## Common Scale Sets on Closed Body Slide Rules

### Mannheim

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 set.

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 = sin q.

### Polyphase, Multiphase

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, K and [S, L, T] "Mannheim Polyphase."

### Rietz

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. 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."

### 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 q ranges from 0.1 to 1 on the C scale.  But the angles on the S scale range from 5o 44' to  90o (not 34' 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. 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'.

### 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.

### Folded Scales

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 off-scale results.