex Slide Rule with Patent Vector and Hyperbolic Scales, 20 inches by Richard Smith Hughes from the Journal of the Oughtred Society Fall 2016

The hyperbolic scales are to be found on many general purpose high-end rules and on many specialist rules used in electrical engineering,

Here now are some areas in which hyperbolic functions appear:

A. If you take a rope, fix the two ends and hang under the force of gravity, it will naturally form a hyperbolic cosine curve.(catenary)
Mathematically, the catenary curve is the graph of the hyperbolic cosine function. Catenaries and related curves are used in architecture and engineering, in the design of bridges and arches, so that forces do not result in bending moments.

We can use the catenary shape to know how much cable to place between two poles in high power transmission lines. Too much cable and it sags too much making it a hazard. Too little cable and it breaks due to high tension as it stretches.

B. Velocity addition in (special) relativity is not linear, but becomes linear when expressed in terms of hyperbolic tangent functions. More precisely, if you add two motions in the same direction, such as a man walking at velocity v1 on a train that moves at v2 relative to the ground, the velocity v of the man relative to ground is not v1+v2; velocities don't add (otherwise by adding enough of them you could exceed the speed of light).
What does add is the inverse hyperbolic tangent of the velocities (in speed-of-light units, i.e., v/c). tanh-1(v/c)=tanh-1(v1/c)+tanh-1(v2/c) This is one way of deriving special relativity: assume that a velocity addition formula holds, respecting a maximum speed of light and some other assumptions.

C. On a map using the Mercator projection, the relationship between the latitude L of a point and its y coordinate on the map is given by y=arctanh(sin(L)), where arctanh is another way of representing the inverse of the hyperbolic tangent function.

D. The wave equation, a fundamental result in classical physics is an example of a hyperbolic partial differential equation, and thus these functions often appear in classical physics problems.
Many kinds of nonlinear PDE have wave solutions explicitly expressed using hyperbolic tangents and secants: shock-wave profiles, solitons, reaction-diffusion fronts, and phase-transition fronts.

E. Fluid Dynamics: Depending on initial conditions, some solutions of the Navier-Stokes equations can be hyperbolic.
The analysis of waves in shallow water also leads to the hyperbolic functions.

F. The pursuit curve (tractrix). The path a shuttle craft uses to link up with an orbiting space station or a missile takes when locked on to a target is repreented by a hyperbolic funtion.

G. Electronics. The output voltage of a differential amplifier made out of bipolar junction transistors is proportional to the hyperbolic tangent of the input voltage. Further, the volume control variable resistors once commonly used on stereo's and TV's followed a hypebolic decrease in resistance as well. If i'm not mistaken, the digital volume controls emulate that hyperbolic curve.

H. Hyperbolic functions also feature in astrodynamics for orbits with eccentricity greater than 1, ie hyperbolic orbits.