Common Scale Sets on Closed Body Slide Rules
The scales on the original Mannheim slide rule were just A [B, C] D.
Hemmi models 24, 34 and 34P (Frederick Post 1443 [1931-41]) used this scale
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.
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
= sin q.
Manufacturers used words like "Polyphase" or "Multiphase" to describe rules
with more than just the basic Mannheim scale set. Keuffel and Esser
called the scale set A [B, CI, C] D,
and [S, L, T] "Mannheim Polyphase."
K, A [B, C] D, L and [S, S&T, T] is the "Rietz" scale set.
It was most popular in Germany but I suspect all manufacturers made a least
one slide rule with Rietz scales. Hemmi Rietz rules have model numbers
in the range 60-79.
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."
Hemmi Rules from WWII to 1955
From WWII to about 1955, Hemmi used Rietz S scales on their Mannheim
slide rules. At first glance the scales on the back of the slide
look like ordinary S and T scales. The T scale is ordinary;
angles run from 5o 44' to 45o and tan
ranges from 0.1 to 1 on the C scale. But the angles on the S scale
range from 5o 44' to 90o
to 90o) and sin
q ranges from
0.1 to 1.0 on the C (not B) scale. There is no direct way to to
read the sine or tangent of angles smaller than 5o 44'.
I suppose there is some computational advantage to this arrangement but
it didn't prove popular and Hemmi reverted to "standard" Mannheim trig
scales about 1955. This experiment occurred on Hemmi models 30/32
(Post 1441), 34R, 34RK (Post 1444K, H-O 1762), 50 (Post 1452, H-O
1771), 50W (Post 1452W), 50-20 (H-O 1772). A few Hemmi models
(43, P43 and perhaps others) started with a Rietz S scale and never changed.
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'.
Inverted Trigonometric Scales
There is a theoretical computational advantage to inverted trig scales.
(TI scale runs from 45o at the left end to 5o 44'
at the right end; SI scale runs from 90o at left end to 5o
44' at the right end.) Hemmi used inverted trig scales on many of
its late model slide rules but the only Post models that use them are models
1445 and 1445P.
In operations on the C and D scale; one frequently runs off scale.
For example, multiply 3 x 4 by putting the left end of the C scale opposite
3 on the D scale; move the cursor to 4 on the C scale and read the result
on the D scale. There isn't any D scale at this point! The
result is 12 but the D scale ends at 10. This problem can often be avoided
if your slide rule has folded scales. In the above example, when
you realize that the answer is off scale to the right, you can read the
product (12) from the DF scale--opposite 4 on the CF scale.
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
If one does a lot of exponential calculations the scale set of choice is
"Darmstadt." The scales are: L, K, A [B, CI, C] D, COS, Sin, Tg and
[LL1, LL2, LL3]. The Sin scale is in degrees and gives the angle
whose sine is shown on the D scale. Similarly, the Tg scale gives
the angle whose tangent is shown on the D scale.
Hemmi made several Darmstadt rules (model numbers 130-139). There
were never any Frederick Post brand Darmstadt slide rules.
Stadia Slide Rules
to Order | Home | Slide
Rules | Hemmi &
Post Parts | K
& E Parts | R/R
Parts | Books
& Literature | Cases & Boxes
| Msc Info
Catalogue Raisonne | Hemmi
Cat II | Frederick
Post Catalogue Raisonne