Modern Mannheim slide rules add [S, L, T] scales to the back of the slide. For an angle q on the S scale on the back of the slide the value of sin q is in the same place on the B scale on the front of the slide. The range of q is 34' to 90o and the corresponding range of sin q is 0.01 to 1. For angle q on the T scale on the back of the slide the value of tan q is in the same place on the C scale on the front of the slide. The range of q is 5o 44' to 45o and the corresponding range of tan q is 0.1 to 1. Note that sines are read from the B scale and tangents are read from the C scale. Trigonometric scales keyed this way are called "Mannheim" trig scales.
Here's the back of the slide from a Mannheim slide rule (Hemmi 50 from before WWII). Note that the S scale starts at 34' (the first angle that is actually labeled is 40') and the T scale starts at 5o 44' (first labeled angle is 6o).
You may have noticed that one can find the sine of angles between 34'
and 5o 44' but not the tangent. For such angles you use
the fact that, to slide rule accuracy, tan q =
Rietz trigonometric scales are of interest. The S scale runs from
5o 44' to 90o (different from Mannheim rules), the
S&T scale runs from 34' to 5o 44' and the T scale runs
from 5o 44' to 45o (same as Mannheim). All three
scales are keyed to the C/D scales. For example, directly opposite
27o on the S scale is 0.454 on the C scale (sin 27o
= 0.454). Directly opposite 27o on the T scale is
0.510 on the C scale (tan 27o = 0.510). Directly opposite
3o on the S&T scale is 0.0523 on the C scale (sin 3o
= tan 3o = 0.0523. When using the S&T scale remember
to insert a zero between the number from the C scale and the decimal point--0.0523
not 0.523.) S scales that run from 5o 44' to 90o
and are keyed to the C/D scales are "Rietz S scales."
Here is a picture of Hemmi hybrid trigonometric scale from a model 50 rule made shortly after WWII. (The rule is marked "Made in occupied Japan.") At first glance it appears to be an ordinary set of "Mannheim" trig scales but note that the smallest labeled angle is 6o on both the S and the T scales. On genuine Mannheim trig scales the smallest marked angle on the S scale is 40'.
Off-scale problems are minimized if the folded scales are folded at the
center of theC/D scales, i.e., folded at square root of 10 = 3.16.
Hemmi made a lot of slide rules with scales folded at 3.16. Frederick
Post insisted that folded scales on their slide rules be folded at p (3.14). Folding at p
means the number under the hairline on the CF scale is p times the number on the C scale. Many engineering
calculations require multiplication or division by p
and p-folded scales provide an efficient way to
do so while still being good in handling off-scale results.
Hemmi made several Darmstadt rules (model numbers 130-139). There
were never any Frederick Post brand Darmstadt slide rules.