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 90^{o} 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 5^{o} 44' to 45^{o} 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 5^{o}
44' (first labeled angle is 6^{o}).

You may have noticed that one can find the sine of angles between 34'
and 5^{o} 44' but not the tangent. For such angles you use
the fact that, to slide rule accuracy, tan q
= sin q.

Rietz
trigonometric scales are of interest. The S scale runs from 5^{o}
44' to 90^{o} (different from Mannheim rules), the S&T scale
runs from 34' to 5^{o} 44' and the T scale runs from 5^{o}
44' to 45^{o} (same as Mannheim). All three scales are keyed
to the C/D scales. For example, directly opposite 27^{o}
on the S scale is 0.454 on the C scale (sin 27^{o} = 0.454).
Directly opposite 27^{o} on the T scale is 0.510 on the C scale
(tan 27^{o} = 0.510). Directly opposite 3^{o} on
the S&T scale is 0.0523 on the C scale (sin 3^{o} = tan 3^{o}
= 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 5^{o} 44' to 90^{o} 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 6^{o} 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.