The less expensive slide rules of simplex type have tended to be made according to several traditional fixed patterns. But duplex slide rules, on the other hand, seem to come in a wide assortment of types.

However, many duplex slide rules have an arrangement that at least tends towards something like this:

K A/B S ST T1 T2 C/D DI P LL00 LL01 LL02 LL03 DF/CF CIF L CI C/D LL0 LL1 LL2 LL3 LL4

It's obvious why slide rules would only tend towards this type of arrangement rather than actually having that arrangement. On the back, each half of the stationary part of the rule has five scales, while on the front, the halves have two or three scales. And, on the other hand, the slide has five scales on it in the back, and six in the front.

Why is the arrangement shown above a desirable goal?

It offers the scale subset

A/B C/D DI

which is an inside-out equivalent of the classic "Mannheim" arrangement of

A/B CI C/D

that allows many functions of three variables to be worked out with a simple operation on the rule.

The trig scales are on the slider. Why is this useful? After a number is multiplied by the sine of one angle, holding the cursor in position allows the slider to be moved so that the result can then be multiplied by the tangent of another angle, for example.

Not all the scales on a slide rule work like the trig scales. While a number on the log-log scales might be used as the starting point for a computation, the end result is then read on the log-log scales again. The L scale is also used for reading a final result, and thus doesn't gain anything from being on the slide.

Why is the L scale on the slide, then, in the arrangement shown above? Because it is on the back of the rule, since it is used for calculations of a type related to that for which the log-log scales are used, and they clearly consume the available space on the stationary part of the rule.

The P scale is shown on the front of the rule, as its function is related to the trig functions.

By adding a few scales, one could arrive at a more balanced slide rule arrangement based on the starting point shown above, for example:

Sh1 Sh2 Th K A/B S ST T1 T2 C/D DI DFM L P LL00 LL01 LL02 LL03 DF/CF CIF R1 R2 CI C/D LL0 LL1 LL2 LL3 LL4

The DFM scale, in conjunction with the L scale, can permit the series of log-log scales to be extended upwards to arbitrarily large numbers. Having the R1 and R2 scales on the slider permits higher-precision multiplication to be performed, although with several steps.

The problem with a slide rule like this, though, is that it has 32 scales. In practice, such rules tend to be unwieldy.

Traditionally, even the high-end duplex rules available in North America had either come with a 3/5/3 distribution of scales - three on each stationary part of the rule, and five on the slider, a 4/4/4 distribution of scales, or even a 3/4/3 distribution of scales. Dietzgen rules tended to have the first, Hemmi and Ricoh rules the second, and Keuffel and Esser rules the third.

Hemmi and Ricoh rules, however, might start with a 4/4/4 layout, and then squeeze in one extra scale on one side of the slider; Keuffel and Esser rules, on the other hand, tended to sneak in an extra scale on one side of the stator instead, and rigidly respected the limitation of the slider to four scales.

With a limited number of scales, it is harder to achieve an ideal layout.

The Deci-Lon slide rule, brought out by Keuffel and Esser in 1961, had a 5/5/3 distribution of scales, giving the arrangement:

LL03 LL02 LL01 LL00 A/B T ST S C/D DI K R1 R2 DF/CF CIF L CI C/D LL0 LL1 LL2 LL3

and, since one has to flip the rule over, around its long axis, to go to the other side, the top becomes the bottom, so there is one wide piece, and one narrow piece, on the stationary side of the rule, not two pieces with five scales on one side and three on the other.

Of the slide rules that could be said to have an optimal scale layout, this one was the one which was best known in North America. This particular slide rule is a highly sought-after and valuable collector's item.

There certainly are, however, other slide rules that had scale layouts which avoided most of the compromises that came with the older slide rules with smaller sets of scales.

A few I would single out for special praise, from among those listed on the previous page, are:

The Graphoplex 690a:

LL00 L T2 A/B T1 ST S C/D P DI LL0 LL01 LL02 LL03 DF/CF CIF K CI C/D LL3 LL2 LL1

The log-log scales are split. Note that the L and K scales are interchanged from their usual positions on a conventional rule; while this does put the wrong one on the front and the back, it puts the right one on the stationary part of the rule and on the slider, which is arguably more important.

Placing the T2 scale on the stationary part of the slide rule preserves both the A/B and C/D scale pairs, This is unusual, but it is very useful for some types of trigonometric calculations. This detail makes this particular rule uniquely optimal for its scale set.

The Pickett Microline 140:

LL01 K A/B T ST S C/D DI LL1 LL02 LL03 DF/CF CIF L CI C/D LL2 LL3

is a conventional rule widely available in North America of smaller size. The Koh-I-Noor 10010 Techni-Log slide rule, also by Graphoplex, had a very similar scale set, but with the L and K scales again interchanged.

The Pickett N500 is another similar rule with the same scale set, although it was less conventional in that it followed Pickett's common practice of pairing scales together from the two log-log series.

Aristo made a large number of slide rules that had scale arrangements I admired. One example, in this range of sizes, is the Aristo MultiLog 0970:

LL00 K A/B T ST S C/D DI LL0 LL01 LL02 LL03 DF/CF CIF L CI C/D LL3 LL2 LL1

Since the stators are both one scale smaller on the front, it is something of a surprise that this rule didn't at least add a P scale, very common on Aristo rules, but that can be explained by the fact that the closely related log-log scales were close together.

A very impressive slide rule from Aristo is their HyperLog 0972:

H2 Sh2 Th K A/B T ST S P C/D DI Ch Sh1 H1 LL00 LL01 LL02 LL03 DF/CF CIF L CI C/D LL3 LL2 LL1 LL0

Although it seems odd for the rule to lack a T2 scale when it has several uncommon scales, as those uncommon scales are in two complete sets (P, H1, H2 and Sh1 Sh2 Th Ch) this is a very well-designed rule. The reverse of the slide, having five scales where the front has six, is spaced apart a bit more generously, but moving the P scale to the rear to make room for a T1/T2 pair is not an option, since it is used with the trig scales.

Note how the Aristo and Graphoplex slide rules place the most-used LL3 portion of the log-log scales closest to the slider.

In comparison to these slide rules, many North American and Japanese slide rules gave up either the A/B scale pair, or the C/D scale pair, on the front of the rule in order to fit in the trig scales on the slider. On the other hand, a number of European duplex rules placed the trig scales on the stationary part of the rule; for example, both the Nestler 0292 and the Faber-Castell 2/83N do this: but the Faber-Castell 2/83N, admittedly, had the good excuse of having to fit quite a bit on the slider to allow a paired set of double-length scales on one side of the rule.

A slide rule with 26 scales, the maximum that might be considered to fit on a conventional slide rule, can present a particularly difficult problem for someone seeking an optimal layout, depending on which scales it is chosen to include. The Deci-Lon and the Graphoplex 690a, as shown above, are two rules that I think were successful in this size range.

If I want to include the same scales as the Deci-Lon, but I also want to place the two R scales on the slider, a first attempt at an optimal arrangment might be:

LL00 LL01 K A/B S ST T C/D DI LL02 LL03 LL0 LL1 DF DIF/CF R1 R2 L C/DI D LL2 LL3

Since the 'Règle des Ecoles' arrangement on the back of the slide rule is intended for multiplication using an inverse scale, in this arrangement I take the unconventional step of placing the inverse scale in contact with the regular scale. In order to place the R scales on the slider, I turn the center of the rule inside-out. This puts a DI scale on the back of the rule instead of a CI scale, and the front already has a DI scale, because of the need to place the trig scales on the slider, but that is only a minor disappointment.

But it is possible to be even a little bolder, to obtain a CI scale and a DI scale on the rule instead of two DI scales, as follows:

LL00 LL01 K A/B R1 R2 CI C/D L LL02 LL03 LL0 LL1 DF DIF/CF S ST T C/DI D LL2 LL3

Moving the trig scales to the back of the rule, where the optimal multiplication scales are, and the R1 and R2 scales to the side of the rule with the A/B and K scales is logical, if unconventional. But the usefulness of this arrangement depends strongly on the scale set used; omitting the R1 and R2 scales to obtain a T1 and T2 scale set and a P scale, for example, leads to the more conventional Graphoplex 690a arrangement, or something like it, as the optimal solution.

As far as I know, no actual maker of slide rules had ever been bold enough to produce a slide rule with this type of arrangement. In one way, however, it is very conventional, in that the scales K A/B CI C/D are all in the same arrangement as on a 'Mannheim' slide rule, and the trig scales are even also on the back of the slider. The resemblance to the European Reitz slide rule is even closer, since the trig scales are of modern type, and the L scale has a typical placement for that type of rule: the Régle des Ecoles on the rear, the two R scales on the front, and the log-log scales, are all added to a typical modern Reitz-based layout without shifting even one scale of its original complement.

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