Hemmi-153 system of trigonometric and hyperbolic scales

The following is short synopsis of the article “The Sun Hemmi System of Trigonometric and Hyperbolic scales” by Brian Borchers and Noël Cotter published in [1].

Hemmi 153 has scale layout

Brief scale description

The Hemmi system contains the following interrelated scales: (as labeled on a slide rule) L, P, Q, Q', θ, Rθ, Gθ, T (or Tθ) L is familiar uniform scale which is normally used in conjunction with D scale to find decimal logarithms of a number. Scales marked "P" and "Q" are square roots of L. They are not logarithmic, which is unusual. Here is how you may find squares of numbers: xP=>L(x²). (This is not a practical algorithm, it is here just to illustrate scale relationship.)

Together scales P and Q allow to find third variable from equation u²+v²=w² when other two are known. Q' is an extension of Q.

Vector length

To find length of a vector with coordinates x and y: 0Q->xP;yQ=>P(l=√[x²+y²]).

For example: x=3, y=4, l=5 0Q->3P;4Q=>P(5).

Trigonometric functions

Angle in radians should be set on Rθ scale, rather than on θ.

Sine: θ(u)=>P(sin(u)).

Cosine: 0Q->θ(u);10P=>Q(cos(u)).

Tangent: θ(u)=>Tθ(tan(u)).

Hyperbolic functions

sinhu: Gθ(u)=>Tθ(sinh(u)).

tanhu: Gθ(u)=>P(tanh(u)).

coshu: |->Gθ(u);0Q->|;10P=>Q(t);tCI=>D(cosh(u).

Approximations

For arguments outside the range of Gθ scale the following table can be used for useful approximation:
 sinhx≈x x<0.1 sinhx≈0.5ex x>3.0 tanhx≈x x<0.1 tanhx≈1 x>3.0 coshx≈1 x<0.1 coshx≈0.5ex x>3.0

References

1. Journal of the Oughtred Society, Vol. 9, No 2. 2000

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