Hemmi153 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
Conversion between degrees and radians:
θ(degrees)=>Rθ(radians)
Rθ(radians)=>θ(degrees).
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≈0.5e^{x}  x>3.0 
tanhx≈x  x<0.1 
tanhx≈1  x>3.0 
coshx≈1  x<0.1 
coshx≈0.5e^{x}  x>3.0 
References
 Journal of the Oughtred Society, Vol. 9, No 2. 2000
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