8.16 The French Connection to a Mystery Rule
Originally posted: 2024 Jul 14
During a recent quest to understand the c. 1900 scale development of various slide rules in the U.S., I learned about the “Beghin” scale layout that was introduced in about 1898-1902 on Tavernier-Gravet slide rules for use in the French engineering schools. Discussed in Cajori’s 1909 text,49 Auguste Beghin – a French author on the subject of the slide rule – suggested in about 1892 the use of a set of “folded scales” on the slide rules used at the French school of science and engineering, École Polytechnique. Modifying the Mannheim design – also introduced by Lenoir-Gravet (later becoming Tavernier-Gravet) – the Beghin layout replaces the A and B scales with folded D and C scales. That is, the scale set would be
DF [ CF C ] D
The Beghin scale set was briefly mentioned in the vignette Scale Wars, as E. Thacher and E. Scofield were introducing folded scale designs in the U.S. during the period from roughly 1880 to 1900, leading to slide rules being sold by Keuffel and Esser and by Deitzgen, respectively, at the turn of the century. Also in Cajori’s text I learned that Beghin had the inverted scale (CI) added to the scale set in about 1902, thus introducing three-factor multiplication to the mix.
A nice collection of Tavernier-Gravet (and Lenoir, Lenoir-Gravet, etc.) slide rules can be viewed at the photocalcul web site. Evidently, over the next 2-3 decades other scales were added to the Beghin system to include the B, A, and K scales as well. (Note that none of the scales are labeled on these early Beghin slide rules; here, we are using the modern scale labels.) A short timeline of the development of the Beghin scale is shown below:50
| Year | Scales | Note |
| 1898 | DF [ CF C ] D | scale set first suggested by Beghin51 |
| 1902 | DF [ CF CI C ] D | Beghin adds an inverted scale52 |
| 1908 | DF [ CF CI C ] D || L ; [ S T B ] | by now, a B scale is added to the slide (back), plus L scale (lower edge) |
| 1925 | A DF [ CF CI C ] D K || L ; [ S T B ] | and by now, A and K are found added to the front |
The basic layout of 1925 for a “standard” Beghin lasted until the 1940s. Note that the fold on CF/DF is at \(\sqrt{10}.\)
As my interest in the Tavernier-Gravet slide rules intensified, I acquired a Beghin slide rule from France that was made in 1929, shown below.
When I first received the slide rule and added its information to my database, I hastily entered the back-side scales using the common combination of S, T, and L and went on with things. A few days later, I started thinking about where I had seen an A scale above folded scales at the top of an early slide rule before. And then I remembered – the K&E Mystery rules! (See A K+E Rule of Mystery and Clark McCoy’s web site.) After looking everything over again, I found my error in the interpretation of the French Beghin scales – there is a “B” scale on the back of the slide, not an L scale. The L scale is on the bottom edge (which I had interpreted in haste as an “inch” scale). I then recalled that the Mystery rule in question also has a B scale on the back of the slide, a feature not seen on the common K&E models. So if we rigorously compare the 1929 Beghin scale layout directly with the c. 1930 K&E Mystery (No. 1) scale set we find:
| Scale Set | Top | Front | Bot | Back | Well | Note |
| Beghin 1929: | cm | A DF [ CF CI C ] D K | L | [ S T B ] | cm | F = \(\sqrt{10}\) |
| Mystery c.1930: | in | A DF [ CF CI C ] D K | cm | [ S L B T ] | F = \(\pi\) |
The only major differences in scale layout between the two are
- the placements of the L scale as noted above,
- the order of T and B is switched,
- the Mystery rule has no scale in the well while the Beghin has an extended centimeter scale, and
- the “folds” are at \(\pi\) = 3.14 for the K&E and \(\sqrt{10}\) = 3.16 for the Tavernier-Gravet.
As the Mystery slide rules date from c. 1930, this appears to be an attempt by K&E to provide an Americanized version of the Beghin rule based upon the French scales being used at that time. And perhaps – as in France – these rules were indeed intended to be used primarily in the college setting due to the success of the Beghin at the École Polytechnique. The University of Washington evidently may have been significant among those to use this K&E model in their Engineering school.
It is interesting that examples of both the Mystery Type 1 (Beghin-style) and the Mystery Type 2 (essentially K&Es model 4053) were both produced from the early 1930s to the late 1940s, many with a U. Washington label, yet were never made part of the official K&E catalog. This makes me wonder if some sort of study was being performed to examine and compare the use of the two scale sets throughout a course of study at a major US university. I wonder what the UW experience was if indeed they integrated the Beghin scales into their introductory slide rule material. I suspect we would have heard about such a comparison had the results been of any statistical significance, if any such study happened at all.
Keuffel and Esser had good connections with Tavernier-Gravet, the latter having produced slide rules for sale by K&E in the mid-1880s. Cajori points out that Tavernier-Gravet had constructed a slide rule with folded scales for a Russian professor in 1882,53 though the concept did not catch on at that time. And this was happening just one year after Thacher received a US patent for his Calculating Instrument with folded scales in 1881. An example exists of a K&E-branded slide rule (a Mannheim) made by Tavernier-Gravet from about 1885,54 and K&E was selling Thacher’s Calculating Instrument by 1887 if not earlier. Though K&E began manufacturing their own rules within the next decade, it should not be hard to imagine that the two companies continued to influence each other throughout the early 1900s as well.
Jay Ballauer also reminded me that the K&E Merchant slide rules had such layouts – Models 4094 (1902 Beghin), 4095 (1898 Beghin, Duplex), 4096 (1898 and 1902), 4097 (1902), 4097B (1902), and 4097D (1925 Beghin). And let’s not forget that the K&E Model 4097D has the same scale set as the Mystery Type 1 slide rules.
So perhaps through the Beghin French connection a bit more light has come to shine on the K&E “mystery”.
