# Types of Slide Rules and their Scales

All slide rules consist of logarithmic scales that can be moved in relation to each other in order to do basic mathematical calculations. The simplest slide rule would have two log scales that you could use to do multiplication and division (see my Slide Rule Introduction page for more info). One of the influential figures in slide rule design, Amédée Mannheim, created a standard slide rule with 4 scales labelled A, B, C, D, in that order (the C/D scales are the standard length log scales). These labels are hardly very descriptive, but they caught on and were perpetuated right up until the end of slide rule production.

As time went by, increasingly specialized scales were added to the rules, increasing the complexity and capabilities of the device. Unlike the basic standard scales, these typically were assigned labels that reflected their function, albeit in a limited way (e.g. K for cubic scales, since C was already taken). Toward the end of slide rule production, many makers began including additional "self-documenting" features at one end of the scales, illustrating the mathematical relationship of each scale to the basic C/D scales. Clearly, this was a great relief for beginners, as deciphering the purpose of the obscure scale labels can be quite a challenge in learning to use the rule.

On this page, I've provided a compendium of the various known scale labels along with categories of rules where they are commonly found. An alphabetical list is provided below for quick and easy reference. Along with a brief description, I've also included the "self-documenting" relationship to the standard C/D scale where appropriate. For detailed instructions on how to use a slide rule, please see my Slide Rule Manuals page.

 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

# Standard Scale Arrangements

Note: Scales are listed from top to bottom, beginning with the top stator. Square brackets [ ] are used to denote scales on the slide portion of the rule.

## Simplex, or Desktop models

Mannheim

Front: A [ B CI C ] D K
Back: [ S L T ]

Originally consisting of A, B, C, D scales, most modern versions of this type of rule also include an inverted C scale (CI) on the front of the slide and a cubic (K) scale at the base of the stator (also known as the body, base or stock of the rule). On the reverse of the slide, you will commonly find sine (S) and tan (T) trig scales and the log (L) scale. Common rules with this arrangement would include the K&E 4053 (illustrated on the right) and Post 1447. Many versions (like the 4053) also come with inch or centimeter rulers, adding to their usefulness.

Rietz

Front: K A [ B CI C ] D L
Back [S ST T ]

Similar to the Mannheim arrangement, but with a few differences in the location of scales. In fact, it's probably a more logically consistent design, as the cubic (K) scale is located above the squared A/B scales, and the log (L) scale is now on the face of the rule instead of with the trig scales. This design also allowed the inclusion of the extended sine scale (ST), useful for both sines and tans. Common on many German-made slide rule models, such as my Faber 57/87 (illustrated) and Eco Bra 1461. These rules also frequently possessed standard rulers as well.

Front: K A [ B CI C ] D P
Back: [ LL1 LL2 LL3 ]
Hard Edge: S T
Slanted Edge: L
Alternate "Advanced" arrangement: S and T scales added to front stator, and L and LL0 scales added to back of slide

Another German innovation, and one of my favourite designs due to its power and versatility. The original Darmstadt design was mainly just the addition of log-log scales on the back of the slide and the interesting Pythagorean (P) scale on the face (see my Faber 1/54 or Nestler 0210 for examples). This required moving the sine S and tan T scales to the hard edge and the log L scale to the slanted edge, which is an unsual place for them (requiring a translating cursor mark). Later on, an "advanced" Darmstadt design was developed with the alternate scale arrangement shown above (illustrated by my Hemmi 130W, shown on the right). Basically, they moved the S and T scales to the face of the rule (along with a rather useless inverted BI scale) and put the standard log L scale on the back with the other log-log scales. This was a rather unusual change, and is actually more reminiscent of the German practice of putting trig scales on the stator face of duplex rules (Japanese duplex rules typically followed the American custom of leaving them on the back of the slide). See below for a comparison of duplex rules. In any case, this type of desktop model also had the advantage of frequently coming with standard rulers.

## Duplex models

All duplex models feature scales on both sides of the stators and slide, with a dual-faced cursor that can reliably relate each side to the other. This greatly increases the number of scales that can be used in combination, although it also potentially increases the risk of error due to misalignment or unintended movement during manipulation. Arrangement of scales differed considerably from model to model, but certain general characteristics were commonly present. Most rules featured additional log-log scales for exponential power calculations (e.g. LL01, LL3, etc.), needed for solving non-integer exponent problems. They also featured C/D scales "folded" at the value of pi (i.e. multiplied by 3.14159...) that are very useful for calculations involving circles and spheres (e.g. CF, DF, CIF, etc.). Some also included expanded square root scales, designated as R1/R2 or Sq1/Sq2, or additional hyperbolic trig scales (e.g. Sh1/Sh2, Th) on "vector" models. Although the variations are too numerous to mention individually, most people tend to have their own favourite arrangement and maker. For comparison, check out some my favourite high-end rules, like the Hemmi-made Versalog, the Nestler-made Staedtler 544 28, the Pickett N4-ES (illustrated), and the K&E 4081.

One interesting note on duplex designs, I've noticed that most American (e.g. K&E, Dietzgen) and Japanese makers (e.g. Hemmi, Relay) liked putting the log-log exponent scales on the front stators, with the trig scales on the back of the slide. This trig placement is similar to the design of most desktop models, which commonly put those scales on the back of the slide. However, German makers (e.g. Faber, Aristo, Nestler) invariably liked putting the log-log scales on the back of the stators, and the trig scales on the front stators. In contrast, many English models featured trig scales on the front of the slide, as do high-end Pickett rules (along with their base 10 log-log scales ... I get the feeling they just liked being different!). Obviously, there's a certain amount of live-and-let-live involved here, but there may be certain advantages to one arrangement or the other, depending on how you plan to use the rule. Personally, I'm not clear about the rationale behind these preferences, so if anyone has any thoughts, please drop me a line. For those of you who are interested, Ron Manley has a section on his The A to Z of Slide Rules page explaining how to use the trig scales in each of these arrangements. Incidentally, like most people, I tend to identify the "front" of a duplex rule by where the maker and model identification labels or logos are placed, which also generally corresponds to the location of the primary C/D scales.

## Circular models

Circular slide rules offer the advantage increasing scale length in a compact space by arranging the scales in a circle instead of on a straight line. At their simplest, they consist of a round disk with dual movable cursors that act sort of like dividers on a Gunter's scale. More commonly, they have a second rotating disk and a single cursor, much the like the common stator/slide/cursor action of a linear slide rule. A good example of a common circular rule is my Concise 28N (illustrated on the right). Circular rules can also be duplex in design, with an additional rotating disk and translating cursor on the back. They were never as popular as linear rules, probably due to the difficulty of keeping the disks and cursors steady while doing manipulations or taking readings. Many circular slide rules are available as additions to some basic flat device or printed table, as evidenced by the E6-B flight computer used by pilots. Various types of cylindrical slide rules have also used throughout history, including the pocket model Otis King.

# Common Slide Rule Scales

This list is hardly exhaustive, and readers are referred to the other relevant slide rule resources listed on my Links and Books pages. Any omissions or errors presented here are entirely of my own doing, and I welcome any comments or additional suggestions readers may have. For detailed instructions on how to use a slide rule, please see my Slide Rule Manuals page.