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Hemmi/"Sun" Model 153

• Type: Hyperbolic Log Log Duplex
• Size: 10 Inches
• Material: Bamboo and celluloid
• Scales:
  • Front: L K A [ B CI C ] D T Gθ
  • Back: θ Rθ P [ Q Q' C ] LL3 LL2 LL1
• HvH: 5062

Trig and Log-Log rule with unique scales for circular functions. The θ scale is marked in degrees from 0-90 in a non-logarithmic manner. It appears to be a projection of the first quadrant of the unit circle onto a line tangent to the circle at θ = 45º. R_θ gives the angle in radians. All trig functions are referenced to these scales.

The P and Q scales are called in the manual "square scales" and are used in the Pythagorean relation √(a² + b²). You can use these scales to determine vector length in higher dimensions as well. P and Q also give sine and cosine against the θ and R_θ scales. Read the sine directly on P; set the left index of Q to that point, and read the cosine on Q under the right index of P. A T (tangent) scale (also calibrated against θ and R_θ) is on the reverse.

When calculating the angle of a vector, one divides X into Y, notes the quotient on paper or in memory, sets that number by hand on the T scale, flips the rule over and reads the phase angle on θ or R_θ.

On the reverse of the rule is a scale marked G_θ. This is the inverse Gudermannian function of the R_θ scale. The Gudermannian function, 2 tan-1 ex - π/2, and its inverse, ln tan (π + 2x)/4, relate the circular functions to the hyperbolic functions in the following manner:

sinh x	=	tan (gd x)
cosh x	=	1 / cos (gd x)
tanh x	=	sin (gd x)

To use, set the argument x on the G_θ scale. Read sinh x on the T scale and tanh x on the P scale. To get cosh x, first determine sinh x. Note this answer and set it on the Q scale then read cosh x on the Q' scale.

Gift of Gretchen Apgar. Dated 1930-1945.