SIDEBAR: The Problems with K&E Rules...

In my own exposure to the vast array of K&E slide rules, I've noticed some problems that pop up consistently throughout my inventory of rules.   While these aspects don't detract from the overall great quality and power of their products, there are some annoyances you need to be aware of when you seek to collect them yourselves.  

The most common problem is the "K&E Rotting Cursor Syndrome," known as "KERCS."  (You know it's a problem if there's an acronym for it.)   Rules made prior to ~1950 had cursor rails that were made of a plastic that didn't hold up well over time and would eventually crumble to pieces.  K&E seemed to rectify this after 1950 with a more durable plastic that also held up better against the typical yellowing of old plastics. Thus, if you are looking at K&E slide rules on eBay, inspect the pictures carefully as a majority of the older rules will have cracked or crumbling cursors.  Once cracked, any disassembly of the cursor for cleaning runs the high risk of watching the cursor rails crumble into your hands.  

Less known about K&E wooden rules, especially when compared with the Japanese bamboo-constructed rules, is that while mahogany is a generally stable wood (often used in the making of guitar necks), it isn't as stable as bamboo.  Some of the rules in my collection show slight signs of cupping across the face of the slide and stator (base) rails, the result of changes in humidity and weather over the life-time of the rule.  This doesn't affect functionality, per say, but I've noticed that it can affect the appearance of a rule.  

In a similar way, because the slide is designed to glide into the stator rails with a tongue and groove design, the mahogany of the slide's tongue is prone to chipping off at the ends, and in many cases breaking off along the wood grain much further down the slide than many would like.  So, an inspection of the slide before purchase is a good idea.  That said, with a donor slide rule of matching mahogany wood, you can typically repair broken slides with some wood glue or super glue.  

Many of the single-sided Mannheim types have a stator rail screwed onto and through the rules backing.  This is not a problem in general, but because the backing is thin, overtightening these screws can wear out the countersunk shoulder in which the screw rests, eventually widening and no longer holding the rail firm to the backing plate.  This can be revealed often times in pictures when there's larger gaps than you'd expect when the slide is in place.  

And because the celluloid plastic is bonded to the wood with glue, separation can occur over time.  While this is often repairable with some super glue and a clamp, neglecting the separation can lead to broken-off sections of the plastic on many samples.   The quality of the adhesives did improve overtime, so older rules are more susceptible to delamination, which is why companies like Nestler and AW Faber actually used screws to reinforce the celluloid laminations of earlier slide rules.

Finally, the smaller, single-sided plastic rules of the 4150-1 and Doric types of pocket rules are a little too flimsy in my evaluation. The plastic backing has easily cracked in my own hands when cleaning them, or can peel away from the stator rails.  Again, a little super glue can fix this, but I find the overall quality in those rules disappointing.  In a similar way, the older Ever-There 4097 and 4098 rules made of "Xylonite" are very brittle now, the toll of 70 to 80 plus years of existence, many of which having warped or curled over those years.  I snapped a slide in two when trying to dislodge a stuck and warped sample myself, so condition is crucial when considering a purchase.  Likewise, these rules have yellowed much more than any other K&E slide rule you will find.  In most cases, the aged yellowing of celluloid can actually be cleaned or sanded away from the surface of the rule without affecting any of the engine-divided scales/fonts.   But I've been unable to make much progress with restoring the Xylonite in a few of my own Ever-There slide rules.  

​Having those concerns mentioned, the great thing about many of the K&E rules is that they can be brought back to life, remarkably so, with a few repairing skills.  You need a completely intact rule (especially the cursor), but cosmetic issues can be corrected indeed.  I've taken a 20" 68-1200 (4081 type) duplex rule that was covered in severe rust-colored stains, and then I used sandpaper, razor blades, and Scotch-Brite pads to bring the celluloid back to it's old glory without affecting the scales themselves...or at least not in a way that hasn't been worth the effort of restoration.  As such, I've bought some pretty cheap, cosmetically-damaged K&E slide rules and turned them back into a display-worthy sample.  Likewise, with use of a cheap donor slide rule, a 3D printer, or via metal fabrication, you can replace missing components, especially cursor rails, on an otherwise pristine slide rule. 

All About K&E (Keuffel & Esser) Rules

Author's note: This, and most articles I write, are living documents.  This article about K&E represents my research and understanding at the time of writing.  There are no targets with this writing, no thesis; no other purpose except to work my way through my own process of learning.  The text will grow, change, and evolve slowly until, some day, a reader finds value in it.  If that is you, thanks for taking the time to read it.  If it's not you, then come back in a few years.   

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Keuffel and Esser is the oldest major US slide rule maker, going back to 1867.  Beginning as a supply house for anything engineering related, they are widely known for drawing supplies, surveying equipment, and of course, their long history and diverse models of slide rules.  Famously, in 1891, they originated the "duplex" design of having scales on both sides of the rule with a dual indicator.  

The key to understanding K&E slide rule production is to pay attention to both their model naming convention as well as their cycles of serial numbers.  I will attempt a healthy description of the histories here, which will also reveal much about the company on the whole.   Likewise, we'll discuss slide rule construction, the major families of rules and their evolutions across model lines, and we'll wrap up with my own portfolio of K&E rules, which are the most numerous in my collection.   The goal is to get the would-be collector up-to-speed on the K&E world of slide rules.  

Model Number Naming Conventions

Model numbers typically take the form of 4XXX-L, where "L" is the approximate length of the slide rule in decimeters (1/10th of a meter).  As an example, the most popular K&E slide rule historically is arguably their 4053 "Polyphase" series of Mannheim-style slide rules.   This family of rules was in continuous production from 1909 until just before the end of the slide rule era in 1970.  These rules came in 4053-1 (5-inch), 4053-2 (8-inch), 4053-3 (12-inch), and 4053-5 (20 inch) variations.  The "dash 2," 8-inch rules are a rather rare size for K&E rules, only offered as an option in certain years of their 4053 and 4088 slide rule lines.  

K&E also used an "N" in front of some model numbers, as in N4XXX, to denote a major change in the slide rule for a new production year.   Presumably, this means "new" to distinguish it from the old version.  (Pickett would later do something similar) They did this to a majority of the model lines in 1925, the first instance of the "N" designation occurring in 1922 with their N4135 "Power Computing" Duplex rule.  And seemingly arbitrarily, they would revert back the original designation in some cases.   For example, the 4080-3 "Log Log Duplex Trig"  replaced the 4090-3 model in 1937, became the N4080-3 in 1948, only to switch back to the 4080-3 in 1954.    The switch to the "N" in 1948 reflected a major change in the scales for that rule.  Another major change in scales occurred in 1954, whereas K&E just dropped the "N" from the model name.  

I said earlier that K&E models are typically a 4XXX type of number. There are notable exceptions.   One exception is when, prior to the year 1900, all of their slide rules (and other items like Thacher and Fuller Calculators and their sector rules) used a 17XX scheme.  Likewise, the earliest K&E products made before 1887 used a 4XX three digit scheme.  Going forward in time, in 1962, K&E shifted away from the 4XXX numbering scheme and began using model numbers of some 68-1XXX variation.  "68" was an indicator of "slide rule," so likely the company shifted all company products to this number format for accounting reasons. 

Thus, you will see something like their 4081-3 "Decitrig" slide rule become known as the 68-1210 "Decitrig" in 1962.  Same rule; different number, of which, by the way, they often didn't stamp on the slide rule itself.  This includes two of their best slide rules, the 68-1261 "Jetlog" and the second variant of the powerful 68-1100 Deci-Lon 10 rule.  It did indicate the model number on the box, however.  Furthermore, there are known instances when a post-1962 slide rule in a 68-1XXX labelled box still contained a slide rule with the old 4XXX model number, likely in the early 60s as K&E was still unloading their inventory of older rules.

The move away from 4XXX series numbers with the dash is frustrating for collectors, as Indicating the length of the rule within the model number with the "dash" is very helpful.  Plus, the 68-1XXX scheme is very difficult to remember.   And I do question the changing of 62 years worth of model familiarity among their consumers.  This couldn't have helped sell more slide rules. 

Albeit a decade later it would hardly matter anyway.  In a way, the 68-1XXX scheme began the countdown towards the hand-held calculator age and the collapse of the company. 

Serial Number Conventions

As for serial numbers, K&E tried to make it difficult for us in that they used a sequential numbering of their rules regardless of the model. So, a 4080-3 model could have a serial number of 123456 while a 4070-3 might be SN # 123457.  These six digit serial numbers, which K&E didn't use at all prior to 1922, would run from 000000 to 999999 and restart to zeros, starting the cycle over again.  There were 4 cycles and 3 "roll-overs" back to zero, occurring around 1944, 1956, and 1967.   Therefore, dating a slide rule will reveal three date possibilities, but usually only one logical choice based on the type of scales, model type, and construction used.  You will discover that within a typical model type, there will be subtle changes from year to year, thus creating a number of "variants" with each rule.   Production rates were fairly linear, with around 66,000 units made prior to World War II, 77,000 per year post-war, and then ramping up to nearly 100,000 per year during the late 50s and 60s until, in 1965, they tapered production back down to around 70,000 units per year. 

There is also an extra wild-card where serial numbering is concerned. K&E used specific serial numbers for certain models as separate from the usual SN# chain.  Most notable in that is their "Ever-There" series of 4097 and 4098 slide rules during the 30s and 40s, and the special "Celanese Celcon" version of the Doric slide rule produced only in 1968.  Likewise, as of 1947, any of the plastic rules are not included within the serial number line-up as they had their own as well.  Thus, it's typically the wooden rules that are more easily dated (not including the Beginner's 4058 rules that never had serial numbers).

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This chart shows the pattern of serial numbers of most K&E slide rules on a year-by-year basis. Note the "rollover" of the serial number as they manufacture each one-million slide rules.

In dating my own slide rules, I typically use the chart above to give me a idea of possible dates, but then I peruse Paul Tarantolo's massive collection of K&E (and other) rules archived over at the Oughtred Society.   This is an amazing collection of m models collected and categorized for us to see.  Compare yours with one of his and you'll know what you have.   Proudly, I have a few K&E rules that he does not have!

Slide Rule Construction

Most traditional K&E rules are of wooden mahogany construction covered in a celluloid, ivory-colored plastic which held the engine-divided (etched) scales.   The exceptions to this are their "beginner's" type of consumer rules that were made of wood, painted white, with black-painted scales (series 4058) and any of their plastic rules; beginning in 1931 with their "Ever-There" family of 4097 and 4098 pocket rules; their "Doric" type of plastic rules beginning around 1950; and most of their more "modern" slide rules like the Jet-Log and Deci-Lon series of slide rules, both of which are highly collectible and highly regarded slide rules made of "Ivorite."  

Certainly as plastic became more practical as a material, it would begin to see progressively more use in the slide rule.   With K&E slide rules, plastic would be more incorporated into their rules, regardless of price, not merely used in their budget rules.  As such, by the end of the slide rule era, most of the traditional wooden construction techniques had either been modified to incorporate more plastic features, the wooden models would give way to plastic rules entirely, or all-plastic rules would be added as the spiritual successor of the older model lines.  

Regardless of construction, K&E made it a point of pride to produce high quality rules, and for the most part during their history they absolutely succeeded. Their slide rules even came with a life-time warrantee, which is unheard of today.  But as a collector I've discovered some problems in this area when judged over the course of time. For more, please see "Sidebar: The Problem with K&E Rules..." at left.  

The K&E Model Map - A timeline and categorization of K&E slide rule models over time - courtesy of Michael Frey of the International Slide Rule Museum.

A slide rule in my collection, the N4082-3 "Radio Special," is a Specialty Rule made by K&E. But it shouldn't be dismissed as "miscellaneous" in our thoughts. Rules like this had a long history in the K&E product line. I will be equally excited to talk about the long history of Merchant's, Stadia, and Radio/Electrical rules later in the article, even if this author wouldn't necessarily see use in them compared to a general arithmetic slide rule.

Slide Rule Families vs. Slide Rule Series

Understanding slide rule construction, which will be discussed when talking about the individual slide rules, we now want to consider how K&E would organize their rules throughout their long history and the way they would market to consumers.  What we will see is somewhat of an evolution in this regard.  

For the most part, from the inception of the company, slide rules were categorized according to their construction, whereby single-sided rules and double-sided rules served as the major differentiation, and within those extremely broad categories, they would be categorized according to purpose (a model line or a scale set).   This is very predictable in the first 50 years of K&E product catalogs.  But around the 1930s, we begin to see certain models of rules sold in "series," a collection of rules with similar build and technology, yet with a diverse range of uses. 

The way K&E thought about "pocket rules" is a good example of this.  During the first 50 years, the company would focus first on a Model line, like the 4041 or 4053, and then decide if they wanted to make a pocket version of that rule.  In doing so, they would essentially cut the 10" wooden slide rule stock in half and build the shorter rule in exactly the same way as the longer rule.  Labor costs in doing so remained high, since production time isn't decreased just because there is only half of the materials.  And as we will see in our discussions of the rules, K&E did NOT make pocket rules any less expensive than their full-sized rules.

Around 1930, with the introduction of plastics beyond celluloid/wood lamination, we see that K&E could (and did) change production techniques to produce their pocket rules, whereas any cost-savings could be passed onto the consumer.   And because of this, product models that traditionally were NOT manufactured with a "pocket" version, could now be made at low production cost for a demanding consumer.   As such, K&E would birth the "Ever-There" rules, a series of pocket models intended to supply users of their traditional model lines a portable option of their favorite slide rules.    

Similarly, K&E produced several other "series" of rules once a technology became available to them, and this does not make it easy to sort individual rules into the traditional "family" of K&E slide rules in which they belong.    Rules in a series are often the completion of an existing model line, or at least their spiritual successor.  And this obfuscates the how we could view the entire ecosystem of K&E rules. 

Article Conventions

As such, attempts to organize K&E rules from today's collector perspective, or to write a comprehensive article about K&E slide rules, is, therefore, complicated.  But previous efforts are substantial and very much appreciated.  For example, Michael Frey of the International Slide Rule Museum (ISRM) did a wonderful job of mapping K&E slide rules over history in his K&E Model Map (see image above).  Here, we see used the three broad categories of Mannheim, Duplex, and Specialty.   It's logical and, for the most part, mirrors the way K&E would have organed the rules themselves.   It does take some interpretation however, which we will do shortly.  But for me, a good discussion about K&E slide rules would not be possible just by moving straight down the K&E Model Map, even though I will stick mostly to that organizational structure until I feel it no longer suits me. 

So my approach in this article will be to itemize rules according to construction, followed by purpose, in the traditional sense.  As such, major sections of the writing will be divided by "family" of rules.  But where a "series" of rules could belong to multiply families, I will discuss the "collection" itself to gain a better understanding of that collection as a series.  And then, if those rules are also the evolution of a traditional model line, I will discuss those also within that second context.  For example, while the Model 4097C of the Ever-There is important to that "series" of rules, it functionally works as the fulfillment of a pocket rule for the Merchant's "family" of slide rules, and as such, will be highlighted in those conversations as well. 

In my mind, this is the best way to focus on the evolution or continuity of a product line, and perhaps give some insight into what K&E was thinking in this regard.   Doing so in any other way would be too granular in terms of their slide rules, which weren't isolated from other rules, but rather part of an eco-system of K&E products.   As we will see, this approach to describing their slide rules works well, even if we run the risk to talking about it in multiple places in this article.  I would recommend getting in-depth looks at a specific rule from many of the wonderful resources on the Internet, including the ISRM, the Slide Rule Universe, or the Oughtred Society.   While you will get some of that here, I want to make some sense of what K&E was thinking with the production of their rules, and not just make it about the rules themselves.

Therefore, it does make sense to divide the discussion of K&E rules into their major families, similar to the K&E Model Map.   From there, I will place my focus on individual model lines within those families, how they evolve, and how K&E likely thought about their overall placement within their slide rule ecosystem.   And within that framework will come specifics about a particular slide rules.   Where scales are concerned, the following convention will be used, using the original single-sided Polyphase Mannheim Model 4053 as an example:  

​Front Side:  Inches // A [B CI C] D \\ K
Back Side:                   [S L T]

Reading across the front side, we have an inch ruler on the top edge of the rule, A on the top rail, B CI and C scales on the slide, D on the bottom rail, and a K scale on the bottom edge of the rule.  On the back side, the bracket indicates that the back of the slide has S, L, and T scales.   While single-sided in build, the "back side" would be considered scales on the usable part of the rule, which is the back of the slide only in this case.   Where applicable, the physical backside of the rule would typically have formulas, unit conversions, or even instructions for a rule's use.  That will be noted where it happens.  

So what follows in the article will be the three major categorizations, Single-Sided/Mannheim-Type, Double-Sided/Duplex-Type, and Specialty Rules. 

Mannheim implies a purposed scale-set; however for K&E rules, especially as listed in the K&E Model Map, it's broadly implied to mean "single-sided" construction as well.  This is why a series of rules like the "Ever-There" would be categorized as "Mannheim," on the model map, even though only one such rule had the Mannheim scale set.   All Ever-There rules were single-sided.  So, in some cases, it will be necessary to talk about a slide rule under multiple family designations.  

Similarly, Duplex is logically about the two-sided design with a cursor/indicator capable of reading both sides of the rule with one setting.  And certainly the duplex design is organized amongst purpose-built slide rules based on scale set, a spectrum moving from basic Duplex >> Polyphase Duplex >>Log Log Duplex >> Vector >> Modern Duplex.    However, as we get to the Modern rules, we begin to see while many rules are built with duplex construction, they have some traits akin to the single-sided rule, most notably the GP-12 and Analon slide rules.  Likewise, a "duplex" format rule like the "K-12 Prep," being only printed on one side and replacing all the Model 4058 "Beginners" rule in the evolution of that product line, should be talked about naturally in the context of the single-sided Mannheim type of rules, as well as in the section of Modern Duplex rules in which it should be fairly categorized.  

Thus, to eliminate some confusion in the K&E Model Map, I add single-sided and double-sided distinctions to the major category label, which becomes the way I think about these general mathematics types of slide rules.  

Specialty Rules can be thought of as "everything else" if you'd like, but that would not be fair to some important families of rules in that class.  The Merchant, Stadia, and Radio/Electrical families are every bit as important throughout our history here and a fundamental part of the eco-system from the beginning of the company.  But it's necessary to classify them in this way since rules in these families didn't always care about using the same construction.  For example, there are both singled-sided Mannheim-style (the Model 4100) and doubled-sized duplex (the Model 4102) "Stadia Family" slide rules.   

Where additional distinctions need to be made, I will redirect the discussion of a family or model line to provide clarity to the reader.   

And now, the K&E slide rules...

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Rules of the Single-Sided, Mannheim-Type

As I mentioned earlier, it's easy to equate the "Mannheim" rule with "single-sided," since historically speaking they typically go together.   But not all single-sided rules are Mannheim.  In Europe, the Rietz and Darmstadt rules dominated historically, and the Mannheim originated over there!  We just know that K&E decided to place its focus on the Mannheim (and it's improvements) when producing a general arithmetic rule for its customers.   

Likewise, the Mannheim scale set can have interpretations on double-sided rules.   As examples, the "Duplex Family" was largely based on a classic Mannheim scale set and the Polyphase Duplex was just a re-imagined Polyphase Mannheim slide rule.   

Likewise, some of the single-sided rules near the end of the slide rule era were duplex-designed rules that simply chose to use only ONE side of the rule.  The other side, like the traditional Mannheims, would contain a variety of conversion scales and useful information.   So whereas you might regard a slide rule like the 68-1400 Analon as a duplex rule (or even a specialty rule), the choice to utilize only one-side of that rule for computations could very easily require me to talk about it in the "Mannheim-Type" section of slide rules. 

The Mannheim Family

General-purpose rules obviously range in capabilities and were classified into several slide rule "families."   K&E's longest running family of rules, the basic Mannheim family, utilized the front size only with A, B, C, and D scales, with formulas filling the back.   S, L, and T were placed on the back of the slide and could be read off a small window indicator on the back of the rule.  Except for the pre-1900 models in this family, the Mannheim rules were mostly marketed towards the everyman, offering basic arithmetic and trig operations.  It appears in 1901 that a major shift in strategy occurred in the company, as K&E started building all of their slide rules in-house, revamping their entire product line-up and marketing campaigns, and pushing toward broader markets, not just the engineers, architects, and builders of the world, but also the casual consumer.  They managed to hit every price point with their catalog of new slide rules, ranging from $1 for their budget "Beginner's rules," to as much as $12.50 for their new 20" Mannheim model.   

In 2022 money, that's $35 for the basic rules and as much as $420 for their flagship Mannheim model.   Note that early on, K&E rolled out a series of Duplex and Specialty rules as well, which we will talk about in later sections. ​

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This is obviously one of K&E's first slide rules, but is it the Model 479? We know little about it. But we do know that their first slide rules were imported to the United States, starting with this Tavernier-Gravet manufactured slide rule. While the original 479 was cursor-less with an ivory slide, we know in 1883 they produced a 10" rule with brass cursor called the 479-2, with no mention of an ivory "slip" in the catalog. I'm betting this is that rule! - Courtesy International Slide Rule Museum.

Historically, the major slide rule models in the Mannheim family include:

Model 479 

Imported from the French company Tavernier-Gravet, the Model 479 (see above) was K&E's first slide rule produced from 1881 to 1886.  

The original, cursor-less 479, offered in 1881, was a 10" model made of boxwood and a real ivory slide.  Known as the "Engineer's Slide Rule," this 479 was later classed as a "Gunter" version of the Mannheim, which featured only the A, B, C, and D scales and no indicator cursor.   This now rare rule (I've seen no actual photographs attaching a old K&E rule with this model number) was introduced with a cost of $3.50, which I'm certain was an absolute fortune back then.  As a historical reference, it appears that K&E stopped using ivory in their slide rules all-together around 1884, as the future 1745 model was said to have celluloid-lamination.  The practice of using ivory in their sector rules also discontinued at that time, but picked up again from 1897 to 1917.

In 1883, the 479 Gunter model became known as the 479-1.  K&E also added two other 479-2 and 479-5 rules.  It's a curiosity, but the dash did not represent the length of the rule (a convention first appearing in 1911).  Instead, all K&E slide rules were a Model 479, with the dash indicating a different slide rule offered.   The 479-2 and 479-5 were 10" and 8" models respectively that included trig functions on the back of the slide and a brass bracket indicator...a true Mannheim rule and the predecessor of the future 4041 models.   
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Model 17XX Series

These slide rules replaced the 479 model in 1887, the new model number reflecting a change to Dennert & Pape as the supplier.  The first two rules in the series were the 1745 and 1746 (10" inch) rules.  The 1745 retained the basic Gunter design, made of boxwood, early celluloid faces with engine-divided scales, and no indicator cursor.  Similar in construction, the 1946, with brass cursor, was a true Mannheim rule, adding S, L, and T scales on the reverse of the slide.   A 20" version of this rule called the 1748 was introduced in 1890, and a 5" version called the 1747 arrived in 1899.  Of the 20" rule, K&E seems to be rather apologetic, or at least cautionary.  Their catalog description in 1890, and as printed below in the 1895 catalog, states...

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Two more rules in the series arrived in 1897, including a glass cursor version of the 1746 called the 1746-1/2 and the first of the "Beginner's Rules" known as the 1749-1, strangely a 10" model.   I say strangely because K&E used the 1749 model number for a wide variety of products by applying a dash.  For example, Colby's and Crane's Sewer rules (see Specialty Rules in a later section) were numbered 1749-2 and 1749-3 respectively, and their 12" ivory and boxwood sector rules were numbered as 1749-7 and 1749-8.  

Finally, and importantly, the 1748-3 was introduced to this series in 1899.   This rule carried the "Favorite" label on the box, and otherwise is not marked with the model number on the rule.  This is described as such in the 1899 catalog.    It is speculated by some that this same rule might have shipped in a box labelled "American," not appearing in the catalogs, or perhaps (maybe likely) existing as the same rule a year or two prior to this.  As such, this boxwood rule, every bit as similar to the "Favorite" Model 4054 series (described later in this section), is thought by some to be K&E's first manufactured rule; the "American" being apropos if that's the case.   It also makes sense that K&E would produce a small number of slide rules in-house in the years just prior to their complete retooling to produce all slide rules in-house in 1901.   This "American" rule is widely known, but rarely found, and the only indication of its existence is because we have samples in a case worded with such.   

A 1744 duplex 10" rule (at a retail cost of $6.50) was also produced in this series (see The Duplex Family in a later section).  These rules were short lived, replaced across the board in 1901 with a complete line-up of new company-made slide rules.   Rules of this series are rare for today's collector, but they do pop up on occasion.  (Note: I do not own one as of this writing).

EDIT:  A note regarding wood choice.  Around 1895, we do know that Dennert & Pape was producing celluloid-faced mahogany rules.   Many of the samples post-1895 of the K&E 17XX rules are indeed constructed with mahogany.  Exact date of the transition is unknown. 

Model 4028

When K&E expanded their lineup in 1901 with in-house manufactured slide rules, the 4028 model succeeded the 479, 479-1, and 1745 "Gunter" rules of previous years.  Remaining cursor-less, this model was discontinued 5 years later. 

Model 4041 Series

Replacing the 1746 and 1748 rules, the Model 4041 led a renewed and expanded lineup of products beginning in 1901.  These rules, and thus the model change, were manufactured in-house by K&E, as all their rules would be from that point forward.  The most expensive Mannheim rule ($4.50 for the basic model), the 4041 was a lovely 10" mahogany rule covered in celluloid with engine-divided scales and glass cursor (evolving over time), also produced between 1906 to 1935 with a finely-divided scale option known as the N4041F (a $3.50 upgrade).  All 4041 rules included celluloid, engine-divided inch and centimeter rulers on the sides (and a centimeter extension rule in the slide well up until 1912).  Prior to 1911, when K&E would move to the "dash length indicator" naming convention, K&E used different model numbers for the same rule in multiple lengths; therefore, the same 4041 rule became the 4031 in a 5" length, the 4035 in an 8" length, the 4045 in a 16" length, and the 4051 in a 20" variety.    The 4041 (and 4035) added the "N" designation in 1925 with a slightly wider frame and endured in this form until 1943, largely replaced in consumer minds with the 4053 "Polyphase" (see next section) which provided more scales at the same price.  ​

Models 4054, 4055, and 4056

Collectively marketed as "Favorites," these basic 10" Mannheim rules are lower cost varieties of the 4041, also part of the huge product roll-out in 1901.   The 4054 ($3.00 retail price) and 4056 ($2.75 retail price) ran during the length of the original 4041 production run (which changed to an "N" designation in 1925).  Dropping all Favorite models in 1925, they reinstated the line in 1927 in the form of a lone 4055 model.   The 4054 was identical to the 4041, complete with celluloid scales, except for its less expensive choice of boxwood and painted-on inch and centimeter rules on the sides.  The 4056 saved consumers even more money by being a bare boxwood, completely painted-scale model.  Both models lacked adjustability.  

Once the 4054 and 4046 were dropped in 1925, the new 4055 design featured celluloid-laminated mahogany with engine-divided scales, but it lacked inch and centimeter rules on the sides.  It was considered a middle-ground option between the top 4041 Mannheim rule and the budget student/beginner rules (see next model) that K&E always offered.  A 10" scale length was the only option for these rules. 

Both of the 4054 and 4056 models would be reintroduced at points in the future.   In 1936, the year that the "improved-glass" cursor was introduced across many K&E slide rule lines, the 4056 was reintroduced as a budget version of the 4055, the $3.50 latter rule receiving the new cursor and the former getting a standard metal-frame glass cursor, saving consumers a dollar.  Otherwise, the rules are identical, including mahogany and celluloid-facing for this new 4056 - a vast improvement over the original, spartan 4056 model.   

In 1944, when K&E finally ended the long reign of the 4041 Mannheim, they also discontinued the 4055, leaving the 4056 as the sole "Favorite" model.   The 4054 was also reinstated as a nice budget option for the N4053 Polyphase Mannheim, which we will talk about in the next section. 

The new version of the 4056 would endure through 1952, when it was discontinued.  I would speculate consumer tastes were shifting more toward the wide-variety of pocket rules K&E offered at the time, including their variety of Ever-There models (see next section), as well as the new line of modern, all-plastic Polyphase rules being pushed out around 1950, known as the Doric models (please see Modern Polyphase Rules in a later section). 

​Model 4058

A "Beginners" or "Student" model for any budget, made of a less expensive boxwood, painted on scales, and a metal-bracketed, glass cursor, the new Model 4058 was also first produced in 1901 and cost a rock-bottom price of $1 (still quite pricey back then).  These rules did not include the inch/centimeter scales on the sides.  Despite being the least-expensive K&E slide rule, it did get a cursor upgrade in 1926 (designated 4058C) and a white-painted face in 1930 (designated 4058W). 

Some form of these wooden rules ran from 1900 to 1961, but they are always consistently underwhelming from a collector's perspective.  While they served the purpose of a budget rule, these were not intended to last long.   Non-adjustable, they would no longer function well in humid conditions and were prone to warping over time.   Boxwood lacks the stability of the more expense mahogany.   Of course, you get what you pay for in that regard. 

Model 4158 "K-12 Prep"
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In 1961, the Model 4058 gave way to this all-plastic (vinyl laminations with heat-pressed markings) slide rule.  The Model 4158 "K-12 Prep", so named to emphasize its intended use in the classroom, was constructed in a duplex form yet printed on only one side - scales on the front and total blank on the back.  It used a duplex-style cursor without a rear glass window, bracketed at the ends with plastic brackets and coming in a plastic slip-case.    It did include more scales than the normal, strictly Mannheim beginner's rules:

K A [ B S CI T C] D L

The 4158 designation was short-lived; however, since all K&E products shifted to the 68-17XX a year later.  From that point until the end of the slide rule era, the 4158 was known as the 68-1892 model.    A 10" scale length was the only option for these rules. 

The rule served the purpose as a beginner's, low cost rule, at $2.25 in 1962.  This was by FAR the least expensive rule in the 1962 line-up, priced $3 less than the next cheapest rule. 

​I find that fact somewhat remarkable.  

Model 4098A

Originally part of the "Ever-There" series of pocket rules, the "A" version of the 4098 became a stand-alone pocket rule in 1936.  Made entirely of Xylonite plastic and using a cheaper frameless Xylonite cursor, this rule was simplified to the same Mannheim scale set used in the 4041 (as well as the 4095 "Favorite" and 4058 "Beginner's" rules).  The revised 4098A was narrower with straight sides, dropping the angular sides of the previous version (and the rest of the Ever-There series).  After the 4041 was discontinued in 1943 - this slide rule was the only true Mannheim rule remaining in the entire K&E lineup until 1953.  At that point customers failed to see the utility of a basic Mannheim scale set rule when slightly better Polyphase versions became available in either the 4161-1 or 4153-1 pocket rules.  ​

Collector's Outlook: Single-sided K&E rules

​Note:  Suggested prices are for slide with case on eBay.  Any extras, especially those considered complete, with box and documentation, can double these prices.
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​Single-sided K&E rules are easily collectible, if by "collectible" we mean easily and cheaply found.   The basic student rules of the 4058 series, followed by the 4053 series, are abundant on used markets between $10 and $20 respectively for the full-scale 10" rule.  The 8" and 20" versions carry a higher price, but can typically be found in the $30 to $60 range depending on condition and completeness.   

The 4041 series is also quite common, likely $15 to $20 for a rule in good condition.   The variations of this rule are where they become much more desirable.   The 5" 4031 and 8" 4035 are difficult to find. The 4031 model comes up on eBay once a month, selling for around $40 on average.  The 4035, maybe once per year for a similar price.    The original 4040 version in this series, that with the brass cursor, only had a run of 5 production years, and thus it's quite rare.  Only two samples have been sold on eBay over the last 23 years, averaging $273.    A sample of the 16" 4045 and 20" 4051 will come up for sale every other month or so, with an average price of ~$50.   Of course there's often another one posted for maybe 3 times that amount. 

Highly desirable is the "F" or "fine-scale" version of the 4041 and 4053 slide rules.  They come up maybe once or twice a year with an average selling price of perhaps $100. 

Others that are moderately collectible are the 4054/4055/4056 "Favorite" series of rules, any of the Ever-There Pocket rules, as well as most of the more modern pocket rules made of the better plastics.  All of these come up very frequently on eBay and can be had for around $15 to $20 or so.  

Anything from the 19th century is basically a unicorn.  Only a couple of 1746 samples have come up on eBay over the last 23 years, selling for around $1000 each.  The early 1749-1 Beginner's rule has seen one sample sold on eBay for nearly $500.  The so called "American" slide rule, in the box, exists in very few collections.   This can cost perhaps $300 to $500.  

I never seen a Model 479, nor it seems has anybody else, though I would speculate that these rules were those imported from Tavernier-Gravet, lacking the model number which could have been known as the Model 479.  

Part of the collection: 8 inch, 10 inch, and 20 inch versions of the N4053 Polyphase Mannheim.

The Polyphase Mannheim Family
  
Introduced in 1909 and pushed out in a variety of scale lengths a year later, the Model 4053 was essentially the big brother of the 4041 model.  Almost identical in most ways, it included two more important scales over the regular Mannheim rule.   Added were a "K" scale to compute cubes and cubed-roots, and a CI (inverted C) scale to permit easier chaining of division and multiplication operations.  This Mannheim version of the "Polyphase" scale set was a response by K&E to offer more scales on the same single sided, wooden platform.   This came from improvements in K&E's dividing engines, which as of the late-ought's, permitted K&E to etch scales in the middle of a rail or slide.  

This family of Polyphase Mannheim rules was perhaps the least changed over K&E's production history, often varying from year to year with minor evolutionary changes, but never shifting model names beyond the addition of an "N" designation in 1925 to the 4053 model name.  The Polyphase scale set itself would find itself on most of the more basic of K&E rules, eventually superseding the basic Mannheim scale set.  It would also be translated for use on various duplex rules throughout K&E slide rule history.

Members of this family include the following:  

The Model 4053

The original 1909 model of the 10 inch 4053 was identical in construction to its contemporary, the Model 4041, based on celluloid-covered mahogany wood and engine-divided scales.   However this rule replaced the centimeter rule on the side with three-decades worth of "K" scale and squeezed in the "CI" scale in between the B and C on the slide.   The cursor had an indicator on the bottom allowing easy reading of the K scale at the edge, a feature that K&E mentioned made the 4053 a "hybrid" of the Mannheim and Duplex style of rules.   

Two years after its introduction, the 1911 models came in all lengths using the "dash" system of model labelling, from an 8 inch rule with a 4053-2 designation, the normal 10" rule relabeled as the 4053-3, and a 20" variant known as the 4053-5.   These models would take on an "N" prefix in 1922 when K&E removed the K-scale from the front edge of the rule and squeezed it (without regard to aesthetics) into the bottom stator rail and gave users back their centimeter scale on the front edge.  Interestingly, during this transition year in 1922, the printing of the "N" on the rule itself varied from outside of the bracket (as in N<4053-3>) to being above the bracketed model number.  By the next model year, the prefix joined the model number inside the brackets.

A finely-divided scale option of the 4053 was also produced (as they did with the Model 4041 Mannheim), which would have been designated as the 4053-3F.   This model was "divided as finely as the 20" rule."  As such it provided the precision of a larger rule.   K&E produced this r ule between 1911 and 1943, but is rather rare.  Perhaps this is because the upgrade to finely-divided scales came with a $3.50 premium!

There would be various other subtle changes to this rule over the years, especially where the cursor was concerned.  And K&E would also produce variants of the rule, including a D4053-3 version in 1954 with decimal-divided trig scales and directly-printed (not an affixed label) conversion tables on the back.   These rules, and all 4053 versions to follow would employ semi-plastic construction beginning in 1954 - the back of the rule in 1954 was now laminated with celluloid as well, making printing directly on the rule possible.  The name of this model would drop the "N" prefix to reflect the major feature change.   More plastic would be added over time, evolving to an all-plastic base with typical celluloid-covered mahogany rails and slide in 1962 and following.  

There would also be discontinued rules in the series, including the 8" N4053-2, discontinued in 1938, and the aforementioned N4053-3F version in 1943.  With the 1962 model year, when K&E changed the model numbering convention for all of their slide rules, the 10" 4053 would become the 68-1617 and the 20" N4053-5 was issued model number 68-1607.  Both rules could be upgraded to a chamois-lined leather case, known as the 68-1622 and 68-1612 respectively, but it did not change the model number on the rule, only on the shipping box and in the catalog.  

Cost for the original 4053 was $5.00 in 1909, or $0.50 more than the 4041 Mannheim.   The K&E product line included both the 4053 and the 4041 for a good 35 years until the 4041 was finally dropped in 1943.   I find this rather surprising since the 4053 largely replaces the capabilities of the 4041 for only a little more money.  By 1921, K&E seemed to realize this when they raised the price of both models - $7.60 for the 4053-3, which was a full $1.10 more than the 4041.  

The Model 4053, first introduced in 1909, would continue in production in some variety until the end of the slide rule era in 1975.   It was easily K&E's longest, continuously running slide rule model.  

Model 4098 and 4097C "Ever-There"

These models of the Ever-There series should be discussed here because they were Polyphase Mannheim pocket rules (but see the next section for a more complete discussion of the series).   For the long duration of the Model 4053, K&E had never produced a 5" pocket version.   Never.  As remarkable as that seems, once new modern plastic rules could be built, it made sense to make a pocket rule based on the 4053 Polyphase Mannheim scale set.  The Model 4098 was introduced in 1931, followed by the 4097C in 1936.  We will talk more about these rules in the next section, since it makes more sense to discuss them in the context of the entire "series."  

Model 4153

K&E worked hard to find new plastic rules to replace the Ever-There series.  They were successful    In 1957, two new rules arrived, designated as the 4153-1 and 4153-C models.  They were exactly the same rule, but K&E began to offer a few pocket rules with the option of a leather-covered metal clip on the included leather slip-case.  Those model variants would have a "C" suffix to differentiate them from the clip-less models.   In 1958, cost for the 4153-1 would be $5.50, with $6.25 for the model with the clip.  

Construction of these rules were very similar to many of the pocket rules K&E was producing at the time.  These rules, with "Ivorite" (ABS plastic) and small clip-on "unbreakable" cursor, were a good platform for many of the scale sets being used, including the classic Mannheim, Polyphase Mannheim, a "Merchants" pocket rule (see Specialty Rules later in the article) and their new "Modern Mannheim" series of rules (see Modern Polyphase Rules).  In this case, the 4153 would have the classic Polyphase Mannheim scale set, and like the Ever-There, while it's nothing close to being like a Model 4053 in form - unlike the pocket rules in the past where it seemed that K&E just cut the longer rules in half - it most certainly was in function. 

Model 4054

Earlier I mentioned that the original Model 4054 "Favorite" Mannheim model was discontinued in 1925.  When it was reinstated in 1944, its role shifted to a more advanced model, a sister rule of the N4053-3.  As such, this new Model 4054 variant would be marketed as a "Polyphase Slide Rule" and no longer associated with the "Favorite" series of slide rules.  With improved glass cursor, the new 4054 rejected its basic Mannheim heritage to sport the same 4053 scale set, missing only the inch and centimeter scales on the sides.  Equipped with mahogany instead of the original boxwood, this $5.50 model saved users $2.00 over the N4053-3 flagship Mannheim-type (single-sided) rule in 1944.   Even so, that's almost double the price of the original 4054 model that was first introduced in 1901.  That's inflation for you!

The Polyphase version of the Model 4054 would last until 1953.  It would seem that company and consumer interest was shifting toward the all-plastic rules at this price point, particularly the pocket rules, as K&E would discontinue no fewer than SIX of their traditional Mannheim and Polyphase based models between 1950 and 1954.  
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The Ever-There Series

Named the "Ever-There" series of slide rules, K&E introduced this product line of pocket rules in 1931.  K&E's first all-plastic rule, made of a celluloid known as "Xylonite," and with a frameless Xylonite cursor, the two models in this series were meant to give users of K&E single-sided rules a light-weight pocket option for the full-sized rule they already had, namely, for those with either "Polyphase" and "Merchant" slide rules (more on the Merchant Family of rules can be found below among Specialty Rules).   

Other than the 4031 Mannheim pocket rule and their pocket duplex models, the intent of the Ever-There series was obvious, to provide a cost-conscious solution to all future pocket-sized rules, regardless of the scale set. 

Models 4097 & 4098:  Initially, the series began with only two models, the Model 4097 and the Model 4098.   These cost the customer $3 and $4 respectively in 1931.   When compared to the 5" wooden pocket Mannheim 4031 at $5, the Ever-There's make financial sense for both K&E and the consumer.   

Almost identical slide rules in construction, only the scales were different between the 4097 and 4098 models.  The Model 4097 mimicked the Model 4094 "Merchant" rule, a 10" specialty single-sided rule introduced the year before.  The Merchant's models will be discussed later (see Specialty Rules below), but the 4097 did away with the A & B scales (and the S, L, and T scales on the back of the slide) in favor of folded C & D scales (CF & DF) and an inverted C scale (CI) on the slide.  The idea of this was to allow rapid chaining of numbers more efficiently, with fewer moves of the slide and the ability to avoid off-scale computations.   The 4097, as with the 4098, did make use of the rule's back side to give 5" and 13mm rulers.  

The Model 4098 stepped in as a true Polyphase Mannheim pocket rule, filling the gap of a 5" pocket rule that the 4053 series failed to offer - at that point there were 21 years of 4053 production and the smallest rule offered was the 8" Model 4053-2.   In fact, the 4098 not only used the same scale set as the 4053, it put the scales in the same place, except the Ever-There, as mentioned earlier, put the inch and centimeter scales on the back.   

Model 4097B, 4097C, and 4097D

The Ever-There series proved to be much thinner, lighter, and budget-friendly than all slide-rule technologies that preceded them.   And this was shown in sales.  Therefore, five years after their introduction, K&E revamped the Ever-There line-up in 1936 to offer more choices to consumers and to increase their profit margins by selling these easy-to-manufacture slide rules. 

Dropping the 4098 from the lineup, it would become redesigned as the 4098A (as mentioned earlier) to give the original Mannheim users a cheaper option - it worked, as the $1.75 would be competitive enough to make K&E discontinue the 5" 4031 four years later.   They would keep the 4097 as the only slide rule in the Ever-There family, but produced three variants of the 4097 to meet consumer needs in three separate markets.  Interestingly, the models would be known as the 4097B, 4097C, and 4097D. 

K&E would never offer a "4097A" rule, presumably because the "A" designation went with the 4098, honoring the 4098's legacy in the previous Ever-There lineup.  This new 4098A version would be known simply as the "K&E Pocket Slide Rule."  

The 4097B (with a 1936 cost of $3) essentially retained the functionality of the original 4097 slide rule, using the "Merchant's" scale set of their 4094 slide rule. 

The 4097C (with a 1936 cost of $3.75) was redesigned to become what the old 4098 was, the true Polyphase Mannheim rule it had been for 5 years.

The 4097D (with a 1936 cost of $4.25) would become, so it seems, a "hybrid" of the other two 4097 rules, merging both the folded scales and the Polyphase scales onto a one-sided slide rule.  K&E accomplished this by altering the Polyphase scale set, moving the B-scale to the slide (joining the S, L, and T scales), and replacing it with a CF-scale, and then squeezing in a DF-scale just below the A-scale on the upper stator rail.   

To me, the intent is obvious...to provide a cheap pocket version of the 4088 Polyphase Duplex rule (see Polyphase Duplex Rules below).  Comparing the two, the only scale that the 4097D lacks is a CIF scale (folded, inverse C-scale) as found on the 4088 line of rules.   And while K&E did manufacturer a pocket version of this Polyphase duplex rule with their 4088-1 model, it was exactly twice as expensive as the new 4097D Ever-There in 1936 prices ($8.50 versus $4.25 respectively).   As a result, the 4088-1 would be discontinued within three years of the introduction of the 4097D.  

Note:  There is much more to be said about this scale set historically, which I will do in another area of this article (please see Sidebar: The 4088, 4097D, and the Mystery Rules).
 
As far as construction, the new Ever-There series did evolve from their preceding lineup, going to K&E's new-improved glass cursor, which they began using in all their slide rules beginning in 1936.   They also tweaked the shape of the rules slightly, enough to appear dimensionally different from original 4097 and 4098.  It is not known if K&E repackaged any of their old 4097 and 4098 stock into new 4097B and 4097C boxes, but it's definitely something I would have done.   However, this will have to remain an inquiry in need of future research on my part.

Finally, it is important to talk about the Ever-There series in terms of the competition that stood against K&E during this time.  German-based Dennert and Pape introduced all-plastic slide rules in 1936 marketed under the name "Aristo."   But at that time, U.S. and European markets were completely separate.  

The chief competition for K&E state-side came from the Frederick Post Company, who began to compete for market-share with their sale of 4" and 5" Bamboo pocket rules, imported from the Hemmi company in Japan.  Post would attack strongly in that market, producing no fewer than SEVEN pocket models by 1937.  And, not surprisingly, Post would market these rules as both "Mannheim" and "Multiphase" rules.   If that's not a direct attack on K&E's, then I don't know what is!

As a collector, I much prefer using the Post/Hemmi bamboo pocket rules, something I freely admit.  Though this speaks more to the longer-lasting nature of a bamboo constructed rule as compared to an early plastic rule some 80 years later.   (I just love my bamboo Hemmi-built rules!)

Sidebar:  The 4088 , 4097D, and the Mystery Rules...

During a 20 year period, from 1930 to 1950, K&E produced a preponderance of rules that didn't have a model number.   I will speculate later on what I feel could be the origin of some of these rules, which are known by collectors as K&E Mystery Rules, but there are a few statements we could make about the Mystery Rules in general. 

First, these rules aren't necessarily rare, so you would be mistaken to think they were not produced in generous numbers.   We can see this today by how often these rules come up for sale, which in my opinion seems to be "not often" rather than "rarely." 

Second, K&E did not show these Mystery rules in their catalogs, as they did not have a model number.  We can judge from this that these rules were likely developed for a specialty or custom market, where K&E did not want the general public to think these rules were offerings they could purchase.  

Third, and more relevant here, is that the Mystery Rule scale set is almost identical to the Model 4088 Polyphase Duplex, which predated the Mystery variants by two decades or more, and the 4097D Ever-There pocket rule which would be introduced a few years after the first known instance of a Mystery rule with that same scale set. 

We do know that K&E produced many custom slide rules for a variety of purposes and industries, a practice that Pickett would pick up on a half-century later with their own specialty types of rules.  Many such K&E rules are so rare that their existence is likely lost to antiquity.   But we should not be surprised that K&E might make slide rules for a variety of universities, among other places, made to order.   in fact, a very curious rule came up not long ago on eBay, one in which definitely had my favorite Facebook slide rule group more than curious (see below). 

This is speculation, but I believe this hints to the purpose of these Mystery rules.
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My thinking is that the 4088 Polyphase Duplex rule was something many students would want.   But it was a $7.00 slide rule when it was first introduced in 1913, which is around $210 in today's money.   That's a lot to ask of a college student!   So it makes sense to provide such clientele with a rule that offers a similar scale set on a more cost-friendly platform.  Many of the "mystery type" rules were made on a Mannheim 10" construction, like the 4041, which existed at this time at a $4.50 price point.   I can see K&E producing these rules "out of catalog," stripped of model numbers, and offered only for sale in select university book stores.  While $4.50 retail is still expensive, I could see K&E offering such rules closer to cost, since they would fall into the hands of students who would likely put money back into K&E's coffers at some upon graduation.   As such, offering students a $2 slide rule (around $50 in today's money) with a desirable scale set of the 4088 model shouldn't be outside of reason.  

We have every indication that K&E understood the importance of quality and brand loyalty in all of their products.   And we shouldn't be surprised that perhaps the company didn't mind being charitable when it suited them, giving back to an education industry who would  produce more engineers and builders of the future.  

As I said, this is speculation, so I have a couple of reasons to question this theory...

First, such Mystery rules were produced sporadically, but in sufficient numbers between the years ~1930 and ~1950.  My first inclination is to question why it took 17 years or more for the 4088 Duplex to exist before K&E provided an alternative rule using the same scale set?   And if I'm a university professor wanting a cheaper, yet powerful slide rule for my students, I'm more inclined to inquire about the Model 4092 Log Log Duplex or 4093 Log Log Vector rules, which were K&E's flagship models for many years.  At least this is what I would have wanted as an engineering student.   

But perhaps it's logical to change our idea of who is pursuing whom?   Instead, I think this might speak to K&E's desire to reach out to the education markets as opposed to universities coming to them.   At that point, knowing that the 4092 and 4093 have too many scales for a conversion to a one-sided platform, then I can see K&E pushing the 4088 instead.   

I think more puzzling is that K&E began to print Educational Catalogs and price list in the 30s, seemingly during the early production these Mystery rules.   That speaks to K&E strongly focusing toward the educational markets, as I suggested.  But it does seems strange that the Mystery rules do not appear in those catalogs, neither in 1933 nor 1936.   Of note, however, is that the Model 4088 is indeed listed while the 4092 and 4093 is not!   But again, I can reason a way around the exclusion of the Mystery rules if K&E was also providing at-cost rules to engineering students in select engineering programs around the country.  At this point, you would not want general education markets (and the general consumer market) to think such slide rules were available to them as well.

This is where the Ever-There 4097D comes into play.  When K&E decided to merge their B and C variants of the 4097 into their "D" version of the rule in 1936, we had already seen this combination of scales in not only the 4088-3 Polyphase duplex rule that had been in regular production since before the 1st World War, but also in those Mystery rules produced for universities.   

Thus, with the 4097D, it's easy to see that K&E was simply providing a budget-friendly, single-sided version of the 4088 for the regular consumer.  At half the price of the duplex rule in 1936, this likely would have had a market demand.   You might think that this rule would have stopped the manufacture of the "Mystery rule" based on the same scale set, but serial numbers indicate that those were likely made up until ~1950.   This would seem to indicate that K&E was indeed comfortable providing even lower cost wooden rules to select clientele, as a $4.25 Ever-There was still a hard ask for a college-going student.  ​

But at the time, the Ever-There's were innovative rules, gaining a solid foothold in the pocket rule market in 1931, and redesigning them in 1936 to match the diversity of the 1936 Post offerings.  The 4098A matched the lowest price of the Post offerings at $1.75.   And the 4097s sold well enough that K&E was not compelled to match price with the imported Post/Hemmi pocket rules, none of which cost buyers more than $2.70 according to the Post 1937 Catalog.   K&E simply knew that their $4.25 4097D was going to sell a lot slide rules, especially since it was half the price of their 4088-1 duplex rule that pretty much did the same things.

However, if there were ever a K&E slide rule that could be considered of questionable quality, it would have to be the Ever-Theres (see SIDEBAR: The Problems with K&E Rules).   The Xylonite plastic was not long-lasting.  It became brittle over time, prone to yellowing and warping.  K&E knew the plastic was not ideal and would seek alternatives to replace it after 15 years.  This is a historically important distinction, as its the impetus for some of the corporate decisions made after World War II in the late 1940s.  

Therefore, the Ever-There series, beginning in 1931 and revamped in 1936, would hang on until the early 1950s, when at that time newer plastic versions of these rules were finally introduced.  Those models would be classified simply as "Pocket Slide Rules."   But the Ever-There's had served their purpose of supplying all K&E family model lines an affordable, pocket version of their popular slide rules.  ​

SIDEBAR: The Doric Family of Rules

Around 1948, when there hadn't been a K&E Catalog since war-time 1944, several all-plastic rules began to be offered by the company with a 9XXX model number designation.  These were different than the Ever-There series, with a different type of plastic altogether.   They were labeled "DORIC" on the rule, on their boxes, and in their documentation.   

It's difficult to find good research on what the Doric rules are and why they were introduced as a series.  The 1949 catalog classifies the "Doric Family" with only three rules, while we know that many others existed around the period of 1950.  Moreover, this 9XXX series rules would be converted to 4XXX rules by 1952, only some of which would persist in carrying the Doric moniker on the rule itself, and none of which are identified as a Doric Family rule in 1952 catalog.   It's confusing, to say the least. 

But in total, there are 7 known rules planned by K&E to carry the Doric designation in production and 2 others that were described and barely exist in reality, likely as prototype rules.  A summary list of these rules is listed here, with deeper descriptions of many of them found with the main body of the article. 

• 9068 - 5" Polyphase Duplex (1948):  Described in the 1949 catalog; became the 4168 in 1951 and the 68-1555 in 1962; labelled as Doric on all versions of the rule up until 1975.
• N9081-3 - 10" Log Log Duplex (1948): Described in the 1949 catalog; became the Model 4181-3 in 1952, while dropping the Doric label
• N9081-1 - 5" Log Log Duplex (1949): Described in the 1949 catalog; never known to exist as an actual rule; perhaps the forerunner of the 4181-1 "Jet-Log Jr." 
• 9061-1 - 5" Polyphase Mannheim (1950): Became the 4161-1 in 1951, dropping the Doric label.
• 9050-1 - 5" Merchant's Rule (1950):  Very few known samples.  Became the Model 4150-1 in 1951, carrying the Doric label for maybe one year.
• 9071-3 - 10" Polyphase Duplex (1950): A plastic version of the 4070 with a Doric label, with its only catalog appearance in a part's list; disappeared entirely by 1952. 
• 4168 - 5" Celanese Celcon (1968): A custom rule made of a special green resin, produced only that year, based on the 9068/4168 platform; used Doric name on the rule. 
• 10000 - Prototype 5" 9068 rule (1946): Thought to exist prior to the 1948 build of the 9068; appeared only in a catalog part's list.  A 10000/10002 manual exists, but a rare number of known samples of the actual rule.  
• 10002 - Prototype 10" 9068 rule (1947?):  See Model 10000 description.  Very rare, with two samples known to exist. 

Pictured top to bottom, the N9081-3, 4168 Celanese Celcon (originally the 9068), the 4168 Polyphase Duplex (also the 9068), and the 4150-1 Merchant's Pocket Rule (formerly the 9050-1). I expect to find the 9071-3 and 9061-1 at some point to complete the collection of known, actual Doric rules.

"Doric," in classical architecture/design meaning "simple of form" or "non-ornate," clearly indicated that K&E's new line of all-plastic Doric rules were intended to be nothing fancy.  In fact, every major family of rules by the late 40s had a Doric representative slide rule and all of those had the same, basic black font, different from K&E's standard font.  Interestingly, the term "Doric" would also be used with other products offered by K&E, most notably their drafting tools.  However, there is no mention of Doric "anything" in their 1944 catalog, so it seems to be a name that K&E was determined to associate with a variety of their products by the next catalog 5 years later.  

As such, It is difficult to know if the Doric rules were introduced as a cost savings option, or as a "budget rule," mostly because the quality of the rules were quite high.  Many assume so, though I am not so sure. 

The common element of the Doric rules is their construction of what has been called by many sources online as a "white Xylonite" plastic, which would be the same description as traditionally given to the Ever-There series by K&E in their catalogs since the beginning of their production and continuing through this time.  But these rules are clearly not the same plastic as the brittle, yellowing, and somewhat sticky Ever-Theres.  Possessing many of these rules, my hands would declare them a different formulation - the Dorics are of much better quality plastic; far more durable,  resembling more the plastic construction in their later "Ivorite" (ABS plastic) rules albeit notably heavier.
​ 
Whatever we call the Doric plastic, these rules seem to be built of a very different and superior plastic, and in the only K&E catalog that describes the Doric rules, they are called "plastic" as opposed the "white Xylonite."  Certainly this is not a mistake, as K&E want to draw a distinction between these rules and the Ever-There rules.  

And I believe this might give a hint to the purpose of the Doric series in the historical product line of K&E rules - to be a transitional, short-term model between the older "Xylonite" plastics traditionally used, and the newer ABS-based "Ivorite" plastics that would dominate the later years of the K&E product models.    It would seem that K&E knew what they wanted to do in terms of product placement and design with the more modern rules, and they appeared desirous of replacing the Ever-There model line. 

I do see some inconsistencies in the rules carrying Doric labels, however.  Not all of them are the same.   While the 9083-1 feels like the newer, modern rules made of "Ivorite," particularly close in feel to its 4181-3 successor (I note that the 4181-3 is a little more light in weight), the 9068 feels very similar to all successor rules, so much so that I couldn't tell them apart if I couldn't see the model numbers.  As such, I feel like K&E was indecisive of a plastic composition to go with during the short appearance of the Doric slide rules.  

I would speculate that when the Dorics were first produced around 1947, they were still a couple of years short of the "right formula."  While it was a wild improvement over the Ever-Theres, it would seem that 1950 was when K&E felt like they got it right and thus began changing the 9XXX models into 4XXX models, all of which were all constructed out of the same Ivorite plastic from that point forward. 

In total, this was a transition of 3 to 5 years, where it would seem 1947 marked the earliest of the Doric rules, and 1952, the latest.    And I do NOT believe it's a coincidence that the older Ever-There rules were discontinued completely sometime around 1953 and 1954.  Out with the old brittle plastic, and in with the new.  

So why not keep the "Doric" label?  I think that's quite simple.  If the Dorics were intended to become the pocket versions of so many of their other families and models of slide rules, then they should be modeled after those rules aesthetically.  Meaning the "Doric" name would no longer be appropriate if future rules needs red scales again, or if they would intend the rules to look like others in those families of rules.   

To add to the conjecture, the "prototype" models of the Doric 10000 & 10002 were less "plain Doric" than the others, including the use of red inverse scales and even ornamented with a red star on both the slide and the cursor.  Perhaps the formula of Xylonite being used didn't work very well with the red ink?   We've seen many rules from a variety of makers have inks that leaked or bled into the rule.  In such a case, "Doric" would be a good way to hide that these white Xylonite rules could only be manufactured with black ink.  

As for the "Doric" label, only the 9068/4168 rule would keep that identity permanently, even up to the end of the slide rule era in 1975.  Perhaps this is to pay homage to that legacy?   It was, after all, the original Doric, with 4 historic slide rules based on the 9068 platform.  

As such, many collectors think of the Doric as synonymous with the lone 9068/4168 style pocket duplex rule, the only one left standing in the 70s, despite the fact that 9 such rules were originally "Doric" in heritage. 

However we think of the Doric rules today, it was these slide rules that either directly spawned or inspired all "Modern Polyphase" and "Modern Duplex" rules that would follow in the 1960s and 1970s.   And as such, when we discuss the newer "modern" rules, we fall back on the Doric Family of rules as their legacy.    ​

From the 1952 catalog. K&E would begin to rethink their "family" naming scheme for their slide rules. This is likely because if a consumer wanted a pocket rule, it would be easier to show all of their offerings on a single page. In 1972, the same catalog page would grow up to 6 distinct pocket rules, regardless of their construction.

The Modern Polyphase Family

The history of the Mannheim platform evolved and endured over almost the company's entire history, from their basic out-sourced rule in the beginning, to high-quality rules built in-house, to the advanced functionality of the Polyphase rules, and through the modern era of the company.  Regardless of the rule, by the 1950s, the Polyphase Mannheim had become the desired scale set for non-duplex rules.  

With the introduction of better plastics, most all slide rule manufacturers dabbled with all-plastic construction at some point.  For a company like Pickett, plastic rules were mated to their entry-level, entry-priced types of slide rules.  While a company like Sun Hemmi would eventually use all-plastic construction for even some of their most sophisticated rules.  

K&E would be more like Hemmi in this regard.  When the technology became available to make high quality rules out of plastic, the company had no problem applying it to their high end rules.   We see this especially in the Modern Duplex Family of rules to be discussed later, whereas K&E's most revered slide rules of all-time are all-plastic in construction - the Jet-Log, Deci-Lon, and Analon rules if you were curious. While K&E did produce some budget rules from plastic, they were always well-constructed, with some heft and substance.  Thus, plastic doesn't have to be the equivalent of "cheap."  

But even the venerable, single-sided Mannheim/Polyphase types of rules could benefit from an all-plastic refit.  So K&E evolved these into all-plastic and semi-plastic models over time.  This is easily noted in the early 50s, when K&E discontinued six of their Mannheim and Polyphase Mannheim models within a 5 year span - those with "Favorite" and "Ever-There" legacies - and began focusing on a variety of new "Ivorite" plastic rules.   In fact, through the 1950s, the only wooden rules of the non-duplex types were the N4058W "Beginner's" rule and the flagship Mannheim-type rule, the Model 4053-3, and even this latter rule became more increasingly plastic over the 50s.  

We could very well be describing these rules as part of an "All-Plastic Family" of K&E slide rules; but I resist doing so for two reasons.   First, there would remain a separation in functionality between rules of the single-sided (Mannheim) type of construction and those of the duplex style.  As such, we need some differentiation when discussing rules of both types.   And second, K&E would still maintain a price structure with their rules already established historically in their product lines.  While K&E could normalize the production cost of all their slide rules by virtue of all-plastic construction though common manufacturing methods and tooling, it's not like we would expect the company to significantly lower the price of their flagship duplex rules, even if they COULD have been priced closer to the more basic single-sided rules. 

Therefore, when writing about K&E slide rules and attempting to describe a family of rules with the "Modern" moniker, then we are mostly certainly talking about their common all-plastic construction.  In fact, with the later (or modern) era K&E slide rules, we start to see some blurring of the lines between their single-sided and double-sided rules.  We already mentioned this in our discussion of the "K12-Prep" (68-1892) rule's replacement of the Model 4058 beginner's models, but this was true of another model, the "GP12" (68-1565), as well.  Even the famous 68-1400 Analon rule, while very much of duplex construction, was in all practicality and functionally, a one-sided rule. 

Note:  Such a single-sided rule in a duplex format is typically known as a "simplex" rule, which Pickett had made popular.  While the GP-12 could be talked about here as a Modern "Single-Sided" rule, it most certainly does not resemble a Polyphase in any form.  It because of its obvious relationship to both the Deci-lon and Analon rules, I will discuss them collectively in the later section called Modern Duplex Rules. 

Lastly, approaching the modern era, it became clear that K&E wanted to recategorize how their slide rules were presented.  While the words Polyphase, Duplex, Mannheim, Log Log, Deci-Trig, et al, would always carry on in their titles for individual rules, they would no longer be listed in "family" designations within their catalogs.  Instead, as we would see in the 1972 catalog,  they would group them as "Pocket Slide rules" or "Limited Purpose" or "Special Purpose" or the "Deci-Trig Group."    This shifting of labels is logical since most slide rules at this point would have common plastic construction, or a blurring of the traditional Mannheim (single-sided) versus Duplex (doubled-sided) point of view.  So rather than listing them as model lines within these broad categories, K&E wanted users to think in terms of how they would be practically used instead.   We can see this in how all K&E "Pocket Slide Rules" were listed in the 1952 catalog (see image below). Previous catalogs would have never grouped single-sided and duplex rules on the same page like this. 

Model 4161 Polyphase Mannheim 

​This model was first introduced in 1948 as the 9061-1, which was part of the Doric series of rules (see Sidebar: The Doric Family of Rules). However, by 1951, the 9061-1 was renamed the 4161-1 (again, see image) as a 5" pocket rule made of an ABS plastic known as "Ivorite," dropping the Doric label.  This is the same plastic that would be used in all of their plastic rules toward the end of the slide rule era.  As a single-sided Polyphase Mannheim rule, its scales were as follows: 
 
​Front side:  K DF [ CF L CI C ] D A
Back of slide: T ST S C

Evidently this became a popular slide rule for K&E, as a 10" 4161-3 version of the rule was added to the product line in 1954.  It's like an all-plastic version of the classic 4053-3 model, except it added folded scales.  The 4161-3 would share almost all of the scales with the 4088 Polyphase Duplex and 4097C Ever-There rules, but add L and ST scales.  As such, it would be the most powerful implementation of the Polyphase Mannheim scale set to date: 

Front Side:  INCH // K DF [ CF L CI C ] D A \\ CENTIMETER
Back of Slide:  T ST S C

Both 4161 model rules rules would prove popular.  Solid, clean, and durable, with an "unbreakable" metal frame cursor, the model name would become 68-1586 for the 5" and 68-1576 for the 10" in 1962, marketed as the "Modern Polyphase."   Five years later in 1966, both rules would be reclassified as the "Jet-Math Modern Polyphase" rules, even though they had been mostly unchanged for 15 years.  

In 1967, K&E would terminate the model 68-1586 (4161-1) in favor of the same rule - since 1956 they had been producing the SAME 5" rule in a leather case that came with metal clip.  This rule, known as the Model 4161-1C (for clip), would be called the Model 68-1581 in the 1962 reorganization of model numbers.   Therefore, at the discontinuation of the 4161-1 in 1956, buyers no longer had a choice to buy the rule without the case clip.   Priced at $11 in 1962, the rule was very well priced, 50% to 80% less when compared to two other rules from different families built from the same form factor. 

As for the 10" Model 68-1571/4161-3, it would continue in production until 1972, when the company filed for bankruptcy, and remain for sale until 1975 when K&E would close its doors. 

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The innovative Duplex design allows access to both sides of the middle slide simultaneously. If the indicator/cursor covers both sides of the slide rule, then a single setting of the slide can allow access to double the number of scales versus the single-sided, Mannheim design.

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​Rules of the Double-Sided, Duplex-Type

When K&E licensed the rights for the "duplex" design from William Cox in the early 1890s, they knew that it was a great way to add more scales, and thus it would be something other slide rules makers would desire to have for themselves.  By locking up the exclusive rights to the design, K&E set themselves up for early success, especially once they started selling to a hugely growing, revolutionary, industrial market in the U.S.  And it was a name to be trusted - K&E surveying equipment and drawing supplies had already built half of Manhattan, including the Brooklyn Bridge, so rapidly expanding the product lines of slide rules couldn't have failed. 

K&E would produce a variety of rules of many different types or "families" since the early 1890s.  And unlike in Europe, which didn't really appreciate the duplex design of rules until closer to the modern era, Americans seemed to like and appreciate a good duplex slide rule.  Pickett, a company which started post World War II, would never make a rule that WASN'T a duplex design.  And, of course, this is largely due to the success of K&E's efforts in that product market. 

What follows are the K&E duplex rules broken down by historical "family" type.  

​The Duplex Family

Certainly, slide rules based on the duplex design had the potential to be powerful rules, especially in light of what they evolved to become throughout the 20th century; however, I do not believe K&E achieved this with this first Duplex Family of rules. 
​
In fact, I would argue that K&E did not use the extra real estate in this Duplex family of rules to good effect.   This is due, somewhat, to a technology-limitation early on, mainly because of a lack of capability of their dividing-engines to etch scales across the entire surface area. 

But I think it's more of a design decision.   K&E's first approach was conservative; safe.  They would stick to what they knew and produce simple arithmetic and trigonometric Duplex rules.   The only thing groundbreaking would be the way the slide rule worked, not necessarily what more could be accomplished with it.  

The earliest rules in these series used identical stator rails front and back, giving buyers an option for differing slides, either an arithmetic slide for simplified numerical operations by incorporating inverted scales OR a trigonometric slide that provides sine and tangent scales.  With either configuration, I would argue that the redundancy of scales is a wasted opportunity.  But more on this later...

K&E did eventually increase the number of scales on the 1909 "T" (trig) versions of the 4071 series of rules, but at that point there would be only ONE scale dissimilar to the Polyphase Mannheim Model 4053, which favored a K-scale instead of the Duplex rule's BI scale.   So, in essence, the Duplex family would never give an option for a wider range of capability over rules already in their stable.   

As such, this family of rules was K&E's shortest lived, introduced with the Model 1744 series in 1895 and ending around 1916 (or perhaps 1917).   Even so, all rules in this family are rather collectible (please see Collector's Outlook for this series). 

The two main series in the Duplex family are described below...

Models 1744, 1744A, 1744B, and 1744 1/2

Also known as the Wm. Cox rule, the 1744 was the first "duplex" rule ever produced by any manufacturer.  They would use Dennert & Pape as the supplier, with a construction similar to the imported Mannheim rules which were made of celluloid-covered boxwood, and with cursors made either of brass or metal-framed glass.   It was added to the 17XX line-up of rules (which included the Mannheims) in 1895, costing buyers $6.50.  This was a basic implementation, with A [B C] D on one side of the rule and A [BI CI] D on the other.  Marketed as an "Arithmetic" rule, this 10" model would allow for three operations in one position of the slide.  Simultaneously, K&E produced a 1744A trigonometry option for the same price, replacing the BI and CI scales on the back of the slide with S and T scales (sine and tangent), as well as a "scale of equal measures," or L scales (logarithms) squeezed in between.  And for a $1.50 more, K&E sold a 1744B variant which shipped with both slides from the other two models, made to be interchangeable.   

Interestingly, K&E offered a 1744 1/2 version of this rule, the 1/2 denoting "half-size."   This would be their first "pocket" duplex rule.  You might think you could save money getting the half-sized rule, but it was priced equally to the 10" 1744 at $6.50.  That combination of price and size likely didn't sell very well, so I'm not planning on finding this rule in the wild anytime soon.  

For the collector, these rules are rare, and thus very desirable.   They are also hard to identify since, as with most of K&E's 19th century slide rules, none came with the model number on the rule.  

Rules in the 17XX series, including both their Duplex and Mannheim rules, were the only slide rules offered by K&E from 1887 to 1900.  But a year later, everything would change.  

The 4071 series

When K&E increased tooling for all their slide rules in 1901, their Duplex Family expanded immensely, becoming their most emphasized and produced slide rules.   The 10" Model 4071 was the main rule in this series of 10 slide rules, all of which were variants of original 1744 series, and all available through the 1901 K&E catalog.  These rules, now made of celluloid-covered mahogany with engine-divided scales, typically came in a choice of brass indicator OR clamshell glass cursor, a choice of arithmetic-only or both-slide (arithmetic and trig) models, and in 5", 10", and 20" varieties.  

Among the 10" models, the 4070 (brass indicator) and 4071 (clamshell glass cursor) directly descended from the arithmetic 1744 rule.  As such, these had A [B C] D scales on the front side and A [BI CI] D scales on the back, meaning there would be no reason to ever invert the slide.  At an introductory price of $8.00 for the glass-equipped 4071 and $6.50 for the brass-equipped 4070, these represented K&Es flagship line, and thus their highest priced slide rules.  

The previous Model 1744B, shipping with dual arithmetic and trigonometric slides, became the new 4075 and 4076 rules in this lineup.  Again, both 10" rules, the former with brass an the latter with glass, and both shipping with interchangeable arithmetic AND trig slides.  The option with both slides costs an extra $1.50 over the single-slide 4070/71 models.   Note that the 1744A trig-slide only variety was no longer represented, so if you wanted trig scales you would have been given little choice but to spend the $9.50 for the Model 4076 with the extra trig slide.  

K&E also offered a full selection of pocket Duplex rules in this series, based on the original 1744 1/2.   These rules were the arithmetic-only 4060 (brass) and 4061 (glass) versions, as well as the 4065 (brass) and 4066 (glass) which shipped with both arithmetic and trig slides.   Again, K&E offered no discount with these rules over their full-sized brethren. 

Finally, the first 20" duplex rules were included in this series.   These were the 4078 and 4079, again with brass and glass cursors respectively.  These only offered the arithmetic slide option, but they still set back the buyer $16.50 for the 4078 and $18.00 for the 4079.    The latter would cost around $630 in today's money, or about the price of your iPhone.  At least they didn't have to recharge the slide rule.  

This series of rules would morph slightly over the next 15 years, beginning with 20" options that came with interchangeable, double slides to add trig functionality.  These models, introduced in the very next catalog in 1903, were the 4080 (brass) and 4081 (glass) rules priced at $20.00 and $21.50 respectively.  These rules would be renamed when the line-up was completely revamped in 1906, and those model numbers would be reused by K&E nearly 30 years later for two of their most important slide rules in company history (see Log Log Duplex Rules in a later section). 

Speaking of 1906, while recovering from a fire to one of their Hoboken, NJ warehouses the year earlier, K&E was undeterred from making a plethora of changes to the 4071 series of slide rules, streamlining some of the models while also adding many others, including a variety of 8" and 16" rules.   Additionally, K&E shifted away from shipping both slides with their rules, going back to single-slide, arithmetic and trigonometric options. Where they did this, the models were denoted with an "N" suffix.  These retained the $1.50 higher price tag over the arithmetic-only rules despite no longer shipping with two slides.   This is because K&E found a way to add the trig scales to the arithmetic slide, with A [B S C] D on the front side and A [BI T CI] D on the back, with the L-scale on the edge of the rule.  The cursor was modified to provide an indicator for the rule's edge.  This was true for both cursor options, which remained the same for the models, except "metal" was substituted for brass in the non-glass variants.  And like the models before it, all slide rules would come with a sewed-leather case and directions.   ​

The left building was K&E's new office building in Hoboken, NJ as built in 1906, next door to their largest factory on the right.

And this is what it looks like today, as shown on Google Earth.

Because of the number of models in this series, 20 in total in 1906, it is easier just to list them by length...

• The 4060 ($6.00) and 4061 ($7.50) 5" inch rules did not change, but the 4065 and 4066 rules became the 4065N and 4066N, with arithmetic and trig slides respectively.  Those rules are priced, once again, at $1.50 premium over the arithmetic-only, 4060 and 4061 slide rules.  
• A new series of 8" rules was added, given model names of 4067, 4068, 4069 and 4069 1/2.  The 4067 ($6.00) and 4068 ($7.50) arithmetic rules came with metal and glass respectively.  The trig-only rules, the 4069 (metal) and 4069 1/2 (glass) carried with them a $1.50 higher price tag,  Why the "1/2" suffix?  In this case, they ran out of numbers, as the 4070 starts off the 10" rules. 
• The 10" rules, which bares the family name, were the 4070 and 4071 models.  These did not change from the previous model years, other than the conversion of the brass cursor to "metal."   The 4075N and 4076N (note the added "N" suffix) were converted to the new all-purpose single-slide "with trig scales," as explained in the 1906 catalog.   Like the 5" and 8" rules, these 10" slide rules retailed at the same prices. 
• Also new to the family were 16" rules, the 4082 (metal/arithmetic), 4083 (glass/arithmetic), 4084 (metal/trig), and 4085 (glass/trig).  Prices were $14, $15, $17 and $18 respectively.  (Note that the 4082 and 4083 model names would be repurposed in later years).  
• For 20" slide rules, the 4080 and 4081 "both-slide" models were changed to the 4088 and 4089, now becoming trig rules.  Curious is the lack of the "N" suffix here, where apparently the change of number was good enough?   The 4078 and 4079 arithmetic rules of the previous line-up would also get new model numbers, now called the 4086 and 4087.   These rules actually got a price-cut, with the 4086 and 4087 priced at $16 and $17, while 4088 and 4089 were $19 and $20. 

1909 would feature one last revamping of the 4071 series, whereas the entire lineup would dispense with the metal cursor option.  As such, all model numbers originally with brass or metal cursors would disappear, reducing the total number of rules offered in the 1909 version by a half.   

Of these ten rules, five of them would change model numbers and add a "T" suffix.   This designation now indicated the rule was of the "trigonometric" variety.    As such, there would only be two rules for each length, with a single number model, both with and without a "T."  

• The 5" rule would now exist as the 4061 and 4061T models, priced at $5.00 and $6.50 respectively.  
• The 8" rule became the 4065 and 4065T models (a number borrowed previously from the 5" trig rule).  These models were priced the same at the 5" rules.  
• The Model 4071 and 4071T took permanent grasp of the both the series name and the 10" selection of rules, again priced identically the 5" and 8" versions.     
• The 16" variety, now known as the 4083 and 4083T rules, were priced at $12 and $15 respectively. 
• The 20" rules were condensed down to the 4087 and 4087T models, at $15 and $18 each.   These prices represent a $2 discount to the previous price and 16% off the highest historical price of this family of 20" slide rules.  I believe it's quite obvious that K&E was aware of how exorbitant some of these prices were and worked conscientiously to reduce cost to the consumer, which in the case of the longer rules, was chiefly commercial customers.  

This series of rules would disappear entirely a decade later.  Listed in the 1916 catalog, there would be a 5 year gap before K&E's next catalog in 1921.  While there is no catalog to show exactly when the line-up changed, it is clear from collected samples today that there were no more production dates after 1917.   Yes, this is before K&E's use of serial numbers in their rules in 1922; however, a production number was etched into the ends of earlier rules, and thus dates within a model's lifespan can be estimated from those batch numbers.   

I mentioned earlier why I felt this model line-up could have been short lived, and certainly, it seemed K&E was never content with it, constantly trying to improve it, but never really addressing the real truth of the matter, namely that the Duplex Family of rules didn't really give additional power for the user to do calculations that couldn't be done with already existing K&E slide rules.  This is something that could have been fixed - quite simply, there's no reason to have A and D scales on BOTH sides of a slide rule with a capability of being read on EITHER side of the rule with the same setting of the cursor.   I find it curious why it took K&E so many years to understand that?   

I suspect that K&E just didn't see the urgency.   If they are selling slide rules by the thousands, then what would they care if a slide rule wasn't as good as it could be?  Or perhaps by 1909, they'd be trapped by their own product placement, not willing to compete with their flagship slide rule, the Model 4092 Log Log rule, introduced the same year (see the Log Log Duplex Family in a later section)?

Even so, from the perspective of history, for all the potential that the duplex technology possessed, the 4071 series and the Duplex Family of rules on the whole were rather unremarkable.

This would change with the introduction of the Model 4088, the first rule in the improved Polyphase Duplex family, which we will talk about next.  
​

The Polyphase Duplex Family

In 1913, four years after the introduction of the Model 4053 Polyphase Mannheim slide rule but also four years prior to the death of the 4071 series, K&E introduced what they called the Polyphase Duplex slide rule in a single 4088 model.   This number is not to be confused with the brass-cursor 4088 model (4071 series Duplex Family) that was discontinued in 1909.  The new 4088 model was introduced in two lengths, an 8" version predictably designated the 4088-2, and the full-scale 10" version known, of course, as the 4088-3.  Before we discuss each of these models though, let's continue our exploration of what K&E offered to that point and why the Model 4088 represented the more complete fulfilment of the promises of the duplex design. 

The 1913 catalog, which is copyright in 1912 incidentally, has a few addendum pages pasted within to reflect newer products in 1913 at the time the catalog went to press.  The new Polyphase Duplex 4088 rule was described on one of those addendum pages.   The first sentence reads, "The Polyphase Duplex Slide Rule is a combination of the Polyphase and the Duplex Rules, with the addition of several special scales."   

I think this statement is disingenuous, at best. 

If K&E were honest, they would admit their Duplex family of rules was not meeting the potential of the duplex design.   Remember, this line-up was revamped in 1909, only then giving it a near-equivalence in scales to the 4053 Polyphase Mannheim.   There was simply no room with existing Duplex rules to innovate with other scales.   So when K&E describes the new Polyphase Duplex in the above catalog statement, the latter half of their statement is actually the real part of the story.  As such, this was K&E's opportunity to make the most of what a duplex rule could do, to the point where K&E would kill production of the original Duplex family within 4 years of the introduction of the Polyphase Duplex.  

That said, so what were the "special scales" K&E mentioned?   These would be the folded scales of the C, CI, and D scales.  This is the first appearance of the CF, CIF and DF in any K&E slide rule, which were valued because now users didn't have to worry about off-scale computations - a solution will always be on the rule with folded scales regardless of which index is used.   Likewise, because the scales were folded over pi (3.14159...), then values on the D-folded scale were also the product of pi and the D scale setting.   This made quick work of problems that dealt with circles.   

Specifics of the models in the Polyphase Duplex Family of the rules are as follows...

The Model 4088 Series

 The 4088-3 (10") and 4088-2 (8"), as introduced in 1913 with the same "column cursor" pushed out to all K&E slide rules, contained 11 scales.  This is the same number as the 4071T duplex rule that was currently being produced and which it would eventually replace. So what makes it better?   

In this new rule, K&E does away with the redundant scales.  Gone were the A and D scales on both sides of the stator rails.  Instead, the front side of the rule would be configured as DF [CF CIF C] D and the rear of the rule would be setup as K A [S T CI] D L.  It's an improvement over the Polyphase Mannheim and Duplex families because of the addition of the folded scales, but also there is a K-scale which was missing from every Duplex rule K&E had produced.   Certainly, this is the best arrangement of scales on any K&E slide rule to this point, even better for general math computations than the Model 4092 Log Log Duplex rule introduced in 1909 (to be discussed in the next section).  It would become even better when K&E added a B-scale on the back of the slide in 1922.

Perfectly timed with the ending of the Duplex Famliy line-up, K&E added a 20" Model 4088-5 in 1917 and then a 5" Model 4088-1 five years later.  Very few other changes were made to this rule over time, except for the cursor modifications they made simultaneously with all other slide rules.  Those changes included the switch to the frameless metal cursor with metal rails in 1915, followed by a switch to plastic rails in 1916. 

This model would continue well into the 1930s.   In 1936, K&E revised the 20" rule and added an "N" designation.  They supplied this N4088-5 with the "new-improved" cursor like all other K&E rules - this new cursor with a metal rim around the glass kept users from breaking off the corners of the glass by overtightening - which was indeed a god-send. It is unknown why the new prefix was used for this rule since no other such "improved" cursor rules added the "N" to the model name.   

K&E would cease production of all versions of this rule in 1939, replaced by similar, but more powerful Polyphase Duplex rules that we'll talk about next.   
The Model 4070 Trig and 4071 Deci-Trig

In 1939, K&E discontinued all models of the 4088 and replaced it with these two slide rules, both with 10" scale length.   Not to be confused with the original 4071 Duplex family of rules that were discontinued more than 20 years prior, these new rules increased trig capabilities to give more accuracy for angle measures that are less than 5.73 degrees.   Because sine and tangent are mostly the same for such angles (actually approximating the radian equivalent for that angle), then a single scale of greater resolution can be used to evaluate both of those functions.   This would be known as the "ST" scale on this rule, which ranges from 0 to 5.73 degrees (on the Model 4071 Deci-Trig) rather than the typical 0 to 90 degrees on a normal S or T scale.  

The secondary benefit of this is extra precision on the S and T scales, meaning that those scales could now begin at 5.73 degrees instead of the normal 0 degrees.  As such, by essentially continuing the S and T scales where the ST leaves off, you gain at least a half a rule's worth of precision for angles bigger that 5.7 degrees, since half of the typical S and T scales normally take up half the rule.   So, in essence, the ST and S/T scales are folded scales when used together.

Now this was somewhat of a new scale for K&E.  It would mark the second occurrence of an ST scale on a K&E rule, following the 4080 Trig/4081 Decitrig Log Log models introduced two year earlier, as we will see in the next section.  

This ST scale (also known as the SRT scale) would become a fixture of all but the most basic of K&E rules.   

The overall scale set is something we could call the "New Polyphase Duplex."  It is as follows...

Front Side:  DF [CF CIF CI C] D L
Back Side:  K A [B T ST S] D DI

So, why two new models?  Taking a cue from the Log Log Duplex Family of rules (see next section), K&E gave buyers a choice with how they wanted their trig angles expressed.  Those who preferred a traditional degrees, minutes, and seconds (DMS) format might have chosen the Model 4070 Polyphase Duplex "Trig" version of the rule.   But the trend toward a decimal degrees rule was strong by this time, whereas the Model 4071 Polyphase Duplex "Deci-Trig" could be preferred.    This distinction with angle input preference had proven successful when first introduced in the Log Log Duplex models in 1933, so it made sense to do the same with the Polyphase Duplex rules in 1939.   

The 10" new models immediately superseded the 27 year old 4088 model in all lengths, except for the 8" 4088-2 that would hang on for one more year, I suspect because they might still have had stock available.  But I find it intriguing that until the Doric rules were introduced in the late 40s, a good decade passed by where the 10" 4070 and 4071 were the only Polyphase Duplex models available.  In fact, this was a decade where there were ZERO K&E duplex rules manufactured in sizes less than 10".  And as I said earlier, I suspect that the Ever-There series had something to do with that.  A pocket version of a 4070 or 4070 duplex model would have cost double that of the Model 4097D Ever-There that sported the same, yet improved Polyphase scale set. 

The European Influence on Keuffel & Esser

An aspect that I believe becomes obvious upon digging deeper into K&Es product line and the historical timing by which they rolled out certain innovations is also something I haven't seen talked about very much on the Internet, and that's the level of influence that European slide rules and their markets undoubtedly had on K&E.  As German immigrants, founders William J. D. Keuffel and Herman Esser leaned strongly on happenings in Europe during the mid-to-late 19th century and brought some of those practices to their own company, which they founded in 1867, to the United States.   

While K&E was the leading seller of slide rules in the United States for the entire duration of the company's existence - challenged early by Dietzgen and Post/Hemmi and later by Pickett for market share - K&E was known as the leading innovator in the States, especially in their main business of surveying equipment.   But they also would hold a variety of important patents regarding slide rule technology.  They were an always evolving, diverse company, able to make money on a variety of products.   In some cases, ruthless, especially in their dealings where licensing agreements are concerned, an example of which can be read about here. 

Regarding their early foray into slide rule production, we know they leaned heavily on what they could learn from European slide rule makers. Companies like Dennert & Pape (Aristo), Nestler, Marc/UNIS, and Tavernier-Gravet predated K&E by several years, particularly in slide rule production.  After all, Great Britain had been developing and using slide rules since the 1600s and, by consequence, became popular in use throughout Europe much earlier than in the U.S., beating the US in popular use by a good 20 years.  The original Mannheim was in use by French military in 1859, whereas it wasn't until 1881 that Americans began to be exposed to the Thacher Cylindrical rule.

Similarly, Dietzgen, AW Faber, Frederick Post, and Hemmi each setup shop prior to the turn to the 20th century, all of which were finding their way to the slide rule market as contemporaries to Keuffel & Esser.  The market for slide rules accelerated rapidly, and there were many makers for Keuffel & Esser to monitor.  

When I think of early K&E slide rules, I picture the Mannheim-type of single-sided rules, such as the 4041 and 4053 models of celluloid-covered mahogany, all built in New Jersey within K&E's own factories.  But most might be surprised to know that their earliest rules were actually imported from Europe. These include their Model 479 rules, likely imported from Tavernier-Gravet, and their Model 17XX series, which were imported rules from Dennert & Pape.  K&E would not produce their rules in-house until the 1901 catalog entries, which included the aforementioned single-sided 4041 Mannheim series and the duplex 4070 Polyphase series of celluloid-covered, mahogany rules. 

​The timing of this is rather astonishing.  Dennert & Pape had introduced such rules in Europe beginning in 1886, with boxwood being their wood of choice, soon followed by mahogany.  K&E was quick to respond with their first 17XX series models, the 1745 and 1746, introduced in 1887, only a year after Dennert & Pape. It is reasonable to believe that this was a coordinated effort from the beginning, where both companies would flood the US and European markets simultaneously with those slide rules.  Dennert & Pape were, after all, also German. 

The use of celluloid on slide rules was a European invention, and Keuffel & Esser were quick to understand its advantages.  Their own boxwood models with celluloid facings first appeared in the K&E catalog in 1890.  Interestingly, the model 1748, a 20" version of that slide rule shows up in the catalog two years later.   In 1895, they would add a Model 1749 20" Stadia rule with celluloid facings and, importantly, the model 1744, which was their first duplex rule (from the William Cox patent of 1891) that began its existence with celluloid laminations.    Even their 1745 bare-bones "Gunter" model acquired celluloid facings that year.  

​These rules, of course, are the precursors to the K&E-made, 4041 series of mahogany rules that would come out in 1901.  

This technology was obviously superior to anything that preceded it, making slide rules more accurate and much easier to read once engine-divided and painted in relief.  Quite simply, Keuffel & Esser did not hesitate to bring it to the States.  Interesting that by 1901 with the introduction of the new product lines, K&E simply referred to the celluloid laminations as "white facings" - they had become that ubiquitous. 

The reason for a European partnership is simple.  K&E's factory, which produced surveying equipment as K&E's main business, was not tooled for the additional production of slide rules.  They needed a partner and Dennert & Pape fit the bill.   Although consumer interest in Europe remained in the single-sided or "simplex" designs based on the traditional Mannheim, Rietz, and Darmstadt scale sets, in the United States there was no extensive history to dictate consumer preference.  So being quick to market, rather than taking years to build up their own production capabilities, was the first mandate for Keuffel and Esser.  They understood the importance of giving consumers many slide rule options, and they sold in large numbers, particularly their duplex rules.   

And this wasn't only a focus of their slide rule business.  From their inception, K&E would rapidly produce an extensive catalog of goods.  While their main lines of products were made in-house, the vast majority were out-sourced to other suppliers.  And by the turn of the 20th century, K&E had thousands of products available in their catalogs, with store-front/distribution centers in four major cities across the U.S., including New York City, Chicago, Saint Louis, and San Francisco.

This idea of borrowing ideas, designs, and supply from other companies shouldn't be a surprise.  It's the way of any good business. We saw this with the Japanese maker, Sun-Hemmi as well, noting that they too borrowed from Europe.  Their own Electro 81 rule comes to mind, as it was a direct "reimagining" of the German Faber-Castell 378 Electro produced around the same time.   Similar with the Frederick Post Company, who would never manufacture a slide rule of their own, and instead farming out their manufacturing to Sun Hemmi over in Japan.  Instead of spending money on the research and development (R&D) of new products, it's much cheaper to let somebody else do that, put your own name on it, and rapidly build a customer-base.  And Chicago-based Dietzgen was in no way particular about from whom they would acquire licensing rights, especially from the Charles Bruning Co.  

With any young company getting one's feet wet in a new market, it's just smart business to produce reliable, known products with a history of consumer demand and commercial success.  Where they came from didn't really matter.  

But K&E wasn't always a reactive company where innovation was concerned.  They were very much industry leaders there, especially upon licensing the "duplex" copyright from William Cox, which would become their biggest seller at some point after it was introduced in 1895.  And by 1901, K&E had factories built, tooled for slide rule production.  While it can be argued how much profit the slide rule business actually was for Keuffel & Esser, we do know that they wouldn't have started building their own rules in-house if it wasn't profitable.  

The Doric 9068 (4168) and Doric 9071-3 Models

So when I discussed the Modern Polyphase Family of rules, I made mention of the Doric series of rules which served as a transition line of rules in terms of construction.   Two of those rules, the 5" 9068 and the 10" 9071-3 require discussion here.  

The Model 9068 was a beautiful, well-proportioned pocket duplex rule with a simple Polyphase-type scale set... 

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

I mentioned earlier that the Doric can be thought of as un-ornamented, but this doesn't have to mean "basic" just because it uses no red ink.  The scale set is logical and powerful for as simple as it is, originating, naturally, from the 4088.   The ST scale (in DMS measures) is very convenient, giving 4 significant figures of precision despite being a pocket-sized rule.  While typically called a 5" rule based on length of the actual scales, this rule is actually 12.5 cm in scale length, not 5". This will come into play when I discuss this format of rule as it appear to be cloned by other slide rule makers (see SIDEBAR: The 9068 and the Clones ​below).     

When K&E shifted direction away from providing a long-term Doric family of rules, the 9068, which likely originated from a different "prototype" rule known as the Model 10000 Doric as early as 1946, was reassigned the model number of 4168 around 1950, or most certainly prior to the 1952 catalog.  In 1956, the same rule would be offered with in a sheath with a leather covered clip, with a model number of 4168C.   In 1962, it would become either the 68-1555 or 68-1550, the latter number if the customer bought the version with the case clip.   Finally, in 1968, the non-clip version was discontinued, only offering the version with the clip until the end of the K&E era in 1975.   It would be the only rule to retain the Doric label over time, despite no other Doric references to slide rules since 1949.

The 9068 Doric model was offered at $8.50 in 1949, suspiciously priced at $10.35 in the December 1951 price list at the 4168, only to be returned to $8.50 in two months later in the February 1952 price list.  In 1962, the clipped case model 68-1550 would be $9.50, while the clipless version was $1 less; I suppose the former is money well spent if it keeps the rule from falling out of one's pocket!

In 1949, the 4071 Polyphase Duplex would get a plastic, Doric equivalent rule known as the 9071-3 Doric Polyphase Duplex.  Strangely, this rule was never described in a K&E catalog; only in a 1949 parts list.   It would survive until 1952 and then disappear complete.  The existing Model 4071 Decitrig on which it was modeled ran concurrently with this plastic rule, returning to being the sole Polyphase Duplex rule (with the 4070 Trig version) once the 9071-3 disappeared.  Made of Xylonite, but no Xylonite like the Ever-There rules, it is not completely rare from a collector's perspective; it is certainly attainable, and reasonably so.  

Log Log Duplex Family

We now come to the "flagship" family of K&E duplex rules, which preceded the Polyphase Duplex rules by four years.  This line of rules would prove to be K&E most successful line, introduced in 1909 with the Model 4092 and continuing in some fashion until the end of the slide rule era.   In fact, when you consider a "standard" duplex rule to compare with rules from other makers during the high-point of slide rule production in the 1950's and 1960's, it would be hard to argue that the later version, the Model 4081, doesn't deserve top placement on any tier list you might create (see Professor Herning's video, at right, where he discusses these leading slide rules of the era in 1955).

"Log Log" slide rules were introduced relatively early in the history of the device, invented by physician Peter Mark Roget (of Roget's Thesaurus fame) in 1814, almost a full century before the first K&E Log Log rule.  Why so long?  Historians would suggest that there were not practical reasons to have a Log Log rule until the engineering boom of the early 20th century, though I feel the real reason is one of production capabilities.  Prior to 1909, K&E lacked the ability to put the required number of scales on a slide rule.    

As previously discussed, K&E's machine(s) that divided and etched the scales could only function at the physical edge of the material; however, they gained the ability to put scales in the middle of the stator rails in 1909.  Incidentally, they could always put 3 scales on a slide, which would appear to relate to a limit in material thickness with earlier dividing engines - the body of the rule is much thicker than the slide itself.   This new 1909 retooling yielded increased capacity to place up to 3 scales on each rail for a lot of slide rule real estate.  Interestingly, K&E's first order of business was to make a slide rule that permitted the etching of three Log Log scales which allowed for the computation of arbitrary exponentials and roots. 

Note:  Since the C-scale represents the log of a number "x", then a "log of a log" scale for some base "b" allows the computation of b^x power.  Because of the need for precision, these Log Log scales are very fine, typically three continuous scales called LL1, LL2, and LL3, folded to produce a single long scale for computing exponentials from base 1.01 up to 22,000.  (LL1 ranging from 1.01 to 1.11,  LL2 ranging from 1.11 to 2.718 or "e", and LL3 ranging from 2.718 to 22,000.)   Other scales can be added for negative powers as well, typically known as LL01 (LL/1), LL02 (LL/2), or LL03 (LL/3) scales.   Later rules from many makers will have up to 8 total log log scales for your exponential solving pleasure.  ​

Such Log Log rules require three or four scales to be useful, which means that unless a company produced a rule ONLY for exponential computations, then there would not be enough space on the typical, multi-purpose rule, such as the Polyphase Mannheim layout championed by K&E the very same year.   With the duplex-design and newly-found ability to have three scales on a rail, then a Log Log Duplex rule could be produced without sacrificing the utility of their own updated Mannheim layout (only the K scale of the Model 4053 is missing from the new Model 4092 Log Log rule), making for a very powerful slide rule.  

One should note that the original K&E-made Duplex rule, the Model 4071 series, had been closing in on a decade of production at this point.   As mentioned when discussing that rule, I felt that K&E had greatly underutilized that design because they couldn't add more scales and were unwilling to remove redundant scales (A and D scales on both sides, for example).   While splitting that series into arithmetic and trig models that same year in 1909, it's no surprise that the days were numbered for this original duplex design.  Eight years later, which was also 5 years after the superior Polyphase Duplex 4088 Model, the old Duplex family of slide rules would be gone.   By 1913, the pricing of these rules seems to support this:  $6.50 for the 10" Model 4071T (Duplex Trig), $7.00 for the superior 10" Model 4088-3 (Polyphase Duplex), and $8.00 for the new 10" Model 4092 (Log Log Duplex).   There would have been a place for the old Duplex at around $4.50 in this pricing structure, but not at a cost nearer the 4088, which was an improvement in every way.  

​As mentioned, there is a long history of models in this family of K&E rules, so let's dig deep into a model-by-model summary.

The Model 4092 Log Log Duplex

Introduced in 1909, and priced at $8.00, the new Log Log model became K&E's most expensive, flagship 10" slide rule.    Front side scales included A [B S C] D and the back side included LL1 LL2 LL3 [C T CI] D L.    This Model 4092 would remain entirely unchanged (except for the company-wide cursor changes) until 1922, when a 20" model, the 4092-5, was created.  Of course, this necessitated a name change of the 10" rule to the Model 4092-3.   With this new rule came a drastic revision to the scales.   Not surprising based on the success of the 4088 at that point and the increased potential for more scales, the 1922 version of both the 5" and 10" 4092 model incorporated folded scales as well.  Moving all trig functions to the back side of the rule, and reworking the Log Log scales, the new layout was entirely revamped:

Front scales:  DF [CF CIF CI C] D L
Back scales:   LL0  A [B S T C] LL3 LL2 LL1

The addition of the LL0 scale is lovely here, adding the possibility of computing exponentials for bases between 0 and 1.  Granted this scale is somewhat imprecise compared to future Log Log rules that would split these values across more scales (usually called LL01, LL02, and LL03 in future rules); however, this one scale did provide two significant figures of precision.  

A K-scale for cubes is all that remained from the typical Polyphase scales, but it could be argued that a K scale is redundant compared to performing the same computations on the new 4092 rule with the LL0 scale.  For example, 8^3 can be computed rather imprecisely to two significant figures using the K scale of a typical 10" slide rule - cubes with large bases are mostly impractical because precision is poor.   While worse could be said using the LL3 scale of the 4092, by using the LL0 scale instead (which was keyed to the A & B scale of the rule), users can rewrite the problem as (.8 * 10)^3 or .8^3 *10^3.  As such, setting the base at 0.8 on the LL0 and raising to the 3 power yields .51 to a similar level of precision as having a K-scale.  Thus, then multiplying by 10^3 is the same as moving the decimal 3 places to the right, for an answer of approximately 510.   

Of course slide rules are much less effective with larger numbers, so the real power of the Log Log scales occurs when you have exponential bases closer to 1.   For example, something like 1.2^3 can be computed to 4 digits of precision using the LL2 scale on the Model 4092 but only 3 digits with the K-scale. Plus, with Log Log scales, the user can just as quickly get all the powers of 1.2, even decimal ones, just by moving the cursor.  

No matter, this would be fixed in 1924 when K&E added a K scale to these rules, with a change in designation to both 10" and 20" rules, adding the "N" prefix.  Interestingly, these were catalog-only designations...the 20" 4092-5 rule is the only one that added the "N" onto the rule itself; a curiosity indeed. 

The 4092 rules would continue in production until 1939, even hanging on 6 years after the introduction of the next two Log Log Duplex rules, as its scale set was very convenient, powerful, and versatile, especially from the perspective of history.   Later similar rules, from K&E and others, would seem to compete for the title of "most scales" on a rule, raising the complexity of their use.  The 4092 keeps to the essentials.   Likewise, this rule is nicely dimensioned.  I haven't talked much about the form factor of K&E rules, especially the duplex rules, but to help with the look of additional scales, the width of the rule is increased compared to other K&E duplex rules.   For example, the 4092 rules are consistently 4 cm wide compared to the 3 cm of the 4088 Polyphase Duplex.   This 4 cm width would become standard for all of the more powerful duplex rules produced by K&E.  

As such, I find the 4092 a joy to use, especially in the later variants that added the K-scale.  It is beautifully constructed.  Its extra-width and uncluttered arrangement of scales seems to strike the right aesthetic, especially after 1922 when the rules shifted to laminated edges and after 1927 when the scales were printed in a non-serif font.   I own a 4092-3 sample made in 1933, which apparently has been upgraded in the past with the improved-glass cursor.  It's one of my favorite slide rules. 

SIDEBAR:  The 9068 and the Clones

What you see is not an illusion.  The image above shows four rules that look almost identical to one another, and two of them are not K&E slide rules.  The third rule is the K&E Doric 4168 Polyphase Duplex, likely introduced around 1950.   As mentioned in the Doric Sidebar earlier, this is the direct descendant from the original 9068 Doric model.  It would continue in production for 11 years until 1962, get renamed the 68-1555 for 6 years, and then be sold for 8 more years as the 68-1550 with a leather clipped case.    Once K&E dropped the Doric "family" concept, no longer mentioning the word in the 1952 catalog or after, this is the only original Doric rule to retain a Doric label. 

Below this rule in the image is a one-time variant known as the Celanese Celcon (68-1555), made of a green "acetal co-polymer" thermoplastic known as Celcon, produced by K&E for the Celanese company for one year in 1968. According to the ISRM, in an interview with the research director, it is the first use of the Celcon resin in an product.  Celcon, and the Celanese Company, still exists today - they are a Fortune 500 company and the world's leading producer of acetic acid and polyvinyl acetate.  The digression aside, other than the plastic used in both the body and the cursor rails, these pocket rules are identical.  

And now, the top two rules.   We likely shouldn't be surprised that Dietzgen is represented here, as they were known to outsource their production to outside makers, but you'd be mistaken to think K&E built the Dietzgen No. 1776 Redirule as pictured.  Despite the shape, the dimensions are subtly different throughout, including the length of the rule, size of the rail grooves, size of the adjustment screw in the end brackets, and of course the cursor.  The scale length is different too.  The Dietzgen has a true 5" scale length while the K&E is actually 12.5 cm.  The Dietzgen also adds two more scales, those of the "new" Polyphase scale set. 

The top rule is the Bruning No. 2398, and yes, in construction, it is identical to the Dietzgen in every way, with the possible exception of the plastic being used.   While age can account for color variations in plastic, the Dietzgen is most certainly colored different from the factory.  However its surface feels like the Dietzgen while the K&E rules feel different.   Interestingly the Bruning has the same scale set of the K&Es, not the Dietzgen.  

Part of the problem with reconstructing slide rule history is the lack of documentation of agreements, contracts, and production timelines for any of these companies. The best we have is trademarks, product catalogs, and patents in order to understand, in this case, who's cloning who?  

The Charles Bruning Company deserves special mention.  Bruning produced their own rules, but a greater number of them existed as rules licensed to other companies.  As I said, Dietzgen was known for out-sourcing many of their rules, especially the plastic ones, but not to K&E in this case.  Bruning is the licensed contractor here, as well as for Dietzgen's No. 1771 rule, among others. 

The hard question to ask is how Bruning might be related to K&E?  And is their any chance that Bruning produced these 9068 rules? 

The latter question can be answered first - not likely.  Most certainly K&E was held back by World War II in the sense that R&D efforts would take a back seat to customized production with the military as their top customer.   

One aspect of that would be that a company like Bruning would produce "Ivorite" types of slide rules even during the war.   It would not be until ~1950 before K&E shifted to Ivorite construction in their plastic rules.  But the early 9068 Doric, and the "prototype" Ecco 10000 version of it, is reputed to be something else - improved Xylonite (?) - or most certainly different than something like the 10" Bruning 2420 "Polyphase" Mannheim rule made around 1943 (from which the Post 1452P rule was also produced).  

A second aspect, one of supplying to the military, means that production of typical products would have been redirected into other areas more specific to what was needed in the war effort.   We see a glimpse of this in the K&E 1944 Product Catalog...

While we suspect that production numbers of slide rules did not wane overall, as the serial number reference table (see much earlier in this article) shows rather normal estimates of 60,000 rules in 1943 and 77,000 rules in 1944.  But if we were to look at K&E rules in the wild during this period, we wouldn't see much of their normal production rules. 

What K&E did instead during this time is hard to figure completely, but we do know they produced their M4 and M16 Graphical Firing Tables during this time (looking suspiciously like William and Cox Load Adjuster "slipsticks" of the same era) , and likely their Model 4108 "US Military" rule.   Their rules, with some speculation of their origins, will be covered in the section on Specialty Rules later.  But importantly, the serial numbers of these rules seem to match the production table used to date our K&E rules. 

Back to the relationship with Bruning - they produced exactly the same rule as the Model 4108, doing business as the American Blueprint Co.  Again, both K&E and Bruning making the same rules during this time?  Certainly, this is something worth further understanding!    

The idea of "cloned" rules is not anything unusual.  While we are inclined to believe that K&E likely produced rules licensed to Bruning, I would leave open the possibility that it could have happened the other way around.  And if that's the case, then it opens up inklings that Bruning could have produced the Doric series as well.  I don't think so, but who really knows how business was run during the post WWII era?  

The Model 4090-3 Log Log Trig and 4091-3 Log Log Decitrig Duplex

The idea of a slide rule with decimal trig scales had been around for a good while, and when K&E produced one in 1929, a new "vector" rule no-less (see the next section), its decimal trig scales got the attention of other consumers.  Thinking at first that only electrical engineering students could benefit from decimal trig scales, K&E was quite quick to offer them with their other duplex lines as well.  As a math educator myself, doing trigonometry with decimal angles is just much easier for all concerned, not just electrical engineers.   While I did not live back during those times and cannot state with authority that education was trending away from DMS (degrees/minutes/seconds) angles, I think it's obvious to see that K&E was not going to deny consumers the opportunity to choose either option where their duplex rules were concerned.  

And thus was born the concept of the Trig and Decitrig versions of these rules.  

I mentioned earlier that, first, the 4088 would give way to new Polyphase Duplex 4070 Trig and 4071 Decitrig rules in 1939, but here, in 1933, K&E did it first with the Log Log Duplex rules, producing the Model 4090-3 Log Log Duplex Trig and Model 4091-3 Log Log Duplex Decitrig slide rules.  Again, like the 4070 and 4071 that would come later, these rules were identical, only differing in the way the trig scales are divided.   

Unlike the new Polyphase rules, these two new Log Log rules would not replace their original rule in the series.  In this case, K&E kept the excellent Model 4092 and produced a new, revamped Log Log scale set.  It was this new scale set that would be implemented into the Trig and Decitrig options.   So in 1933, and for six years after, K&E would carry three Log Log Duplex rules in their product line.  

The new scale set for both rules:  

Front:  L LL1 DF [CF CIF C] D LL3 LL2 
Back:  LL0 A [B K CI] T S1 S2

And scales of the original 4092 Log Log Duplex:

Front:  K DF [CF CIF CI C] D L
Back:   LL0  A [B S T C] LL3 LL2 LL1

The first thing to observe with the new scale set is the emphasis on trigonometry, as you might expect.  The original 4092 gave up it's additional C scale and the S scale for the S1 and S2.   

These split sine into S1 (0 to 5.73 degrees) and S2 (5.73 to 90 degrees).  Since the sine and tangent for angles in the small angle domain are largely the same, then the S1 scale could be used for tangent evaluation of those angles as well. 

This is the only implementation of the extended sine scales S1 and S2 on a K&E slide rule, though their Log Log Vector Duplex rule introduced 4 years prior had inverted S1, S2, and T scales.  As we will see in our discussion of that rule, there is an interesting backstory there.  But suffice it to say, there might be another reason that K&E would re-label these scales for the next version of this slide rule 4 years later.  

Two rules from my collection, the 4080-5 Trig and 4081-5 Decitrig, top and bottom respectively. As per all Trig/Decitrig models, the only difference is the divisions for the trig scales. The minute marks on the angles of the ST scale on the 4080-5 is the obvious indicator, but note also that all three trig scales (T ST S) are divided differently, degrees etched in 6ths on the top rule and degrees etched in 10ths on the bottom rule.

I do find it curious why the 4092 hung around for six years simultaneously with the new 4090 and 4091 rules, and wasn't discontinued like the Model 4088 Polyphase Duplex when usurped by the trig models?   

​I would suspect that it's because the newer models sacrifice too much for its new capabilities.  First, as a Log Log rule, having all the log-log scales on the same side of the rule (which the new models do not) was important for many customers - it is important to me, which is why the 4092 is one of my favorite K&E slide rules.  Secondly, it is natural and usual to have all the trig scales on the slide, which the majority of K&Es have always had.   While you gain additional precision with small angle trig on the new rules, you do so at the penalty of having the trig scales on the stator rail.   Because these are referenced off of the C/D scale, the user would either need to flip the rule to read from D or flip the slide and perfectly align it to assure an accurate reading.   

Importantly, I would also suspect that because the original 4092-3 also came in a 20" version, the 4092-5, and because such a long-scale version was not offered with the 4090 and 4091 series, then the 4092 in both 10" and 20" were rather safe, at least for a time.  

I consider this layout of the new trig rules poor for what it attempts to do.   Too many trade-offs needed to be made to gain a small amount of accuracy.  And even then, if you wanted more accurate trig evaluation of smaller angles, then sticking with the superior 4092 layout and dividing any small angle by 57.3 will give you same levels of precision (this is essentially what the ST/SRT scale would later do).   In fact, we even see earlier K&E rules, like the Log Log Vector rule (see the next section), add a "radian" mark on the C/D scale for exactly this reason.  

Thus, it should not be a surprise if K&E wanted to replace these rules as soon as possible, which leads us to...

The Model 4080 Log Log Duplex Trig & Model 4081 Log Log Duplex Decitrig

​The run of the 4090/4091 rules would indeed be short lived.  In only 4 years, they would be replaced with these 4080 Trig and 4081 Decitrig models in 1937. However, the 4092-3 and 4092-5 would hang on for two additional years.   Introductory pricing in 1937 for both rules was $11.75 with synthetic leather case, with a real-leather upgrade for one dollar more.   The Model 4092-3, still available, would save you a dollar.

If I hadn't mentioned how terrible the previous 4090/4091 rules were, one might be inclined to think that the 4080/4081 were simply a re-designation of model numbers, as K&E was sometimes bent to do.  However, not only do these rules fix the usability issue of the 4090/4091, the new models add three additional scales from the previous 17-scale models; bonus capabilities as well.   

​The change was well-received, and it is said that this new series of Log Log Trig and Decitrig rules became not only K&E's top seller but was, historically, the top selling slide rule of all time, regardless of maker - considered by many, including Professor Herning above, the "standard" bearer of slide rules.  And I can easily see why it was successful...

​The front side scales would revert almost identically to the excellent 4092 model, also squeezing in the CI scale in the middle of the front slide.  The back side of the rules would look like LL0 LL00 [B T ST S] D DI K.    Here, we see the improvements.   First, the ST scale arrives to the first K&E slide rule.  As mentioned  earlier, instead of an {S1 S2 T} set of trig scales, having {S ST T}  just makes more sense from the perspective of what the S1 or ST can do, namely to give functionality to sine and tangent equally.   And it also puts these scales back on the slide. 

K&E duplex rules, provide some nice added functionality.   The LL00 extends the LL0 scale, splitting exponential bases between 0 and 1 into two scales to give another significant figure worth of precision.  In this implementation, the LL0 is extremely accurate now, covering only bases from .999 down to .905, while LL00 scale handles .905 to 0.    The added DI scale inverts the D scale across the rule, nice for a variety of general math uses.    

Moreover, the organization of the log-log scales are nice, with scales for bases greater than one on the front side of the rule, and scales for bases between 0 and 1 on the back side.   It's really a good design and, as mentioned when I introduced the Log Log Family of rules, the Model 4081-3, in particular, would become a "standard" rule by which others could be compared.  

At this point, there would be nothing new in terms of construction of these rules, as most all duplex rules of this era remain 4cm wide, with celluloid covered mahogany, engine-divided, and sporting the same "New Improved Cursor" as most 1936 and greater rules would have.   Not all is great, however, from the standpoint of the collector, as cursors of this era suffer from a breakdown of the plastic cursor rails (see the first SIDEBAR above).  Thus, while great rules, many of these models can be hard to find with a good cursor (I bought two 4081-3 models before finding a good one).   

Because this rule, in either choice of trig scales, was so successful at the time, both the 10" and 20" 4092 models would be discontinued two years later.  In dropping those rules, the 4080-5 and 4081-5, a 20" rule for each trig variety, would be added in 1939 at the price of $25.30 each, or $27 for the chamois-lined leather case.   As such, these 4080/4081 series rules, in all lengths, became the true functional successor of the Model 4092 series, something that the 4090/4091 abominations failed to do. 

Nothing would change with the 4080/4081 until 1947 - at a 30% to 50% price increase for almost all of K&E's rules - when all versions of this rule in both -3 and -5 varieties would take on the N- prefix when the scales were majorly revised.   The objective of the change would be to split the log-log scales so that inverses could be read directly off of one side of the rule.  As an example, with LL2 and LL02 being on the same side, one could set a number on one scale and read the inverse of it directly from the other.  The utility of this is questionable, though it does give more precision than doing the same with the traditional C & CI scales, or dividing the number into 1.  

As such, the new N4080 and N4081 rules of 1947 would have scales as follows: 

Front Side:  LL02 LL03 DF [CF CIF CI C] D LL3 LL2
​Back Side:  LL01 K A [B T ST S] D L LL1

It's with this configuration of scales that the 4080/4081 model line would dominate the "slide rule era" of the 50s and 60s, except for one minor change in 1954 where K&E would relocate the L scale to the back-top rail above the K scale and add a DI scale to the back-bottom rail.  K&E did drop the N- prefix form the room with that change, but I feel it was a response to match "N-less" model numbers with the new all-plastic version of this rule, the 4181-3, that was likely introduced a year or two prior.  

If you are curious, K&E did change the wood models to match the scale set of the plastic rule.   In 1956, the ST scale would be relabeled the SRT scale, likely to remind customers that the scale could always be used to convert angle measures from degrees to radians. 

There rules would continue being produced, advertised as their "highest grade" slide rules in order to differentiate them from the new all-plastic rules that would arrive as this time.  But it's important to note that when those rules DID arrive with the same functionality in the 9081/4181 (discussed next) and with more functionality in the modern "Decilon 10"  (see Modern Duplex Rules below) for a substantial price savings, there's no question that 4080/4081 would lose some share of the market in this category.   

Visible vs. Laminated Edge Construction

We have explored the majority of duplex rules to this point in the article.  And while construction techniques with the rules have been very consistent, other than the evolutionary changes to the cursors/cursor rails and the addition of plastic in the duplex family,  one of the more distinctive elements you will notice about duplex rules is that sometimes K&E banded the edges of the rule with celluloid - other times they did not.  

Up until 1922, duplex rules were constructed with only the faces of the rule covered with celluloid, with one exception...the original Duplex Family laminated the edge of the rule beginning in 1909.  These versions, those with the "T" suffix, would place the L scale on the celluloid-laminated top edge of the rule with a cursor that could indicate also on the edge.   While that feature only lasted maybe 4 years in that configuration, when the L scale shifted back to the rule's face, K&E didn't bother to remove the edge lamination.   It would stay in this style until 1917 when the entire Duplex Family lineup was discontinued.  

The edge of everything else showed the mahogany construction of the rule.   This would remain true until, in 1922, when the company started using serial numbers; they also began to laminate the edges of the rules with celluloid plastic.   And this is without exception. 

For 30 years thereafter, all K&E duplex rules had laminated edges, where the only place the mahogany could be seen is where the slide meets the grooves of the rails and the ends of the rule where the wooden end-grain is visible.  K&E would put the new serial numbers toward the end of the bottom edge.

And then, in 1952, the edge banding disappeared, or rather was altered to a strip of inlayed celluloid on the edges, where the mahogany shown on both sides of the inlay.  

For the Log Log Duplex and Log Log Vector Families of rule, K&E would clean up the rule faces of all maker-marks (except a new K&E logo at the end of the slide's face) and relocate them to the inlayed celluloid strip on the edge.  Here we typically see the model number, patent numbers, and "Made in U.S.A." black printed text (see image below the Sidebar). 

​At this time, I cannot give a solid reason as to why K&E would go to edge laminations outside of the notion that, perhaps, they felt customers would want it that way.  Or perhaps they just felt it was more aesthetically pleasing?  Or perhaps, with competitor rules like Dietzgen offering complete celluloid-covered rules, they felt some external pressure as well?   

Certainly it was not because they felt it necessary to move the serial number to the rule's edge.  While this is something they did with the edged-lamination, they could have continued that practice when they shifted to the inlays, where instead they used the last three digits of the serial number on the faces of both rails near the ends, as well as on the slide.   Labeling both in this manner assures that the rails and slides are well-mated during production.  

There is some notion that covering around the entire slide rule would prevent seasonal movements of the underlying mahogany, but mahogany is more stable in this regard than most hardwoods (a reason it's used in guitar necks) and edge grain of the wood is less prone to expansion or contraction as the face were it not laminated.  However, I do have some duplex rules in my collection that do show a slight lack of uniformity in rail/slide alignment over eons of time - or even a slight cupping in the edges of a stator/rail.  So it does exist, but the amount at which that becomes objectionable is debatable since any expansion would be latitudinally and not in the direction of the scales themselves.  
   
This could have been a concern of K&E's at which point the company could have taken a "better be safe than sorry" approach.  This notion would support the incorporation of the edge inlay in 1952 as perhaps they did not feel it was necessary any longer?  But more than likely it was because the new inlay design would prevent delamination at the face/edge join of the rule, which is something that many rules of the time did indeed experience (see image below for examples).  

Verdict?   K&E likely felt edge-lamination was desirable and practical for all the reasons I mentioned.  But these were also aspects of their duplex rules that could be sacrificed if unsightly delamination of the celluloid became a bigger issue. 

Delamination at the edges, as shown with the 4088-1 and 4090-3 rules in my collection. If this was happening shortly after they were produced, then the 1952 shift away from lamination of the entire edge of the rule would make sense. It may not affect the rule functionally, but it isn't good for the overall aesthetic.

This is especially true when the 68-1xxx designations came in 1962, the same year the Decilon was introduced.  The 10" 4080-3 would be rebranded as the 68-1318, the 10" 4081-3 became the 68-1210, the 20" 4080-5 relabeled the 68-1308, and the 20" 4081-5 shifted to the 68-1200.  These would be the designations on the rule, though the catalog numbers would show more 68-1XXX numbers depending on the case accessory.  [sigh]

The 68-1308 (4080-5 Trig Model) would be discontinued first after 1966, which should tell you something about which rule was more in demand between it and the Decitrig model (68-1200), which would last until 1972.  Same story with the 10" models, as the 4080-3 (68-1318) would end after 1967, outlasted by the 4081-3 (68-1210) which would last through 1972. ​

The Doric 9081-3 Model

​This Doric model, introduced in 1948 and appearing in the 1949 catalog as one of the three members of the Doric family, was an exact conversion of the Model 4081-3 scales into a plastic rule.  Interesting to me, there was no 9080-3 version of this same Doric rule.   We could make a judgment about the Trig vs. Decitrig choices being made during this era of slide rules.   K&E's own choice of the "decimal degree" version of the rule instead of the DMS version likely demonstrates which version was the best seller.   As such, the Model 4081-3 Decitrig was deserving to be rewarded a doppelganger rule.  

As we've said, the rule itself was short-lived, as it, and the series of Dorics, would be replaced by 1952.  But its own replacement is the one we want to spend time discussing...

The Model 4181-3 Log Log Duplex Decitrig (Jet-Log)

In the 1952 catalog, K&E writes, "Nothing has been found to surpass the combination of selected, seasoned mahogany and xylonite, of which the famous highest quality K&E slide rules are made."    

But if the real intent for the new plastic rules was not known, I think with a little more reading we see K&E's full plan...  

Product positioning is important.    As I've tried to express through the article, any line of consumer goods needs to cover the gamut of possible consumer demand.   Since the inception of their rules, K&E had struggled in trying to lessen the cost of their slide rules in order to meet a demand of the every-man.  By the 1950s, we see the need for slide rules within the markets of the common person, chiefly to students and to workers at normal jobs, and to those who might find it convenient to keep another slide rule in their pockets.   While they would still sell rules to engineers and builders, willing to spend the most for the best, K&E knew that if they could provide plastic rules for ~70% of the price of their "highest quality" rules, and if those rules were functionally equivalent, then they would have positioned all their products to maximize market share. 

As such, the Model 4181-3, first introduced as a repackaged Doric 9081-3 in 1952, would prove to be one of K&E's most important rules.   Priced initially at $15 in the 1952 catalog, compared to the $22.50 price of the N4080-3 and N-4081-3 models, we can see the savings to be had with all-plastic production.

​Shown in the September "Slide Rules" only revision of the 1955 catalog, the rule would be transformed from the familiar Doric N9081 characteristics to that which we would become more familiar, dropping the N- prefix, dispensing with the Doric label, and adding red ink for the labels and the inverse scales.  The catalog description for this rule (and the plastic pocket rules) states that "Ivorite" construction is used.   We see this description also in an edited version of preface that we saw in the previous 1952 catalog (see below).  "Ivorite" is copyrighted here, which might simply mean that K&E, while their rules were already transitioned to Ivorite with the Doric rules, could now be safely called what they wanted all along.   They seem proud... 

Judging between the N-9081-3 and newer 4181-3 versions of this "4181" slide rule, they are slightly different in feel.  The former seems heavier, smoother, and more square, and the latter would appear to be lighter, thinner, and more beveled.  Color of my samples of these rules are slightly different, with the N9081-3 Doric retaining more of a pure white tone, though I would not make judgments whether or not this is true across all samples of these rules.  Regardless, they appear to use a slightly different plastic, both of which are leagues removed from the white Xylonite advertised by K&E as used in their Ever-There rules beginning in the early 1930s.

Of course, as talked about with the Doric rules and as we witness here, it becomes hard to find consensus in what the actual plastic composition is.  My thoughts on this, and a little more about plastics in general, can be found in the Sidebar: A Little About Plastics below.

Sidebar:  A Little About Plastics

In a way, we've been talking about plastic construction the entire article.  Celluloid is a bioplastic, produced from a cellulose substance (like cotton fibers), nitrated (with nitric acid), and then mixed with a plasticizer (like camphor).   This was originally produced in the 1860s.  It wasn't until 1907 when Bakelite, the first all-synthetic plastic, was formed.  

Since then, most plastics begin with a fossil-fuel and then polymerized with a compound of substances, synthetically comprising the plastic being manufacturer.  For example, ABS is an emulsion of acrylonitrile (produced from propylene), a polymerizing butadiene, and styrene.   This, and many other modern plastics, are thermoplastics, meaning that they can be melted, molded, formed, and even recycled to become something else.  

As we talk about the modern K&E rules and their evolution to all-plastic construction, it should be expressed that the actual composition of plastic being used is never explicitly revealed.  This would be true of the plastic industry as a whole.  As plastic became more widely used, their composition and methods would have been a finely guarded secret.  So as collectors of slide rules trying to understand their evolution, it can be confusing for us.  

Our chief indicator is simply what slide rule manufacturers advertise them to be.  Over the 40s and 50s, we see a transference of title from Xylonite to "plastic" to Ivorite in the K&E catalog descriptions.  We know that the "Ivorite" copolymer is a Acrylonitrile Butadiene Styrene (ABS) resin, which has become standardized across industries for a variety of molded components and products ever since.  And we know that "Xylonite," a name first given to celluloid types of compounds (plasticized nitrated cellulose), was a catch-all term used for a variety of celluloid-type evolutions for nearly a century prior.   

Joseph Soper, who began a career at the K&E Salisbury plant in 1966, eventually becoming Plant manager, says in his book, K&E Salisbury Products Division Slide Rules, that the molded Cycolac ABS blanks were produced by a company called American Insulator.  He states that all such slide rules were made of the resin in 1966 and that earlier molded rules would have been made from any of a variety of nitrocellulose-based compounds, likely because they found it difficult to find a place with enough capacity to produce the rules. (Soper, p. 113)   

As such, there would be variance between model lines up until the point at which they could agree on a compound, which probably had more to do with waiting for the plastic industry to catch up to the demand that K&E wanted to fill.  As I speculated earlier in our discussion about the Doric rules, the very temporary aspect of that series could be explained in that K&E wasn't ready (or able) to settle on one consistent formula or supplier. 

Soper confirms that, stating that the ABS resin was in limited use around the country in 1960.  Interpretation of the word "limited" is obviously a matter of degree, so the question does arise, at what point prior to 1960 could K&E find the right supply?  Regardless of timing, it does confirm for us that finally settling on a plastic would have been an evolutionary process.  And with the Doric rules, they all lack a consistent look and feel, unlike the Modern Polyphase and Modern Duplex rules we discuss here. 

For this author, lacking scientific methods for determining exact composition of the rules, the best I have to go by is the feel of the rule in my hands and the way that K&E decided to talk about them.  From that standpoint, it would definitely seem that once K&E settled on the term "Ivorite" in their language use, then we are talking about slide rules of very similar ABS resin composition, in this case from the 4181-3 model up until the latest rules of the slide rule era (1975).  

Scales for this rule were identical to the 4081-3 rule, adopting the scales as it also changed.   The N4181-3 (9081 Doric) version of the rule was based on the 1947 Model 4081 scale set, with the newer 4181-3, that described in the 1955 supplementary catalog, taking on the same revised 4081 scale set that moved the L scale to the top rail and putting a DI scale on the bottom. ​

This new rule would be renamed the "Jet-Log" in 1960, referred to as such in a pocket slide rule brochure in that same year as the big brother of the 4181-1 "Jet-Log Jr.", a very important pocket rule that deserves its own discussion (next).  Two years later it would receive the Model 68-1251 designation.  It would hang around until the end of the era, becoming the sole Decitrig option once all the wooden slide rules disappeared in the 1972 catalog.  Despite filing for bankruptcy and ending all slide rules production that same year, the importance of this rule as K&E's lone successor of the long Log Log Duplex line demonstrates how good this rule had become.  As the end of that line, I felt it important to talk about it here under the Log Log Duplex family of rules.  But it could have very well had been slotted into the Modern Duplex category.   

Model 4181-1 Pocket Log Log Decitrig (Jet-Log Jr.)

The pocket version of the 10" Model 4181-3, this 5" Model 4181-1, in many ways, became the more successful rule.   This is largely because as a pocket rule, it is just really good!

Like the big brother, the Doric version of it was described early on in the 1949 catalog.  But unlike the 9081-3 Doric, the 9081-1 was never actually produced, at least not to our knowledge.   It would come into production in finished form, described in the 1953 Educational Products Catalog, looking very much like a smaller 4181-3 (non-Doric version).   (Note: The 4181-3 in that same catalog is described as the Doric version.)  And by finished form, I mean complete with dual-color scales, red K&E logo, and unbreakable indicator.   There is some indication from internet sources that the 4181-1 initially appeared without a model number on the rule; however, I have not seen to be the case myself and this would also conflict with the aforementioned 1953 catalog that is shown pictured with 4181-1 on the slide.  

The scale set was the same as the 10" version, which made it the most powerful pocket slide rule upon its introduction, regardless of make.  And its for this reason, from 1953 and a decade following, this rule would be K&E's chief competition against the powerful pocket rules being produced in the United States by Post (Pocket Versalog 1461 in 1957) and Pickett (Model 600 Magnesium introduced in the late 1940s).  Priced at $9.25 for the synthetic leather sheath ($1 more for the ungraded leather case), as compared to $15 for the full-sized rule, I'm quite sure there were many consumers that saw the value in going straight to the pocket version of this rule. 
​

Note: The Post Versalog would seem to be a response to K&E's excellent 4181-1 pocket rule, but it would come 4 or 5 years later.  And as a celluloid-covered bamboo rule (by Hemmi), it was expensive to make, priced around $6 more than the K&E rule in 1962.  However, there's an argument to made that the competitive threat was coming from Pickett.  Their Model 600 Pocket Log Log Duplex rule had been in production since the late 40s, and while it's not as powerful as K&Es rule (two fewer scales), it's very likely K&E felt some sense of urgency, and that the 4181-1 was a response to that Pickett model.  The magnesium Picketts were troublesome, which was common knowledge, but K&E wouldn't have viewed any time to spare.  Pickett would replace the magnesium models with aluminum by the late 50s, and the new N600-ES rule would have been very much a competitor against the K&E 4181-1.  

4181-1 Pocket Log Log Decitrig - Front

4181-1 Pocket Log Log Decitrig - Back

As mentioned earlier, 1960 would bring about a new marketing name for the 4181 rules, now referring to them collectively as the "Jet Log" series of rules.  The 10" 4181-3 would be known as the Jet-Log and the 5" 4181-1 would be referred to as the Jet-Log Jr.   Strangely, until 1962 when the rules would receive their 68-1XXX model designations, the version of the 4181-1 with the upgraded sheath with leather-covered clip would be listed as the E4181J.  Even more strangely, only it's successor, the 68-1251 with the upgraded leather case would be called the Jet-Log Jr, with two lesser case options, the 68-1282 and 68-1287, would be known as the "Decitrig Pocket" rules.   These model numbers would not be printed on the actual slide rule. 

In 1968, only the 68-1251 Jet-Log Jr. would be offered, along with the 10" Jet-Log, both outlasting the wooden rules in the Log Log Duplex family of rules, as the classic 4080/4081 legacy wooden rules would be missing from the 1972 catalog.  

​​I find it ironic that K&E would hang on a long as possible to their all-wooden rules, reminding consumers that those were their "highest quality" rules.  In retrospect, it was an effort to convince customers that these expensive to produce slide rules were still worth the highest price in their product line.  K&E seemed to soften from that in 1962, where the catalog became less descriptive and more matter of fact.   By this time, unfortunately for the classic wooden rules, they would not be able to compete with Modern Duplex construction, high quality or otherwise.  The Ivorite constructed rules would prove in the end to be cheaper to produce, and I think quite surprising to many, would still maintain a standard of quality deserving to become their "highest quality" of slide rule. 

But prior to discussing those, there's one more general-purpose wooden duplex rule we need to discuss...

Log Log Vector Duplex Family

This was very much a special purpose slide rule and from that standpoint could be talked about among the Specialty Rules, but staying true to ISRM's K&E Model Map, we will discuss it here with the other duplex rules.  

This history of the origin of the Log Log Vector Duplex is most unique among all other K&E slide rules.  Being a scale set licensed from an individual, there would be decades of negotiations, disagreements, and even a lawsuit over royalties and breach of contract.    (Credit to William K. Robinson for his study linked above.  Much of our understanding comes from this research.)

But what must be said is that the introduction of a Log Log Vector Duplex rule in 1929 was an important development for K&E, not so much because it was necessarily a great seller, but rather what it could do was quite remarkable.  The ability to read hyperbolic functions on a slide rule was nothing new, but the scale set, developed and patented (1924) by Professor M.P. Weinbach of the University of Missouri, could compute hyperbolic functions on the complex number plane, a task at which electrical engineers and students would find extremely useful.   Secondarily, the user could easily convert these functions between rectangular and polar coordinates, a useful feature for a variety of vector operations.   Doing all of this required both hyperbolic trig and standard trig scales on the same rule, since the computations always require a combination of both of scales.    

Weinbach would hold exclusive royalty rights until 1947, which meant that for 18 years, K&E would not have competition in this market from other makers.  Hemmi's Model 255 was made after WWII and could solve hyperbolic trig functions directly, but it was obviously limited in the States.  By 1948, the Pickett Model 4 and Dietzgen Model 1735, both of which had hyperbolic scales, finally challenged K&E for market share.   

Two K&E models spanned the 43 years in which the Log Log Vector rule was offered.  

​The Model 4093 Log Log Vector

Introduced in October 1929 after being on the drawing board for nearly two years, K&E introduced the Model 4093-3 Log Log Vector Duplex.  The rule was constructed much like all other duplex rules of the era, with celluloid-laminated mahogany construction and with frameless glass indicator.   The rule featured the circular K&E logo on the upper right stator and patent notifications on the lower right stator.   Upon introduction, the scale set was as follows:

Front side: L LL0 DF [CF B CI C] D LL3 LL2
Rear side: Sh1 Sh2 Th [SI1 SI2 TI] D S T

For the uninitiated, the front scale set will look as familiar as any Log Log slide rule, bearing some similarity to the Model 4092.  However, the rear of the rule would be like nothing seen before.   An engineer would have recognized the utility immediately, where the only scale on the rear of the rule that is NOT a trig scale is the D scale.    The top stator, with Sh1, Sh2, and Th scales are the hyperbolic sine and hyperbolic tangent scales.   The slide had extended sine scales for small angle trig, like the 4090/4091 that would come 4 years later, but in this case, these scales are inverted.  And a normal sine and tangent scale are below the D scale on the lower stator.

The New "Unbreakable" Indicator

Yet another new feature was being applied to K&E slide rules of a variety of types was something they called the "unbreakable" indicator, appearing first described in the 1953 Educational Product Catalog with their pocket rules.  This new cursor design would use a clear plastic window replacing the glass in their normal "improved" indicator.   

From that point forward, in addition to their pocket slide rules, it would be applied to many others, including the new 4181-3 Log Log Decitrig, their classic 4053 Polyphase Mannheim, the Merchant's rules, and the 4058 Beginner's rules. It was not considered as luxuriously appointed as their improved glass cursors and so it was not applied to their top-end duplex rules, at least not early on.

Introduced in 1962, despite not being their "highest quality rule," the Decilon 10 (68-1100) would go down as their best, most powerful rule. Of interest here is the clear "unbreakable" cursor, a feature introduced in 1954 on all K&E rules except for their "highest quality" wooden duplex rules. By 1962, K&E would use it on all their slide rules.

This was not the first attempt at a place cursor by K&E.  Their first Ever-There rules were equipped with an all-plastic cursor based on the same transparent Xylonite used in their drafting triangles and protractors.   While good transparency is less important with such tools, it's obviously important for slide rules, and therefore use of the clear Xylonite was short lived.  The material was naturally yellow tinted and would become increasingly yellow over time.   In 1936, with the introduction of the new Ever-There lineup, K&E would supply these rules with the improved glass cursors. 

Of course by the mid-50s, suitable materials for highly functional transparent cursors were plentiful.  My guess would be acrylic, which polishes very clear and naturally resists UV light, and thus remains clear over time. 

An important characteristic, being very much a rule designed by electrical engineers FOR electrical engineers, is that the trigonometry on this rule is all in decimal degrees.  As such, this is the first K&E slide rule to use decimal degrees instead of DMS measures, even prior to the first "Decitrig" rule, the 4091-3, Log Log Duplex, which would arrive three years after.  The instruction manual, written by Weinbach himself, supplemented the rule.  

In short, after much debate with K&E about the design of this slide rule, Weinbach got exactly the rule he wanted to produce almost six years after his scale set was first patented.   Upon introduction in 1929, the 4093-3 was $16, or $16.85 for the nicer leather case version.  This was $6 more than the Model N4092 Log Log Duplex, just as reference.   K&E would drop the price to $12 by 1932.   According to Robinson, Weinbach's royalty was 5% per slide rule.  

​A longscale 20" version, the Model 4093-5 Log Log Vector Duplex, was added in 1931. Pricing of this rule made it the most expensive offering among all K&E slide rules, priced at $32 ($33.50 for the good leather).  Of course this is a serious premium for the superior resolution rule.   And it would stay that way, raising to $35 by the 1938 price list.  This made the 4093-5 K&E's most expensive slide rule by at least 25%, even outpricing the expensive specialty rules, such as the N4096 Desk rule and the 4102 Surveyor's Duplex that we will discuss later.   Needless to say, K&E must have felt they would sell at that hefty price.

I mentioned some degree of contention between Weinbach and K&E that would eventually lead to a lawsuit in 1944.  The initial conflict actually started with the Model 4091 Log Log Duplex Decitrig first produced in 1932.   Weinbach did not like idea that K&E was producing another decimal rule, which at that point was a feature exclusive to his Vector rule, as was the extended sine scales.  His concern, somewhat unfounded, was that engineers and students would forgo the more expensive Vector rule for the new 4091 Decitrig.  Of course, K&E felt the concerns were unfounded as well.  Their argument was that what made Weinbach's rule unique was the ability to do complex hyperbolic trig problems, and not necessarily have the monopoly on complex standard trig problems.   However, this could be the impetus for K&E to change to an ST trig scale when the 4090/4091 rules were upgraded to the 4080/4081 models 4 year later. 

This did not make Weinbach waver in his position though.  Although K&E would dispense with the extended sine scales for the 4080/4081 series upgrades, Weinbach also felt that he was owed a royalty for 4 years of the 4091 Decitrig for using mostly identical scales to his rule.  In Weinbach's own notes, he  mentioned that since K&E was using decimally-divided trig scales in the Decitrig rule (half of all his copyrighted trig scales of the Vector rule), that K&E should pay him half-royalties akin to Vector rule contract, or 2.5% per rule (Robinson, pp. 21 & 22).   According to Robinson, K&E would not concede on the Decitrig, but would offer to make the minimum annual royalty payments for this own rules $650 beginning in 1934, which didn't seem to appease Weinbach despite only making around $250 on average the previous 4 years.  In the middle of the Great Depression, and being guaranteed the price of an average car each year, I'm inclined to think Weinbach might have been standing too much on principle.  

However, he would be worn down by the end of the year and conceded to the agreement in December, 1934.  K&E, not completely philanthropic as they tried to appear, did want Weinbach to agree in writing to the loosely held termination date of their royalty agreement in 1947.   

The parties, according to Robinson, would be mostly silent for 4 years.   But in May 1938, Weinbach decided to contact K&E about a new idea for a slide rule.  He was disappointed to be turned down, but there was a "bright side" to the letter.  He was informed that K&E wanted to update the 4093 with a major scale revision and asked for Weinbach to give input.  He was sent a prototype rule to evaluate and was asked if he would update the instruction manual in light of the new rules, which would be called the Model 4083-3 Log Log Duplex Vector (note the added "Duplex" to the name).    

Weinbach loved the new design, submitted a couple of minor suggestions, and updated the manual extensively to provide more examples to demonstrate the increased capabilities of the new rule...​

The Model 4083 Log Log Duplex Vector

The quality that Weinbach loved most about the new scale set for the new 4083 was that it would improve the scales to make it more capable in a general math sense.  He saw this rule as one that non-engineers might actually buy, and any opportunity to cut into the sales of the Decitrig was favorable to him. 

The Model 4083-3 Log Log Duplex Vector went into production mid-1939 at a cost of $13, or $14 with the case upgrade.   Interestingly, a 20" Model 4083-5 would be made ready with the roll-out as well, not something that K&E was always ready to do with their non-standard sized slide rules.    

Of note, K&E asked Weinbach for approval of a price decrease for the larger rule.  According to K&E, the rule could be discounted $5 because the new scale set required much less machine setup as compared to the old rule.  Weinbach approved.  The Model 4083-5 would be priced at $30, or $31.70 for the plush leather case.  

The Log Log Duplex Vector rules in my collection. The lower 10" rule was likely made just before the move to inlayed celluloid edge in 1952. The upper 20" version shows a serial number placing it right around 1962. This slide rule came in a 68-1424 box with a blue/green case, which is the synthetic leather version.

As for the new scale set...

Front side:  L  LL1  DF  [CF  CIF  CI  C] D  LL3  LL2 
Back side: LL00  LL0  A [ B  T  ST  S] D  TH  Sh2  Sh1

The obvious change is the replacement of the SI1, SI2, and TI scales with non-inverted T, ST, and S scales.  Being functionally the same, the new standard trig scales are borrowed from the 4080/4081 rules and would become commonplace for all future K&E rules.  The hyperbolic trig scales were moved to the bottom rail to make room for an A scale, which the earlier rule lacked.   So easy squares/roots could be calculated, as well as basic multiplication and division on the back side of the rule without need to flip the rule over for hyperbolic trig functions.   This general math enhancement, as well as the improvements to the log log scales, are what excited Weinbach about the new rule.   

​With the relationship seemingly repaired, Weinbach easily made money beyond the $625 guaranteed minimum royalty, something that happened the first time in 1936 with the 4093 version of the rule as the economy was leaving the Great Depression.  By 1940, the new 4083 would earn double those royalties for Weinbach, where Robinson estimates in his writing that maybe 3000 rules were being sold on average between 1939 and 1942.  

It's at this point, in 1943, when the relationship goes bad between Weinbach and K&E, under very strange circumstances.  I spare you the narrative of everything here and point you to the Robinson paper, but a condensed version of the story is enlightening...

Weinbach received a visit by a couple of gentleman in 1943, presumed to be from K&E (according to Weinbach's later statements).  The gentlemen inquired about the copyright of the rule, what the contract between he and K&E looked like, and when the agreement would end.   Weinbach offered the information freely, with too much worry about it.   Shortly thereafter, Weinbach revealed to K&E the meeting with K&E in good faith, not really thinking much of it.   In return letter, K&E expressed their dismay, calling Weinbach's disclosure into question, and stated that they were stopping royalty payments immediately.  Promising to reveal their associations later (something that would never be stated explicitly), the two gentlemen were certainly from a competitor, likely looking for information as to when their own company could freely produce a Vector rule of their own.  While not obvious to Weinbach - this shows naivety that I find surprising - it was certainly quite obvious to K&E!

As a consequence, and through a chance run-in with an old friend who happened to be a patent attorney, Weinbach decided to file suit against K&E, not only for breach of contract for withholding the royalties, but also to sue for back-royalty payments for all the Deci-trig sales over the previous decade.  Weinbach never let go the idea that K&E was making money off of his technology without compensation.   Robinson estimates the total profit loss to Weinbach was around $190,000, which was a very large amount of money at that time.  Of course that figure comes from the fact that the Model 4081 Decitrig had become their best seller, and would be for decades later.   This amount also falls in line with the damages claimed in court documents. 

The story does not end well for Weinbach.   Although his agreement was to end in 1947, the number of sales of the Vector rule increased massively with the return of American soldiers after the war - Weinbach stated himself that enrollment in his university program tripled in 1945 - and therefore he would miss out on his largely royalty payments, which would have likely matched his salary from his job.  That same year, Weinbach lost his wife, as well as suffered from health problems that would put him in the hospital for lengths of time.  

K&E, surprised by the lawsuit, responded with everything delay tactic they could, in a way which would greatly increase court costs for Weinbach, but also seek to win the battle by attrition.  We know the lawsuit never went to trial.  Robinson suggests that a settlement was reached, and through interviews he ascertained that Weinbach's portion of the settlement was likely around $10,000, which is a figure that likely replaces the lost royalties of his own slide rule.   Weinbach would die himself mid-1946, with the original royalty agreement ending in 1947.   Not coincidentally, both Pickett and Dietzgen would produce American rules that incorporated hyperbolic trig scales in 1948.  

Author's Aside:  I make no serious judgments about the conflict based on the limited sources we have, though the letters brought to light by Robinson do give me some general impressions about it.  First, K&E was a big business, something easy to see here, and their only concern is the financial ledger.  When I think of all the outsourced products in their entire catalog, the hundreds of contracts and dealings that K&E had made over 70+ years at that point, the significance of one man, one contract, meant very little to them.   Second, I feel that Weinbach acted too often like a man with more resources than he actually had.  Often naïve; often unappreciative; even sometimes greedy.   Robinson estimates nearly 24,000 of his slide rules were sold between 1929 and 1943, with total royalties paid out to Weinbach at approximately $14,000; this, for a very niche product.  Considering the economic era in which these events happened, Weinbach would have had a very comfortable and meaningful life as a college professor.  As a professional educator myself, I would consider myself having the perfect career if I were making another 25% to 50% extra income and the satisfaction of having my intellectual property out there as a service to others.   

This, of course, was not the end of the Model 4083 Log Log Duplex Vector.  1948 would also signal a redesign of the 4083.  Adding the N-prefix to both 10" and 20" models, the rule would shift around the log log scales and add another.  It would also move all notations on the rule's front onto the top edge of the slide rule (see right).   

Note:  We have no price lists between 1944 and 1947.  There 1947 catalog shows a substantial price increase to most of their slide rules, especially with their duplex families of rules. The price of the 4083-3 (basic case) slide rule jumped from $13.50 in 1944 to $20 in 1947 and again to $22.50 in 1948 for the new N4083-3 model.   The 20" version would jump from $31.50 to $48.50 over that same time frame.  Even the nice leather case (an option that puts as "S" suffix on the model number) would cost an extra $3 and $5 for the 10" and 20" upgrades respectively.   While I'm not an expert economist, nor an expert on the post-World-War-2 era, I would find further study of this inflation intriguing.  

Like the rest of the K&E duplexes, the N4053 models would also shift to the inlaid celluloid edge and stylized K&E red logo in 1952, with another $2 price hike as well.    

Another major change to the scale set would occur in 1954, significant enough to drop the N-prefix from the models.   As you might recall, all of the Log Log rules (both Duplex and Vector families) dropped the N- prefix that year, which I speculated earlier might have been because of the new 4181-3 introduction of the Ivorite rule that lacked the N, where K&E revamped the Log Log Duplex rules to match its scale set.   In the case of the Vector rule, the scales were shifted to place the Sh1, Sh2, and Th scales to the top of the back stator rail.  The L scale was removed and a DI scale added below the D scale.   New green cover manuals would also come that year (for all their rules), with an additional 8-page supplement to support the new DI scale.

The following year, in 1955, the ST scale would be relabeled the SRT scale, with no other changes to the rule.   It would be packaged with a 6-page supplement to the manual describing SRT functionality.  

The typical 1962 renaming of the model numbers occurred, with 68-1439 assigned to the 4083-3 and 68-1434 assigned to the 4083-3S (upgraded leather).  New numbers for the 4083-5 and 4083-5S are 68-1429 and 68-1424 respectively.  No changes to the rule otherwise.  Prices are $26.50/$29.50 for the 10" rules and $59.50/$65.50 for the 20" rules in the 1962 price list.  

The 20" Vector rule would disappear in the 1967 catalog, with the 10" disappearing from the 1972 catalog.   A long run for a very good, versatile slide rule, with the most interesting of histories! 

---

Modern Duplex Family

​By the mid-1950s, K&E had transitioned successfully to all-plastic construction for many of their slide rules, keeping the wooden rules around out of tradition, legacy, customer expectations, and cachet.  It is not, after all, like the wooden rules were some how made lesser by the plastic revolution.   They were as they always were, long-lasting, powerful, and beautiful tools.  

But they were not high profit items.   Wooden rules, from the first day, were costly to build, especially in smaller sizes where the labor hours were mostly the same regardless of the size of the rule.   This mean lower profit margins for wooden pocket rules.  The few that K&E did produce over time (e.g. Models 4031, 4053-2, and 4088-1) would likely have been sold nearly at cost, just so they could fill market expectations.   This fact has led some to believe that slide rules were never a very profitable venture for K&E.   I would disagree, instead choosing to say that some slide rules made them money while other ones did not.  

I will be talking about all of the following slide rules in rather glowing terms, just as I discussed rules like the 4181-1 and 4181-3 Log Log Vector rules earlier.  And certainly, being ABS plastic rules, they deserve honorable mention here.   But they, like the rules that follow, have one important aspect in common, that being their lower cost to produce.  If K&E did not find slide rules all that profitable BEFORE this time, then they certainly would have profited afterward.  

Because the new 68-1XXX model number convention of 1962 lacked all manner of intuition, K&E helped customers out by publishing this conversion list in the 1962 catalog.

That said, the performance/value ratio of the Modern Duplex family rules are off the chart.   K&E could have sold these rules at prices near (or above) their "highest quality" wooden duplex rules.  Instead, we will see that these prices will greatly undercut those wooden rules.   

But before we discuss the three rule in this family, its important to note that K&E was preparing for another rule to go into production in the 1970s known as the "Ke-Lon."   Only a mockup prototype of this rule exist, but Ke-Lon, prototyped on a Deci-Lon body, was essentially a Deci-Lon with three added scales for hyperbolic trig.  All "Lon" scales would have moved to the front of the rule.   The idea of this slide rule would have been to produce the "ultimate" slide rule in the sense of unifying all previous Duplex families into one flagship model.  However, in my opinion, it would have been just for cachet only, impractical in the sense that few people would find the Vector functionality useful.  But K&E did drop the Log Log Vector family in completely in previous years, so perhaps it had sense to them to revisit them on this new proposed slide rule?  No matter, K&E would be in worse financial trouble by the early to mid 70s, so this rule never really had a chance (see Sidebar: The End of an Era).  

But at least the slide rule market was still good for K&E in the 60s.   So let's talk about these Modern era rules that DID find success...

The Model GP-12 (68-1565)

"GP," meaning "General Purpose," it is clear what K&E intended with this slide rule: to supply a higher-quality duplex rule than the K-12 Prep rule, with more capability, but not equating to a large price tag.   The 1966 K&E catalog, the year it was introduced, clearly intends this toward general math/business/science applications.  Note: The rule is marked with a 1964 copyright, so a 1966 introduction might not be totally accurate, lacking a 1964 or 1965 catalog. 

Without knowing this upfront, it would be one of the most curious slide rules of the modern era in my estimation.   It might still be.  Regardless, it is certainly a very functional and attractive rule.

The Model GP-12, also mostly unknown as the 68-1565, is of a duplex design, made of Ivorite, but it would be considered a simplex rule (a duplex rule using only one side for scales).  This is only mostly true, however, as we will see.  It has a lighter feel than other full-size plastic rules, like the 4181-3 Jet-Log and the Deci-Lons (described next).  This is largely due to the use of plastic end brackets rather than metal. 

The top and bottom edges are grooved to accept a single-sided, clipped-on cursor that rides in the grooves, but it could have easily been supplied with a full-duplex cursor if they wanted to put indexable scales on the reverse side.  Instead, the back is populated with useful formulas and conversions...with a couple of surprises!   

Front Side:  Sq1 Sq2 DF [CF T SRT S CI C] D A K
Back Side:  single decade log scale [centimeters  inches] L scale

For the front side, it seems like it was created by taking as many scales as you can off of the Deci-Lon rule to fit them into a one-sided rule.  Perhaps it's no coincidence, but the arrangement seems to pattern the Aristo 901 Junior  {DF [CF CIF CI C] D A K} only with the double square root and trig scale thrown in.  

But the back side of the rule is quite different.  It is populated with "Conversation Scales and Factors," but it has a logarithmic scale on the top rail and an L linear scale on the bottom, with centimeter and inch rulers along the edges of the slide.   These scales, however, are not calibrated to the front face, nor is a cursor needed.   How they work can be seen below...

The back of the GP-12. Note the two uses. On the left, note the inch/cm indicators on both rails. Here the bottom indicator is aligned to 4 inches, whereas the top indicator reads ~10.15 centimeters. So, the GP-12 allows for direct inch-to-centimeter conversions. Or, since these scales are on the slide, it can be removed and used as a 10" ruler. On the right side, note the indicators are now on the slide, pointed toward the scales on the rails. As set here, the number 3 is indexed to the bottom log scale. In this case reading ~.477. As such, the log(3) = .477. Experienced users will also recognize that .477 is the mantissa for the log (3 * 10^x). As such, log(30) = .477 + 1 = 1.477, log(300) = .477 + 2 = 2.477, or log (.3) = .477 - 1 = -.523.

As such, the rear of the rule is more than merely conversion tables.  It provides what is essentially an L scale for computing base-10 logs and exponentials, and gives easy inch/centimeter conversions.  And while it's not designed to do so, because the linear scale on the bottom rail is divided up in inches like the ruler above it, it can be used to do easy addition (and subtraction).  For example, in the right image above, the rule is positioned to see that 446 + 925 = 1371, among other sums involving 446.  

Price for the GP-12 is unknown since there are no published price lists for anything later than 1962.  However, in 1962 prices, there is a large gap between the K-12 Prep rule at $2.25 and the 4181-3 Jet-Log at $21.  Those being the two least expensive full-sized duplex rules, K&E likely felt this slide rule was needed to fill that gap.   If I were to estimate a cost upon introduction of the rule in 1966, I would place it roughly equivalent to the Model 4161-3 (68-1576) Modern Polyphase in terms of materials and capabilities.  This $11 classic Mannheim-designed slide rule (in 1962), also with all-plastic construction, had 12 scales (as did the GP-12, disregarding the back side), yet it was very much a different rule.   So I feel like K&E would have naturally priced the GP-12 around $11 to $13 in 1966 without too much concern that it would steal sales from the Model 4161-3.  

The more that the GP-12 is used and held, the more that this user can appreciate the overall utility it provides.  I find the rear of the slide rule to be more useful than I first imagined, perhaps because of its simple functionality.  Additionally, the rule is really attractive, feels great in the hand, and it gives the impression that it's more akin to a top-end rule than it's price would have certainly been.  And speaking of top-end rules...

On the short list of the best pocket slide rules ever made - the Deci-Lon 5 as introduced in 1962.

The Deci-Lon Series

Shifting to a new model numbering scheme wasn't the only thing that K&E accomplished in 1962.  They also introduced, arguably, their best slide rule in the history of the company.  It came in both pocket and full-sized versions; the Deci-Lon 5 68-1300 and the Deci-Lon 10 68-1100. 

The Deci-Lons were the product of everything K&E had learned about making slide rules.  From the modern Ivorite plastic construction, unbreakable cursor, and a very powerful, 26-scale set, these functional rules could have very well taken over the entire industry.  But what made this slide rule really wonderful is its form.  This was K&E's exit from modernity, coming to age into what a future slide rule should look like. 

This rule throws out tradition entirely.  Gone are the right angles and the conservative approach to design. 

​Now, angular, deep-bodied, muscular, and even angry, as if it has something to say.  Colored vertical lines on the ends of the slide declare, "Hold me here."   We've seen K&E produce innovative rules before, even giving them bold names like Decitrig and Jet-Log, but this shouts its name on the rule itself. 

Deci-Lon!

Exploring the scale set, we have 13 scales on the front and back each: 

Front side:  Sq1 Sq2 DF [CF CIF L CI C] D Ln0 Ln1 Ln2 Ln3 
Back side:  Ln-3 Ln-2 Ln-1 Ln-0 A [B T SRT S C] D DI K

There is nothing new here.  It's all been seen before...just not THIS MUCH of it! 

Basic arithmetic operations are handled with ease with all the Polyphase Mannheim scales.  Efficiency of these operations are improved by a full suite of folded and inverse scales.  Squaring, Cubing, and rooting functions can be found with high precision with the A, K, and a pair square root scales.  Trig functions are handled by the full {S SRT T} set, in Decitrig divisions (hence the "Deci-" in Deci-Lon).  Exponentials and logarithms powerfully executed with eight log log scales based on "e" known as Lons (hence the "-Lon" in Deci-Lon). 

It's the perfect, complete general math rule with a cool name in the prettiest package imaginable.

But such a slide rule would not be useful for many if it were priced out of reach.   Whereas one might think that K&E's best slide rule would command the highest price, this is not the case with the Deci-Lon.  Being a modern rule, based on modern, finely molded plastics and inexpensive production techniques, the rule could be offered at a much lower price point than their traditionally "highest quality" mahogany-based slide rules.   The 10" Deci-Lon 10 68-1100 entered the 1962 catalog with a $25 price tag.   This is $4 greater than the Ivorite 4183-3 (68-1251) Jet Log and $3.50 less than the wooden 4081-3 (68-1210) Log Log Duplex Decitrig.   So from the standpoint of usefulness, I'm not sure how the venerable Decitrig could compete?  But perhaps it did not need to?  The Deci-Lon was most certainly a hit.

As for the pocket rule, the Deci-Lon 5, with a 68-1130 model number, it was priced exactly half the price of the full-sized rule at $12.50.   This would have been a very popular option, as by the 60s pocket rules were very much in vogue, as the longer "slip-sticks" weren't as popular as they once were.  And the 5" Deci-Lon gave buyers even more reason...it's every bit as powerful as the larger rule.  Where some precision is obvious lost, many would believe that the trade-off in portability, and the huge discount compared to the 10" rule would have been an easy choice.   And keeping in mind the popularity of the 4181-3 Jet-Log, K&E would now have a complete array of six pocket rules, all reasonable priced, by the time of the Deci-Lon in 1962.  

Keep in mind this is a long way from the days when K&E offered a wooden pocket rule (Model 4031) at the same price of its full-sized version (Model 4041).   Because production for the modern plastic rules are quick regardless of size, then they can be priced more in-line with the cost of actual materials used.   This was not the case with the wooden rules, which required longer production times across the board, which meant the pocket rules also had to be priced to cover the labor involved.   When you recall that very few pocket rules were made by K&E until the 50s, and that one of their longest running models, the 4053 Polyphase Mannheim, didn't receive a pocket version until the Ivorite rules in the late 50s, then it should be clear the advantages of plastic when 6 pocket rules appear in the 1962 catalog. 
  
The Deci-Lons would stay in production for a decade, largely unchanged, until the early 1970s when K&E removed the vertical stripes (grips) from the ends of the slide and ceased putting the actual model number on the body of the rule.  The latter point is not unheard of, as there were many instances in the company's history when a slide rule did not show a model number.  But the disappearance of the vertical stripes, which in my opinion are one of the interesting and distinctive features of the Deci-Lons, obviously requires some reasoning.  The popular story there is that the mold for one of the Deci-Lons was damaged on one of the slide's ends, so instead of fixing the mold, they dispensed with the vertical stripes altogether.   Honestly, I am not sure I concur with this explanation as I write this, since a new mold would be needed to omit the stripes anyway.  But I have a right to change my mind upon further research.  

---

Sidebar:  The End of the Era

If you were an engineering student at a university in 1972, you would have entered your first year with a slide rule in hand.  By the time you graduated, your slide rule would have been stowed away in your desk, giving way to the new Hewlett-Packard or Texas Instruments hand-held calculator that you were required to buy.  

The end of slide rules happened that quickly - and without remorse.  There was no asking whether or not switching to electronic calculators was a good idea?  Or whether something would be lost by the dismissal of the slide rule?   All that mattered was that computations formerly done on the slide rule could now be done with 8 digits or more of precision, all without having to track the decimals yourself.   

Speed?  People of the era would say that they could do computations on the slide rule just as quickly, but being a generation removed from the era myself, and only now learning my way toward slide rule proficiency, the calculator is most certainly faster and more convenient.  

Precision? How precise do you really need to be?  We round our answers routinely anyway to the hundredths or thousandths anyway. 

As a high school math educator for 28 years, I now see where we have missed the slide rule pedagogically.  Numeracy among students is far worse than it ever has been.  My students give little regard to magnitude and significance to the output of calculator computations, not knowing if they got the question right until the see lack of red-ink on their graded exam.  

Much of this is could have been prevented during the analog to digital transition.  Traditional pedagogical methods could have held, withholding calculators from young students until they understood the principle of "garbage in, garbage out."   Our school system could have hammered the rules of numbers and their operations solidly prior to letting them be lazy with a calculator.  Or better yet, educators could have kept teaching children the slide rule to give them a better sense of numbers, operations, proportions, fractions, arithmetic rules, and reasonableness of solution within math problems.  

I recall a time early in my educational career when the importance of "manipulatives" were stressed, to supplement understanding of material with hands-on, visual reinforcement.  And while today's "apps" on our smartphones can somewhat do this, it's in no way standardized in an educational system still trying to push TI graphing calculators on the kids, all because of a huge lobby at the state level to assure that they do not go away.  Where were the slide rule lobbyists in 1973?  

Despite going public on the NASDAQ in 1965 and advertising their biggest sales year in 1966, Keuffel and Esser rapidly declined shortly afterward.  Slide rules were in no way their big money maker, but these were still going strong into the 1970s.  It was their technologies in other fields that were being supplanted by improvements.   All of their traditionally analog products were quickly becoming obsolete.  Their surveying instruments, transits, and theodolites were giving way to digital equivalents, which would be 5 to 10 years ahead of pocket calculators.  So while we tend to think the digital world began with pocket calculators, that is only true for the general consumer like us.  Within industry, digitalization had already begun it's transition. 

K&E would continue on making slide rules until 1975, selling from existing stock in 1976.  In 1982, they filed for chapter 11 bankruptcy.  They hung around for 5 years before selling their existing intellectual properties and products to the Azon Corporation in 1987.   After other transactions through the years, K&E's optical tooling technology found its way into the hands of Brunson Instruments Company, a competitor to K&E in that field since 1927.  Today, Brunson still sells a line of products based on the original K&E designs.  

Many of K&Es products have become collectibles, not only their slide rules.  But while K&E would always reap the most profit from the rest of their consumer goods, they would be best acknowledged by the general American public for the long history of excellent slide rules.  

The Analon (68-1400)

One of the more unusual slide rules ever produced, and certainly the pinnacle for slide rule collectors, the Analon was introduced in 1967 as a slide rule for doing dimensional/numerical calculations within Engineering-Science applications.  Simplex in design (duplex with single-side use only), the Analon was very similar to the GP-12 in build, with gold-painted plastic bracket instead of the GP-12's black.

With A, B, C, and D scales in their traditional places, the rule was populated with 7 other scales: three U scales on the slide and 2 V scales on each of the rails.   These U and V scales had any of 30 different dimensional and physics variables (with a legend for these on the back side of the rule).  These could be used to confirm the formulas that comprise those variables.  In other words, it helps keep dimensional units straight when you do lengthy computations.  At a more basic level, a formula like F (force) = M (mass) * A (acceleration) can be confirmed on the Analon by placing the C-index at "M" on the bottom rail's V scale, moving the cursor to "A" on the slide's U scale, and reading "F" off of the same V scale, thus confirming the formula.   

Without getting too much into the design history and theory, to make this happen with 30 of the typical variables in science & engineering fields, while keeping all the scales spaced out and readable (increasing accuracy), was quite the feat.  As such, the scales are no longer logarithmic, but rather affixed linearly with a confined 10" space.  In other words, it's easy to image starting with scales from scratch and positioning the F, M, and A variables on the rule so that they perform the same example above.  But how do you place 30 other variables for all the other formulas you might need to confirm while keeping them equally spaced on the scale and, importantly, keeping them on the 10" rule?    For more on this, I encourage you to read Cliff Frohlich's article for the Fall 2014 Journal of the Oughtred Society.  

Design considerations aside, as remarkable as they are, it's the collectability of this rule that has made it popular.  In truth, it is a "holy grail" for collectors.  I still do not own one as of November 2022, though I've passed over  a few that have come up on eBay, as I'm not ready to spend $300 or more on one.  But why so expensive?  

The Analon was only produced for one year, 1967, in very limited supply, perhaps 600 to 1000 samples.   It is said that double that amount were manufactured, but half of them did not pass quality inspection due to errors with the painting of the rule.   When you couple so few samples of the Analon with its very unique design, its something very desirable for collectors.  For myself, I will remain hopeful to stumble onto a bargain somewhere.

In a very large sense, this is a specialty rule, and had K&E continued in its manufacture after 1967, the question becomes if it would have caught on.  In Frohlich's article, after listing six specific flaws that might have ended the Analon's production, he summarizes it for us thusly: 

"Nevertheless, the design of the Analon was highly innovative. If calculators had not replaced slide rules, other dimensional slide rules probably would have appeared, possibly targeting specialized audiences, such as chemists and even earthquake seismologists. Undoubtedly slide rule manufacturers would have experimented with other layouts and designs. Because there is a certain ‘geek market’ attracted to slide rules with numerous scales, dimensional scales-either like or unlike the Analon’s U and V scales might have been added to some of the more complex slide rules. However, these events did not happen, and the Analon is unique—there is nothing else like it."       
​- Cliff Frohlich, Journal of the Oughtred Society, Fall 2014, p. 27.

Had the Analon not been produced so close to the end of the slide rule era, I could imagine that other slide rules would be produced, the K&E product line would have shifted, and we might be thinking today of a different way of categorizing all of their slide rules historically.  But despite being anything but a general-purpose rule, it was a Modern Duplex family very much akin to the Deci-Lon and GP-12.  It made sense to talk about it here, instead of among the many K&E slide rule descriptions left to cover...

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The Specialty Rules 

The majority of slide rule products sold by Keuffel and Esser in the 19th century were actually highly specialized tools already developed by others.  This was due to K&E's growth focus on becoming a sales distributor of products they felt aligned with their major markets of engineering, construction, and design.  And keep in mind that it wouldn't be until the 20th century until K&E felt there was a market toward the every-man consumer.  They built their foundation on providing quality products and tools to industry, those that would be building and rapidly expanding the U.S. after the Civil War. 

To this point in our survey of K&E slide rule products, we have covered only those slide rule products that resonate with those who practice general mathematics.  While there were certainly engineers in need of a Log Log Duplex slide rule, there are a wide variety of specialty products and slide rules that must be addressed in our effort to be complete.   As we discuss the Merchant's Family of rules, which comprises the 4094, 4095, and 4096 models, it will be easier to discuss them in a narrative form rather than by a rule-by-rule breakdown...

The Merchant's Family 

In 1915, K&E introduced a new duplex rule for those that might be confused by so many scales.   Seriously, this is what they state in the catalog.  They suggest that businessmen, accountants, merchants, mechanics, foremans, and the like need a rule that doesn't detract from what they need the slide rule to do, namely "multiplication, division, and proportions."   The Merchant's rule, Model 4095, was designed to do this.  10" long and built with the same form factor of the Model 4088 Polyphase Duplex introduced just two years before, the 4095 sports a very simplified scale set of DF [CF C] D on the front side and only two scales, the CI and D on the back.   

Of course, this scale arrangement focuses only on C and D scales, which are the multiplication and division scales.  These are enhanced by FOLDED scales on the front side, which allows any such computation to use either index and assure that a product or quotient is not "off the rule," a common happening on rules without CF and DF scales.  And naturally the C and D scales can do conversions and proportions by virtue of being one decade logarithmic scales.  The back side, with only a CI scale on the bottom of the slide and another D scale on the bottom rail, allows users to invert their multiplications to work more like a division problem. 

The idea of only two scales on the back was to allow users who do commonly performed conversions or computations blank space on the rule to mark their own "gauge-points" to align with the index marks.  Seems like a great idea, though it does make for a sparse presentation on the back side of the rule. 

The Model 4095 came with a frameless glass indicator and a "Morocco," synthetic leather case at a cost of $4.50, which was also the cost of the Model 4041 Mannheim.   For further perspective, this is the least expensive duplex rule, with the Model 4088 priced at $7.00 and the Model 4092 at a dollar more.  The Duplex Family of rule, two years before being discontinued, were also available in 1915 at 50 cents more for the base version and $2.00 more for the "T" or trig version.   The rule also beat the Model 4053-3 Polyphase Mannheim on cost, coming in at 50 cents less.  

In 1922, the year of the introduction of serial numbers and the celluloid-wrapped edges, the 4095 was renamed the Model 4095-3 and 5" and 20" rules were added, the Model 4095-1 and Model 4095-5 respectively.   The 4095-3 was $5.50 that year, with the case upgraded "S" version at $6.35.  The 4095-5 and 4095-5S were $13.00 and $14.50.  The 4095-1 only came in as an S version, priced the same as the 10" model (with basic case).   This 4095-5 rule would only last 4 years and is a quite rare rule, with only two of them popping up on eBay since 1999.  

A year after the 4095-5 was discontinued, it would return in a different guise in 1927 as the Model 4096 Desk rule for $18.  Enclosed in a nice wooden case, and able to be worked IN the case, the Mannheim-style rule was much larger in cross-section than the typical Mannheim, with attached metal feet that lifted the rule off of the table at an angle.  A knob on the slide allowed for slide movement.  The scales were a repeat of the Model 4095's front side, with DF {CF C] D.    There were no scales on the back of the slide.   In 1930, the desk rule would become the Model N4096 when it added a CI to the middle of the slide.   This made it a Merchant's model, though it wouldn't not be classified as such in catalogs. 

Also in 1930, the Merchant's Model 4094 was born.   This was a standard Mannheim build out of celluloid-covered mahogany, like the current Model N4041-3, only with the Merchant's scale set.   In fact, it was essentially the same rule as the N4041-3 in all regards except the scale set, DF [CF CI C] D.  Inch and centimeter rules were on the top and bottom celluloid-covered edges of the rule.   There were no scales on the back of the slide.  Price of the rule was $5.00, which was 50 cents less than the N4041-3 for that year. 

For 1939, K&E would add the Model 4096M, which was identical to the 20" desk model without the metal stands and metal knob on the slide.  Thus, it was hand-held.   Priced at $15 in a Morocco case, which was $5 less than the desk model for that year. 

1943 would signal the end of both remaining Merchant's 4095 models as well as the end of the N4096 desk version.  The 4096M actually was rebranded as the "desk" model, but without the case or leg stands.   It, and the Model 4094, would remain until they were discontinued in 1947.   Strangely, these slide rule appeared in the 1947 catalog, but indicated that their were "temp. disc."   The Model 4094 would indeed return in 1948 and the Desk Model 4096M returned a year later, along with the old N4096 version on the metal stands and with the desk presentation case.  Albeit, once the case version was reintroduced, the 4096M version disappeared the following year. 

There had not been a pocket Merchant's duplex model since 1942, and never a pocket model on a Mannheim build.  Of course, in 1949, this is where the Merchant's Doric rule is first introduced.   The 5" Model 9050-1 Doric filled that gap nicely, though as I mentioned in the discussion about the Dorics, this version would not hang around long, being reclassified in 1950 as the Model 4150-1.   This was the same rule as the Doric, keeping the Doric label on the rule for a couple of years prior to dropping the label altogether.   At this point, the 4150-1, 4094, and N4096 would remain in some form until 1972 when all but one of the Merchant rules disappeared.  

In 1954, the 4094 became the N4094, taking on the same transformation as the Model 4053-3 Polyphase Mannheim, gaining a plastic body with printed on conversion charts and the unbreakable cursor.  

In 1962, these rules would receive their 68-1XXX series numbers. They would also be marketed as "Business" rules, with the N4094 being the only "Merchants" trademarked rule.  The 4150-1 and N4096 would have two case options each.   Prices at this point were $5.25 for the 4150-1 (68-1791) and $6.25 for the 4150-1C (68-1786) with clipped case.  The N4094 (68-1775) cost $16.50 with only a synthetic leather option. The Desk Model 4096 cost $29.50 for the 68-1754 with synthetic leather case and $39.50 for the 68-1749 with wooden presentation case.   The 4150-1 would be the only rule to make it past 1972, however.  It would be sold until the end of the slide rule era in 1976. 

The Stadia Family

Next to the original Mannheim family of slide rules, the earliest offered by K&E are rules from their Stadia Family.  Recall that early demand for K&E products was from an industry that was actively building and expanding the country, which meant that land surveying was big business.  A method known as the "stadia method" for surveying was the most advanced method used up until the technological age of laser range-finders and GPS, requiring optical tools for which K&E, of course, would provide.  With these tools, a survey crew could measure the height and distance of any object given any "line of sight" to which the optical system is capable.   The stadia method is still in use today, as the method is simple and accurate enough for a variety of range-finding and topographical applications, without having to spend thousands of dollars on automated surveying tools.

A lengthy understanding of stadia surveying is beyond the scope of this article; however, perhaps we can gain quick understanding of the process in order to know how to use most any Stadia slide rule.   The important variables and factors are shown below, courtesy of Mike Syphers' amazing and comprehensive website.

In order to find both the height and distance of an object, four inputs are needed, with one derived from the first three observations: 

1.)  The angle theta, given by the tripod of the sight device. 
2.)  The angle alpha, given by the fixed measure of the optics, typically .010 radians.  This the fixed angle between two stadia marks in the optical field of view. 
3.)  The delta-y, given by the linear measure of a stadia rod/board with gradations measured in inches, judged through the sight device between the upper and lower stadia marks.  
4.)  The measure of "R" or "r" computed by dividing the delta-y with the angle theta.   This is a simple inverse-tangent function. 

On a slide rule, certainly on any of the K&E rules, an R scale for the sight distance will typically be on the top rail, with an A scale on the bottom rail.  Both these scales are the same two decades of logarithmic scale, though A is marked 1 to 100 and R is marked 10 to 1000.  Given such a rule, situations between 10 and 1000 feet are required.   On the slide are a V scale on the bottom and a combination H (right side) and V scale (left side) at the top.  These signify "horizontal" and "vertical."   The V part of the scale on the upper slide is for measuring large angles of theta up to 45 degrees, with entire bottom V scale allowing for small angles, yielding the ability to measure possible vertical angles in degrees and minutes.   The H scale runs backwards from 0 to 45 degrees, with limited resolution.  

The scales are derived from the formulas for horizontal and vertical displacement (defined later).   The scales increase in angle from left to right, marked in black, but the H scale also has angles marked in red running from right to left.  The black and red angles meet at 45 degrees around 1.5" inches from the right of the scale. 

To use the stadia rule, you would first compute the R input from the inputs given you by the survey measurement, which is the amount of feet measured on the stadia rod divided by the .010 radian standard of the sight device.   An easy example is if delta-y is 2 feet at an angle theta of 15%, then R would be 2/.010 = 200 ft.    On the slide rule, the right index of H/V would be set to 200 on the R scale.  The cursor can be set on the H/V scale for the angle theta (15 degrees in our example).  When set on the LEFT side of the 45 degree mark, the resulting reading (off the R scale) of 50 feet is the vertical height gained over the distance.  When 15 degrees is set on the RIGHT side of the 45 degree mark, the resulting reading (off the R scale) of 187 feet is the horizontal distance to the site being measured.   

For smaller angle of theta, the V scale would be used, as it is finely etched for angles of 3 minutes of arc to around 5.5 degrees.  When this is used, the height is measured off the A scale, at better precision than using the H/V scale.   For smaller angles of theta, the horizontal distance would approximate whatever R was, so that would not be computed on the slide rule.  

More on the math:  When given the hypotenuse (R) of the larger right triangle (in the above diagram), finding distance would typically be R * cos(theta) and height typically R * sin(theta), but the measure of delta-y assumes a perpendicular measure to the line of sight.   Unless it is a level measurement (0 degrees of rise to the target), then there will be error caused by larger angles of theta.  As such, the computations for height and distance change according to some reworking of the trig, a process known as a "reduction."  And this is where the stadia rule is designed to help, as it will compute the actual vertical ((R * (cos theta)^2) and horizontal (R * cos theta * sin theta) displacement, or height and distance.  On most of the K&E rules, this will be labeled clearly by "cos-squared-theta" on the left side of the rule, to remind users that measuring to the left of 45 degrees is the vertical reduction and is labeled "1/2 cos-theta-sin-theta" on the right side, indicating that measures to the right of 45 degrees measures the horizontal reduction.

As such, a crew of two men, one operating the sight device and the other holding the stadia board can map out the topology of any area from any single spot within a sight radius of 1000 feet.   Math is a wonderful thing...and a stadia rule makes it better!

Armed with this discovery, let's look at members of the Stadia Family over the nearly 90 years slide rules were produced by K&E. 

The Model 1749 Mannheim Stadia

​K&E's entry into stadia rules occurred in 1895 with this 20" Mannheim-style rule, Model 1749.   Several items were designated the 1749 at the end of the 19th century, including another stadia rule, their beginner's rule, two sewer rules, and their sector rules, all with dash indicators.  Being the first 1749, the Stadia rule was dash-less.  This slide rule, of course, gave way to the 4XXX series of in-house rules in 1901.  

Priced at $13.50, which was $3 less than the 20" Mannheim Model 1746 they sold at the time, the 1749 Mannheim Stadia was most certainly constructed on the same frame, with celluloid-faced mahogany (probably), imported from Dennert & Pape.  At least this is the likelihood since there are no known lasting samples of this nearly 130 year old slide rule. 

The scale set is unknown, though based on drawings in the 1901 catalog of the successor Model 4100 Stadia rule (see next), as well as known samples of that rule, the 1749 was likely the same.   Instructions, as on the 4100 model, were likely printed to paper and affixed to the back of the rule.   The back of the slide would have been empty, although later models will add a B and C scale for doing general math computations and proportions.  As such, this rule, and all stadia rules until 1913 (when the B and C scales are introduced), would have not included a cursor.   

The Model 4100/4101 Stadia Series

In 1901, the Model 1749 Mannheim Stadia was re-designated as the Model 4101 Stadia.  This 20" model would endure until 1952.  But it would be the 10" version added that same year that would become K&E's longest lasting slide rule, being produced in some form until 1960. 

The scale set, like the forerunner Model 1749, was R [V/H  V] A.  Nothing changed in the way the rule worked.   Directions were spelled-out on the back, printed on paper stock, and glued to the rule.   This rule was also blank on the back of the slide, and was thus without a cursor, as once the index of the slide was set to the stadia measure's R-value, then it was easy enough read the rule from the measured elevation angle. 
 
The rule would change in completely in the 1913 catalog, advertising a reversible slide with a new B scale on the back, located on the bottom of the slide.   This, when used with the two decade A scale on the bottom rail gave the rule a general purpose application for multiplication, division, and proportions.  For the first time with a stadia rule, this catalog pictures and describes a glass indicator.   This cursor would have the metal frame.  

Both rules were offered in a Morocco covered case for $4.50 and $12.50 for the 4100 and 4101 respectively.  K&E did sell cases a la carte at this time, offering a sewed leather option for $.90 and $1.40, as well as a sewed case to accommodate a magnifying indicator.  The cases were $1.40 and $2.10 for any 10" and 20" rules.  The magnifying indicator for these rules was an extra $2 and $2.50.    Other than adding the the frameless glass indicator in 1914 (metal rails) and 1915 (plastic rails), both rules would remain unchanged over the next dozen years.  

Both rules would be greatly improved for 1925.  The Model N4100, priced at $6.50 and sporting a new N- prefix, added an HC scale in the middle of the front slide and a C scale to the back slide.  

The C scale is obvious.  As a single decade, it would now work with the A scale for squares and square roots.   The HC scale, however, is not so obvious.  This scale, very smartly I might add, allows for more accuracy with horizontal measures, akin to what the small angle V scale did for vertical accuracy.   It too, like the bottom V scale, is keyed to the A scale.   It is laid out with angles from ~ 1 degree, 48 minutes up to almost 18 degrees, 25 minutes.  This works as an offset to the stadia measure (R value) used.  As an example, if the reading is for 5 degrees elevation (theta) at 500 feet away (R-value), then setting right of 45 degrees to 5 degrees on the H scale, there's around a foot of error for the horizontal reduction.  However, when setting the cursor to 5 degrees on the HC scale, the A scale reads ~3.77.  As an offset, this number subtracts from the 500 feet R-value to yield a much more accurate horizontal reduction of 496.23 feet.   Remarkable!

As a further bonus, the rule would also add celluloid to the edges, with an inch scale on top and a centimeter scale on the bottom.  Because general math computation are read from the two decade A and B scales, resolution suffers compared to the normal Mannheim, but with the added utility of this N4100 Stadia, it would have been hard to be disappointed.  

The 20" would not add the N- prefix until the 1928 catalog, and while the 1925 catalog states that the 20" rule is identical to the 10", there seems some discrepancy with known samples of the rule, as the 4101 appears to have waited until 1928, not only for the new prefix, but also for the new scales.  I cannot concur however, since the price of the Model 4101 did not change from the 1925 price of $16 when the N-4101 was introduced in 1928.  Curious indeed.  

There were S options for both rules in 1928, with the N4100S costing an $.85 premium and the N4101.  

Other than the typical cursor improvements in the mid 30s, both rules would remain unchanged for two decades.  As mentioned, the N4101 would show as "temp. disc" in the 1949 catalog and subsequent price lists over the next three years.  While some rules come back from that status, the 20" version of this rule never did, but apparently there was enough back stock to sell until 1952, as known samples age to that year.  

The N4100 would continue, but it would see a similar conversion to a semi-plastic in 1956, just as did the 4053 Polyphase and the N4094 Merchant's rules did in 1954.  Essentially, all three of these rules were identical by 1956, with plastic bases.  The N4100 dropped the N- prefix to reflect the major change.  It would also print the identical Conversion Factors onto the back of the rule, with stadia instructions now placed into the front channel beneath the slide.  As strange look, indeed.   Checking price of all three of those rules in 1956...the Model 4100 was $14, which was 50 cents more than the 4053-3 and a $1.50 more than the N4094.  Both the Stadia and Polyphase rules had "S" upgrade options to the chamois-lined leather case for $3 more.  

Apparently, the Model 4100 was discontinued around 1960, but we have no catalogs or price lists between 1959 and 1962 to confirm this.   Introduced around that time was the Model 4143 Kissam Stadia.  It was to the Model 4100 as the 4161-3 was to the Model 4053...an all-plastic spiritual successor.  But the first mention of the rule wasn't until the 1962 catalog when it would be called the Model 68-1486 Kissam Stadia (4143).  Because there are Kissam Stadia rules with 4143 printed on the slide rule, it's obvious that the Kissam rule would run for a year or two as the 4143 before the 68-1XXX switch. 

​This Kissam, other than being all-plastic construction with the unbreakable cursor, was also quite different from the former 4100.  The scales on the back of the slide were removed, lessening it's functionality with general math computations.  It kept the Conversion Factors on the back of the rule, but it removed the stadia instructions entirely, placing them on a separate plastic card.  The front side scales were also changed - { inches // R1 [V1/H/V3 HC V2] R2 \\ centimeters}. 

The R1 scale added another decade, now numbered 1 to 1000.  The top scale on the slide was now a combined V1/H/V3 scale.  The H scale, in red, would index from the end of the second decade, at 100 on the top rail.  Horizontal measures would read right of the 45 marker.  To the left, vertical measures for standard angles was the same.   The right side V3 part of the scale allowed for the measure on sub-minute sized angles, read off the R1 scale, but then divided by 10,000 and subtracted from the R-value to give the vertical reduction.   For angles between a minute and a degree, the V2 provided vertical reduction for those angles with more precision, read off the R2 scale on the bottom rail.  The HC functioned as normal, also read off the R2 scale as on offset to the R-value to yield horizontal reduction.  To make that work, the R2 scale was entirely different from the previous A scale, as sacrifice made - no more general math ability - in order to give much more precision for stadia measures. 

Priced at $19.50 in 1962, substantially more than the 4163-3 ($11) and 4094-3 ($16.50) rules which shared the same form factor, this final version of the Mannheim Stadia rule would last through 1972, ending the long run of a popular slide rule for K&E. 
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The Model 1749-3 Colby's Stadia

Patented by Branch H. Colby of St. Louis, and one of two devices known as "Colby's Computers" licensed to K&E for sale, the Model 1749-3 Stadia is pictured in the 1897 catalog (shown at right).  The rule is for an office desktop, at a massive 50 inches in length.  It is comprised of an upright slide riding in a vertical, perpendicular groove.  Advertised for stadia reductions, computing the difference of elevations between two points if the Stadia reading (delta-y) and vertical angle (theta) are known.  

The July 30, 1895 patent for the device describes a "Log Scale" on the base of the rule to be three decades of logarithmic scale (an extended R scale) numbered 1 to 1000.  The vertical slide is labeled "Arc Scale," keyed to the Log, based on angles of elevation from 0 to 18 degrees relative to the distances on the base scale according to the formula for vertical displacement.   The device has indexes for meters, yards, and feet, covering any unit by which a survey could have been measured. 

The idea is to have surveying crews recording distances and angles of elevation/depression data from a known benchmark, measuring points either in a site plan grid or radiused from the sight device using recorded azimuth measures.   The data could then be brought back to the office where the all of the elevation differences are computed in higher precision than with a standard 10" or 20" stadia rule.   This would give accurate topographical surveys of the fields being measured, whereas a map could be produced - all done from the comfort of the office.  This was traditionally accomplished with reduction tables, which was accurate, but a very tedious process.  Colby's slide rule attempted to fix that.  The extent at which he succeeded is unknown? 

The rule itself, according to pictures of a known sample, appears to be celluloid colored mahogany.  It was sold in a long wooden case and priced at $20 in 1897.  This rule was given the 4125 model number in 1901, but would be discontinued prior to 1904.  Perhaps this short life-span answers my question as to success of the rule? 

The only one of these devices ever sold on eBay happened in the year 2000 at a price of $400.  This collector is holding out hope of finding another one! 

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The Model 4105 Webb's Stadia

A very unique rule, the Webb's Stadia rule was added to the K&E catalog in 1903.  The price was $5, only 50 cents more than the Model 4100.   It's unique in that the Webb's stadia rule is an elongated wooden cylinder with seven paper scales affixed around the surface.  Four scales are labeled "Differences in Elevation," which collectively serve as a single folded scale ranging from 1 minutes to 40 degrees.  At 12.5" long, these scales are the equivalent of a single 50" scale.   Three scales are labeled "Horizontal Correction," which serves a similar function as those for elevation, ranging from just under 2 degrees up to 40 degrees.   Sliding through an outer metal tube with a slit through which the inner cylinder may be read, the slit is labeled with a single scale ranging from 100 to 1000 feet.   

The idea is simple here:  align the index of the cylinder to the stadia measure (R-value) by sliding it through the tube, and then sighting the vertical angle on the correct vertical and horizontal scales by rotating the cylinder to measure the reduction respectively.  No cursor is needed.  Quite nice!   The device shows that such a rule is actually better formatted to a cylinder rather than to a linear rule, especially from the standpoint of increased resolution.  

K&E sold the device for exactly 20 years until it was discontinued in 1923.  Cost for the rule was $7.50 in it's last year, but will cost you more than $400 today for the rare sample that might come up on auction. 

The Model 4102 Surveyor's Duplex

In 1915, despite already producing 3 other stadia slide rules, K&E introduced their first stadia rule based on a duplex form factor.   However, this rule had a few special tricks up its sleeve.  

The stadia functions of the rule cover the back side, giving the same functionality as a typical stadia rule.    

But it's one thing to use the stadia method of surveying any number of points, but they mean very little unless those points relate to a meridian line. So before a survey can be done, precise latitude and longitude as well as an exact north bearing measurement is required.  At night, this is possible by sighting Polaris (the "North Star") and staking out when true north is on the ground.   But this isn't feasible during the day.   In that case, measuring the sun's azimuth and comparing it to it's known bearing provides that information.  Known as "astrometrics," this is a practice still valued by modern surveyors today.  

As for the Model 4102 , this is where the front side, astrometric scales of the rule​ are useful.  Once the altitude of the sun is ascertained by transit observation, the slide rule has scales to compute accurately the sun's azimuth (bearing from true north).   This important measurement allows a survey crew to stake out a true directional grid, as well as conduct a radial survey of the area.  

Front side scales include...
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D, sin d [ cos l & h, C, tan ] tan I, Az

The Chemical Family

4160 Chemist's Rule

4175 Kurtz Psychrometric

4165 Urea Index Rule

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The Sewer Family

4130 Colby's Sewer

4132 Crane's Sewer

4128 Nordell Sewer

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The Steam Power Family

4135 Power Computing

4141 Hudson's Horsepower Computing Scale

4142 Duplex Allan Friction Head

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The Radio/Electrical Family

4082-3 Radio Special

4133 Roylance Electrical

4138 Morrison Radio Engineering

4139 Cooke Radio Rule

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The Sector Family

4175 Ivory Sector

4176 Boxwood Sector

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The Miscellaneous K&E Family

There exists a number of products over the history of K&E that were highly specialized, custom-built, or otherwise distributed by the company.  Many of these were not listed in their product catalogs.   Others were more expensive, produced in short supply, or existing for a very short duration of time.   Most of these existed earlier in the company's history and, thus, are exceedingly rare.   But on the whole, all of these devices will be hard to find as a collector today. 

My own collection does not contain many of the following analog computing devices, but I'll provide descriptions of them nonetheless.  Of the devices I do have, I will be able to speak about them in a much more detailed fashion.  

Thacher Calculator (Models 1740 and 4012/13)

We start with this device because of its importance to history - and not just K&E history.   

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Reckoning Machine
Fuller Calculator
Charpentier Calculator
Boucher Calculator
Sperry Calculator

4090/4091 Universal Rule

4095 Triangular Metal Rule
4096/4098 Metal Mannheim

4180 Fraction Adder
4185 Brunton Slope Chart
4108 Military​
4019 Goodchild Mathematical Chart

Demonstration Rules
68-1955 Deci-Lon Projection
Beseler Projection Slide Rule





Custom Rules (out of catalog)

D4053-3 Military Rule (with decimal trig scale)

K&E 4104 - 20" 
K&E 4110 Power Trig
K&E Lumber rule
Reymond Leather Calculator
Belleville Spring Washer Computor
Beatley I-Q Slide Rule

Brick-o-Meter
National Union Radio Reactance Computer
M4 & M16 Graphical Firing Table
E-6B Aerial Dead Reckoning Computer



No. 8858 Special War Time Issue
Syrup Rule
Short Base Triangulation Rule

4010 Stima Calculator

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My Portfolio of K&E Slide Rules...

General-Purpose Rules 

Specialty Rules

Here are K&E slide rules that allow for basic to complex evaluation of mathematical computations and functions. I will also include "engineering" rules here, since that designation typically includes hyperbolic trig scales which, as far as I'm concerned, is still computational mathematics; however, where that applies, I will make note of that.   

Expand the titles below to see pictures and specifications...

Full-Scale Rules

These K&E slide rules are "specialty" rules because they were designed or marketed for a specific purpose in mind.  Such a rule will often include one or more scales for a specific application, whether a finance formula, chemistry conversion, electronics functions, or even unit conversions.  Or it might be something "different." 

Expand the titles below to see pictures and specifications...

Full-Scale Rules

N4053-3 Polyphase mannheim

N4070-3 Polyphase Duplex trig

N4080-3 Log log Duplex Trig

N4081-3 Log Log duplex decitrig

N4083-3 log log Duplex vector

4088-3 Polyphase Duplex

4090-3 log log duplex trig

4092-3 log log duplex

4041 mannheim

4054 Polyphase Mannheim

4055 Favorite

4056 Favorite

4058 Beginner's Rule

68-1100 Deci-lon 10

68-1210 log log duplex DeciTrig

4181-3 Log log duplex decitrig

N9081-3 Doric log log duplex decitrig

68-1565 GP12

68-1892 K-12 prep

Pocket Rules

4150-1

Front

Back

Pocket Log Log Duplex
Scale Length: 5"
Purpose: General-Purpose
# of Scales: 22
Country: USA
Material: Aluminum
Date: 1968 to 1975
​Condition: C3 (like new, but with case only) 
Front Side Scales
LL1, LL01, A [B, ST, T, S, C] D, DI, K
Rear Side Scales
LL2, LL02, DF [CF, Ln, L, CI, C] D, LL3, LL03

4163-1

4181-1

68-1300 Deci-lon 5

4097C Ever-There

4097D ever-there

68-1261 Jet-log Jr.

4161-1

4053-2 polyphase mannheim

4168 pocket polyphase duplex

Long-Scale Rules

N4053-5 Polyphase Mannheim

68-1608 Polyphase Mannheim

68-1200 Log Log duplex decitrig

N4092-5 Log log Duplex

4083-5 Log Log duplex vector

4080-5 LOG LOg DUplex Trig

4081-5 Log Log Duplex Decitrig

4100 Stadia Rule

4139 Cooke Radio Rule

4082-3 radio Special

Mystery Polyphase Mannheim rule

4095-3 Merchant's rule

N4100 Stadia rule

4143 Kissam Stadia

title 7

Pocket Rules

68-1555 Celanese Celcon

Front

Made in USA for Douglas Aircraft - 1952
Scale Length: 5"
Purpose: Aeronautic Engineering
# of Scales: 16
Country: USA
Material: Aluminum
Date: 1952
​Condition: C3 (like new with case) 
Front Side Scales
Dynamic Pres., Temp Rise [ Static Pres., Pres. Alt, Density Alt., C ] D, True Mach. TAIS
Rear Side Scales
Impact Pres., Std. Temp., Pres. Alt. [ Temp. Venier, Pres. Alt. ] True Mach No., Dial Airspeed. ] Blank

Long-Scale Rules

N4102 Surveyor's Duplex Rule

Scale Length: 20"
Purpose: Stadia Surveying
# of Scales: 15
Country: USA
Material: Celluloid-covered mahogany; metal end brackets.
Serial Number: 598125
Date: 1937
​Condition: C5 
Front Side Scales
D, sin d [ cos²α & h, C, tan ] tan I, Az
Rear Side Scales
VERT (½ sin 2α), A [ B, CI, C ] D, VERT (½ sin 2α)/HOR (cos^2​α)

Model 4012 thacher Calculator

Scale Length: 20"
Purpose: High-resolution computation
# of Scales: 1
Country: USA
Material: Celluloid-covered mahogany; metal end brackets.
Serial Number: 2758
Date: 1910

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 Demonstration Rules

68-1929 Deci-lon 10 Demonstration rule 6.5 ft