8.23 Scale Wars

The turn of the 19th century into the 20th was a period of significant fluctuation in the design of slide rules and their scales. Not only was the mechanical structure of slide rules rapidly developing during this period, with celluloid laminations and adjustment screws and self-adjusting mechanisms, but there appears to have been a significant amount of investigations into various scale layouts as well. In particular, I would like to contrast and compare what was going on in the U.S. companies Keuffel and Esser, Eugene Dietzgen, and Kolsech at this time as good examples of the slide rule development trajectory.

8.23.1 The Duplex

To start, we first recognize that the Manheimm layout of the standard A, B, C, and D scales was patented in France by Amédée Manheimm in 1850, followed by his patent of the modern “cursor” in 1851. As briefly discussed in Comments on Scale Development, the Manheimm layout was a relatively slight adjustment to the Engineer’s rule scale set, which itself was a slight adjustment to the Carpenter’s rule scale set. For almost 50 years no new ground-breaking feature was introduced to the slide rule layout. While the Carpenter’s and Engineer’s rules also had scales labeled “A, B, C, and D”, it was the Mannheim rule and its popularity that enshrined these labels with their modern-day meanings: logarithmic scales of two decades (A/B) or one decade (D/C) on the stock/slide of the rule. Meanwhile the use of these instruments continued to grow throughout Europe and in America where, other than a few Carpenter’s rules, the slide rules being sold were being imported from Europe.

But soon a new development occurred. In 1891 William Cox, of England, patented his concept of a “duplex” (double-sided) slide rule while working as a mathematical consultant to K&E in the U.S.85 This revolutionary step in the slide rule story simultaneously connects any number of scales on the front and back of a double-faced slide rule. Cox assigned his patent to K&E and at this point K&E became the first major slide rule manufacturer in the U.S. Their production line soon included both the standard Mannheim rule as well as the new Duplex slide rule. And as the U.S. patent has a 17-year period, it was quite awhile before another manufacturer could produce a double-sided duplex rule of this nature to compete with K&E. The first Duplex rules hit the market in about 1895.


An early Duplex rule – K&E Model 4071.
An early Duplex rule – K&E Model 4071.


As is common in such steps forward, the Duplex rule allowed a person familiar with the standard Mannheim to continue using the front face of the Duplex as before. But now, extra scales could be added to the back of the rule that are also connected to those on the front. The first application was to add reversed, or inverted, scales on the back that could be used for additional multiplication and division of numbers with only one setting of the slide. For example, suppose we want to multiply 1.6 times 7.8 times 3.7. We can set the 1.6 on the CI scale against the 7.8 on the D scale on the back of the rule. (See cursor in the top image below.) Simply turning the rule over, we can locate 3.7 on the C scale using the cursor. (See bottom image below.) The number below on the D scale is the product of the three: about 46.2.


Three-number multiplication with the Model 4071.
Three-number multiplication with the Model 4071.


Through the use of the Duplex, multi-numeric operations could be performed not only more quickly, but also with less error in the result due to the limited times that the slide would need to be reset. This gave K&E’s new slide rules a great selling point and helped to quickly put them on the map world-wide.

K&E’s patent language was very well written and strong enough to keep any other manufacturers from introducing a slide rule with the front and back connected through a cursor. It meant that competitors had to find other ways to perform operations with three numbers on a single setting.


8.23.2 The Multiplex

In the meantime, things were brewing just 800 miles to the West. An ex-K&E employee, Eugene Dietzgen, was creating his own slide rule manufacturing operation in Chicago, Illinois. In the early 1880s Dietzgen started his business career as a Midwest Representative of K&E based in their local headquarters in Chicago. He eventually teamed up with a manager of the A.H. Abbot & Co.’s mathematical and drawing instruments department, Otto Luhring, to form the company Luhring & Dietzgen in 1885. In 1891 Dietzgen bought out Luhring’s shares and started Eugene Dietzgen and Company. By 1893 Dietzgen opened his own manufacturing plant to produce T-squares, drawing boards, surveying equipment, and other relevant supplies and the company was renamed the Eugene Dietzgen Company.

Then, in 1898 Dietzgen acquired a U.S. patent (No. 606,388) that was granted to John Givan Davis Mack in that year (the “Mack Improved Mannheim Simplex Slide Rule”). Mack was a professor of engineering at the University of Wisconsin in Madison.86 The Mack Improved slide rules, which have pins and springs embedded in the slide rule to maintain alignment, were a successful improvement to the general slide rule design and were sold from 1902 to 1912, giving Dietzgen national attention. While Dietzgen is noted as a major slide rule maker, they bought their parts mostly from German sources and performed final assembly of their rules in Chicago. K&E, on the other hand, was making their own slide rules completely “in house”.

Now during this period, in 1904, Leon Walter Rosenthal of New York City was granted a patent for a scale arrangement with a special scale on the slide. On most slide rules the “B” scale runs from 1 to 10 to 100, or two decades, from left to right. On Rosenthal’s B scale, the numbers run, left to right, from 10 down to 1 and back up to 10 again. From his knowledge and experience with the K&E Duplex patent, Dietzgen no doubt recognized the significance of this new concept. The Rosenthal patent (No. 767,170) was acquired by Dietzgen that year and was immediately applied to their slide rule repertoire with a new model, the 1762. While the K&E Duplex rule provided a new method for multiplying or dividing three numbers with one setting, the rule was also quite expensive. The Rosenthal simplex design provided the same function but without having to turn the rule over. One could argue that the decade of a B scale is half the length of a C or D scale, and hence not as accurate. While this might be true, for those not requiring that level of improved accuracy the same function can be acquired with the rule lying on the table. The new model could be produced through a simple scale change on an otherwise standard Dietzgen slide rule.

Below is an image of a Rosenthal rule with its special “B” scale. Its distinctive feature is how the numbering emanates from the center of the B scale, increasing to the left and to the right from the central “1”. To distinguish the two parts of the B scale, the left-side of B is called the Br scale (“r” for reverse, or perhaps for “Rosenthal”, or perhaps for both), and the right-side of the B is called the B’’ scale. (For comparison, in the Dietzgen literature the standard A and B scales are often treated as having two parts, the A’ and A’’ scales, and the B’ and B’’ scales, each designation referring to a left or right “decade”.) But note that the scale is simply labeled “B”, just like any other Mannheim rule. One needs to look closely at the numbering to see the difference. Dietzgen dubbed their rule the “Multiplex Slide Rule” and variants of this model were available from 1905 until about 1926.


The Rosenthal scales of the Dietzgen Model 1762.
The Rosenthal scales of the Dietzgen Model 1762.


For comparison, let’s perform our earlier calculation using the Dietzgen rule. The operation is illustrated in the image below. Since dividing by a reciprocal is the same as multiplying, we can line up 1.6 on A’ with 7.8 on Br to perform the first multiplication. The result would be on the A scale at the central index of the B scale. (In our case, 12.5.) Then, simply finding 3.7 on B’’ multiplies this first result and gives the final result on A’’: about 46.2.


Multiplying 1.6 times 7.8 times 3.7, using the Rosenthal scale.
Multiplying 1.6 times 7.8 times 3.7, using the Rosenthal scale.


8.23.3 The Triplex

But Dietzgen wasn’t the only one looking to compete with the new Duplex rules. At about the same time, another company, New York’s Kolesch & Co., was also getting involved in this pursuit.

During the turn of the century, Dennert & Pape and Nestler were the world leaders in slide rule production. Working together, much of the stock used to make their rules was being produced by D&P, and Nestler had the world’s best engine dividing machines used to engrave the stock with highly precise scales. The tooling of their dividing engines could be changed fairly rapidly, yielding fast turn-around times and the ability to produce small batches of rules for outside customers. A discussion of a Nestler prototype for Dietzgen of a Rosenthal slide rule is described in the vignette, Dietzgen Rosenthal Nestler Prototype. And both Dietzgen and Kolesch sold slide rules with their respective names on them, but which were obviously produced by Nestler.

One of the slide rules sold by Kolesch, called the “Triplex”, was introduced in about 1907. Like Dietzgen’s Multiplex, the Triplex (which does not appear to have been patented) used an inverse scale to set up a three-factor multiplication problem with one setting. However, this reverse scale was a single-decade scale of similar length of the C and D scales and hence did not give up on accuracy.


The Kolesch Triplex Slide Rule.
The Kolesch Triplex Slide Rule.


Whereas the Multiplex has the “Br” scale as part of the B scale arrangement, the scale set of the Triplex is simply a different take on traditional scales. In terms of modern scale labels, it goes like this:

A D [ CI C ] D K  


Once again let’s perform our earlier calculation using the Triplex rule, which is illustrated in the image below. We line up 1.6 on D (upper) with 7.8 on CI to perform the first multiplication. The result would be on the D scale at the index of the CI scale, which is also the index of the C scale. Hence, we are in position to perform the next multiplication by finding 3.7 on C, below which we find the final result on D (lower): about 46.2.


Multiplying 1.6 times 7.8 times 3.7, using the Triplex Slide Rule.
Multiplying 1.6 times 7.8 times 3.7, using the Triplex Slide Rule.


The Triplex discussion in Kolesch’s book, How to Use Slide Rules,87 emphasizes the use of the “additional triple scale” which today would be called the K scale. This is likely how the slide rule got its name, though just four paragraphs later 3-factor multiplication is also described:

The arrangement of the four middle single scales also permits the multiplication of three numbers in one setting. Thus, for instance, to multiply 2x3x6 set 2 on inverted scale under 3 on adjoining rule scale and read 36, the result, on D scale under 6 on C scale.
In like manner the arrangement often conveniently permits consecutive division with one setting.



8.23.4 The Duplex Redux

Keuffel and Esser continued to give their Duplex slide rules ever more powerful functionality. Besides the basic Polyphase Duplex rule introduced in the late 1800s, they introduced their new Log-Log Duplex rule – the 4092 – in 1909. The 4092 had the standard A,B,C, and D scales on the front but, along with the new log-log scales, it had CI, C, and D scales on the back of the rule. This allowed for three-factor multiplications and divisions in much the same way as with the Triplex.

Now when multiplying three numbers together, the order of the multiplication can typically be chosen so as to not have the answer fall off-scale. For example, if we multiply three numbers between 1 and 10, we put the smallest number on D, say, and the largest number on CI, and then the middle number is more likely to be available on C, with the answer on D. However this often is not the case, and when extended to further multiplication of more factors it can become harder to avoid having results that fall off-scale, in which case the user must “reset” the index, or perhaps start the calculation all over again. Such a scenario was largely overcome by the introduction of a “folded” scale, which is simply a similar logarithmic scale that is offset from the standard scales. If that offset is half the length of the standard scale (i.e., physical length, not numerical length), then if a number being sought is off-scale, it should be available on the folded scale. Such a solution to the “off-scale” problem was introduced by K&E in their Thacher’s Calculating Instrument, which uses two logarithmic scales offset from each other. The same solution to this problem was introduced by K&E in 1913 with their new Polyphase Duplex rule, the 4088, which included scales DF,CF,CIF,C, and D on the front and CI and D scales on the back.


The K&E Model 4088-2 (c. 1916).
The K&E Model 4088-2 (c. 1916).


The “fold” of the scale is at roughly the half-way point of the standard scale from 1 to 10. Exactly half way would be at “\(\sqrt{10}\)”, or 3.1623, and some slide rules have the fold at this value. However, K&E and other manufacturers realized that having the fold at a value of \(\pi\) = 3.1416 would accomplish the same thing, but would provide a scale that could be convenient in other calculations as well. This scale arrangement with CF and DF scales allowed for more factors to be multiplied with a minimum number of re-settings of the slide. The 4088 also eventually came in four different scale lengths of 5, 8, 10, and 20 inches. (See The Long and Short of It.)

8.23.5 The Dietzgen Phillips

Not to be left behind, Dietzgen continued to play with alternate scale layouts for awhile. In 1919 they introduced their Phillips Slide Rule – Model 1759 – which included an inverse B scale which they called the R scale. (Later, a more standard name for such a scale would become BI – inverted B scale.) An example is shown in the image below.


Dietzgen’s Phillips Slide Rule.
Dietzgen’s Phillips Slide Rule.


From the Dietzgen catalog,

The reciprocal scale “R” is an inverted, double logarithmic scale of the same unit length as the “A-B” scales to which it is referred. This scale permits rapid reading of divisions and inverse proportions, and enables three factors, such as 97 x 98 x 99, to be taken without resetting the slide. It is an unique scale found only on the Phillips rule.

The Phillips eventually came in 8, 10, and 20 inch lengths. It also continued to perform main calculations using the A/B and R scales, rather than using C/D.

8.23.6 The Duplex Redux Part Deux

But then, in 1922, K&E revamped its Log-Log rule into the 4092-3 and 4092-5 (10- and 20-inch versions). The new version included the powerful DF [ CF CIF CI C ] D arrangement from the 4088. So in addition to having the Log-Log scales on the back side, this slide rule also facilitated the multiplication of any five factors with only two settings of the slide.

As an example of the use of the 4088/4092 scale set, let’s take the product 6.3 x 2.5 x 0.17 x 9.3 x 2.8. Consider aligning 6.3 on D with 2.5 on CI. (See circled region in image below.) The intermediate result would be at the index of CI, which is also the index of C. So, moving the cursor to 0.17 on C gives the next intermediate result below on D (2.68). We can now re-adjust the slide to align 9.3 of CIF with the cursor. (See red line in lower image below.) Next, moving the cursor to 2.8 on CF the result will be under the cursor, on D: 69.7.


Five-figure multipication with the K&E 4092-3.
Five-figure multipication with the K&E 4092-3.



8.23.7 The Calculex

As the above series of events was beginning in the US, J. Halden & Co., Ltd. of Manchester, England, was designing and selling drawing and surveying instruments. Their popular “Calculex” slide rule, a circular pocket-watch style slide rule that was introduced by Halden in about 1910, was also apparently participating in the Scale War. The Calculex has a reversed scale on its back side and so has the capability to perform three-factor multiplication with one setting. (See image below.) In its instruction book, under the topic of “Multiplication”, it states,

If there are three factors to multiply, … On reverse scale, set the first factor on A to second on B. Turn over to front face and read result on B opposite third factor on A.


Three-figure multipication with the Halden Calculex. To perform our previous example, on the back side set 1.6 on A against 7.8 on B (reversed scale); on the front, find the result 46.2 on B opposite of 3.7 on A.
Three-figure multipication with the Halden Calculex. To perform our previous example, on the back side set 1.6 on A against 7.8 on B (reversed scale); on the front, find the result 46.2 on B opposite of 3.7 on A.



To round out the discussion, note that in Germany, the Rietz scale arrangement (DRGM 181 110) had been established in 1902. This was an important innovation and historic step in the standardization of scales and scale arrangements. However, Dennert and Pape did not add an inverse scale to their “System Rietz” until 1926.88 Nestler, on the other hand, deployed their Model 34 with a reciprocal scale in 1911, though it was a specialty rule with low production numbers.89 The Nestler catalog of 1911/1912 describes this model, with design features for electrical engineers, as allowing for multi-factor multiplication. By 1914 Nestler was selling models for which inverse scales could be added to the slide rule at additional cost. From the 1914 catalog:90


Wording from Nestler’s 1914 catalog. Catalog image courtesy of Andreas Faßbender. English translation image via Google Translate.
Wording from Nestler’s 1914 catalog. Catalog image courtesy of Andreas Faßbender. English translation image via Google Translate.


Several Nestler models began having reciprocal scales as part of their standard arrangement by the early 1920s.


Back in the U.S., The Frederick Post Company, also of Chicago, was selling German-made rules. Their Model 1452 – the Mannheim Multiphase Adjustable 10” Slide Rule – included a CI scale and was introduced in 1925. From the Post catalog (1925, 1927, and 1930),91

The inverted scale of the Multiphase Rule permits performing multiplication and division, or both, with three numbers in one setting.

Post also introduced a duplex rule in 1925, the Model 1458 – Multiphase Duplex Slide Rule – which included CI on the front and DF, CF, and CIF on the back. These rules were produced by KHH. Though popular in America, the duplex slide rule was not as popular in Europe and did not begin to catch on there until the mid-1930s.92 It was also about 1930 when Hemmi began manufacturing rules with CI scales as well as duplex slide rules, at about the time when they started producing the Model 1458 for Post. Interestingly, Dietzgen did not produce its first duplex slide rule until the 1940s, while Kolesch – to my knowledge – never sold a duplex rule, certainly not one with their name on it.


8.23.8 The Winner

In the end, it appears that K&E won the Scale Wars of the early 1900s. Their early Duplex rule was a natural extension of the Mannheim, and users didn’t need to learn how to use a new split-B scale (nor make mistakes by thinking it was a standard B scale). Guarded by their strong patent, by the time competitors came up with new simplex designs K&E was beginning to introduce even more powerful duplex slide rules that included multi-factor-friendly scales. Even the popular Versalog slide rule, introduced by Post in 1951, used the 1913 DF [ CF CIF CI C ] D arrangement of K&E on its front side. As a matter of fact, the present slide rule collection has 44 slide rule models made by 12 different makers that have this exact scale arrangement. And the duplex slide rule – eventually found on a multitude of makers’ platforms – continued to adorn the most advanced slide rules until the end of the slide rule era, some 90 years after its introduction.

All of that being said, I do find the 1910 Calculex an interestingly competitive option for the multi-factor multiplication application. Its two-sided pocket watch-style design is essentially a duplex and, being circular, it does not have the need for “folded” scales. And it fits in one’s pocket to boot!



From Mike Frey’s magnificent K&E Timeline, the first Duplex rules were offered in 1895, with a larger proliferation of Duplexes being sold beginning in 1901. A Dietzgen time line has been documented by Bruce Babcock, wherein we find that the first iteration of the Multiplex rule was sold by Dietzgen in 1905, with the second iteration available in about 1910. The Kolesch Triplex, from their catalog and book, was offered beginning in 1907-1908. And the early Phillips rules were introduced in about 1919. In Germany, though Nestler, KHH, and D&P helped produce the new US slide rules, they did not include reversed scales on their own rules until the newer versions of the Rietz layout in about 1911 (Nestler) and 1926 (D&P).93 Some history of the early Calculex can be found in an article by Peter Hopp.94


Scale Wars Time Line
Scale Wars Time Line



Note that much of this article has been compiled from information first written in the vignettes entitled Dietzgen Rosenthal Nestler Prototype, Carpenters vs. Engineers, and Rule Change 4092-3, as well as information found at other locations within the web site.




  1. See articles Jour. Oughtred Society 10.1 p29 and 13.2 p20.↩︎

  2. See https://en.wikipedia.org/wiki/John_Givan_Davis_Mack.↩︎

  3. D. Petri-Palmedo, How to Use Slide Rules, Kolesch & Company, New York, 1908.↩︎

  4. Hans Dennert, “DENNERT & PAPE and ARISTO Slide Rules 1872 – 1978”, Jour. Oughtred Soc. 6.1 p5 (1997).↩︎

  5. Andreas Faßbender, private communication.↩︎

  6. Ibid.↩︎

  7. See https://sliderulemuseum.com/srtco/PostCrossRef/SubPages/1452%5b1925-30%5d.html.↩︎

  8. See Dieter von Jezierski, “The Duplex Slide Rules of Faber-Castell”, Jour. Oughtred Soc. 17.2 p17 (2008). ↩︎

  9. See “Slide Rules of the Major Makers” on the Oughtred Society web site.↩︎

  10. Jour. Oughtred Soc. 8.2 p 45 (1999).↩︎