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Table of Contents

Introduction
Chapter 1: Understandings and Conventions
Chapter 2: Rules of the Single-Sided, Mannheim-Type
Chapter 3: Rules of the Double-Sided, Duplex-Type
​Chapter 4: The Specialty Rules
​Chapter 5: Miscellaneous K&E Rules
Chapter 6: Out of Catalog, Custom Rules
​Chapter 7: My Portfolio of K&E Rules

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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 wonderful collectability 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 celluloid cursor rails that didn't hold up well over time and would eventually crumble to pieces. This begins with the outgassing of the nitric acid from the celluloid, which rusts the cursor spring and compromises the spring's anchor going through the rail.  Eventually, the spring breaks away and the celluloid rails become brittle as the camphor plasticizer is also forced out of the celluloid.*  

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 compromised, any disassembly of the cursor for cleaning runs the high risk of watching the cursor rails crumble into your hands.   If cracks in the rails are not obvious in the pictures, then note any rust stains either at the cursor or along the slide rule rail itself.  If this is noted, avoid purchasing the rule. 

K&E seemed to rectify this condition after 1950 with a more durable plastic that also held up better against the typical yellowing of old plastics. 

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. 

* There is some debate about what causes the rusting of the spring in these cursors.  While camphor does sublimate out from old celluloid eventually, it is a well-known rust inhibitor, and as such would not be the cause of a compromised cursor spring.  Of note though is that the camphor is the plasticizer of the nitrocellulose structure, and as such the leaching of camphor would certainly make for brittle celluloid.   Some feel that "celluloid rot" can cause other rot in other pieces of celluloid, but I find this untrue in observation...the rule itself, also covered in celluloid, never seems to suffer from the breakdown of the cursor rails beyond some rust marks from the cursor spring. 

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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.

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All About Keuffel & Esser Rules

Author's note: This, and most articles I write, are living documents.  This article (or book) 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.   

To click to specific Chapters on this page, use the Table of Contents at left.

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Introduction

If you are a lover of slide rules, or merely a connoisseur of trinkets, then it would be difficult to find something more collectible than rules made by Keuffel and Esser.   Not only is this because of the enormous variety of rules to be found in circulation today, but because of the vast number of model types and styles of rules manufactured by K&E over the 100 plus years of its existence.  

Speaking as a slide rule collector myself who, like most, begin the hobby not as a respecter of anything​ particular, swallowing up any rules we can get our hands on, it's not until we begin to accumulate a few K&E rules that we realize that we could spend most of our lives acquiring, researching, cataloging, and writing ONLY about K&E.  The history is that extensive.   I have found myself charmed by their rules; enthralled with understanding K&E product evolution and design decisions; curious as to the nature of their consumer demographics and preferences; and wondering about K&E marketing, company leadership, and the business successes and failures surrounding these slide rules. 

For those who live in the United States, count yourself fortunate that K&E slide rules, among other collectible K&E products, are easily and quickly found.  For the most part, particular K&E models dominated sales for them.   Such models, like the 4081 Log Log Deci-Trig and the 4053 Polyphase Mannheim are in no way rare, nor particularly valuable.  But because each of those models have existed for 70 years or more, and because K&E never felt their rules could not be improved in some way, a collector like me might end up with 20 to 30 of EACH of these models, realizing that no two of them will be exactly the same.  And because of this, the evolution of changes can be mapped in a way that would satisfy Charles Darwin. 

I am thankful that K&E was never satisfied with the status quo, because I can collect an enormous number of meaningful slide rules without needed into pay exorbitant prices for them.   That said, I could do that if I wanted as well!  Historically, there are rare slide rules of substantial value.  Such can be the result of either extremely limited production or historical significance.   And when these slide rules are unearthed, there is a large group of dedicated K&E collectors who want to add it, not only the rule itself, but the story it will tell. 

And I believe it's a exciting time to become a K&E devotee, as we are now two or three generations removed from people who actually know what slide rules are.  There is the opportunity for even ultra-rare K&E slide rules to come out of hiding, as family heirlooms turn into dispensable curiosities of "what is this thing and why do we still have it?" 

So unlike any other makers, including Pickett, Hemmi, Faber-Castell, and Aristo, Keuffel and Esser - both the slide rules and the company - sparks curiosity in me.  So much so that what follows here will be a written love-affair of not only the slide rules in my collection, but information and understanding about every other slide-rule related item produced or sold at some point by K&E.

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Chapter 1: Understandings and Conventions

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.  

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. 

In my writing, it should be noted that my thoughts about K&E and their rules comes firsthand, from my own gleaning of the available product catalogs, price lists, and supplementary product brochures, mostly as made available by Clark McCoy at his important resource website here.  This, when coupled with a study of my own collection of K&E items, including slide rules and product manuals, yields a tremendous amount of understanding, especially in terms of product time-lines, product "families,"  and the evolution of the slide rules within individual model lines.   This provides the initial framework of my writing, whereas I might conjecture about items within the broader context of what I've already gleaned.   At that point, especially where many of the more rare models are concerned, extra research is conducted to fill in gaps of understanding or specifics about a product.  This research begins first at the aforementioned ISRM, as well as the rich resource of materials made available by the Oughtred Society, in which I am a member.  There are also many websites from fellow collectors available on the internet from which extra information can be gleaned.  And finally, where applicable and available, any books or other published works are consulted.  An appendix (see the end of the article) to all of these writings is added for any support materials used, with references noted within the text itself. 

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, especially the ISRM, the Slide Rule Universe, or the Oughtred Society.   While you will get some individual product depth 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 five major categorizations organized in separate chapters:  Single-Sided/Mannheim-Type, Double-Sided/Duplex-Type, Specialty Rules, Miscellaneous Rules, and Out of Catalog/Custom Rules.   

The chapter covering Single-Sided/Mannheim-Type rules will transition across the families, Mannheim >> Polyphase Mannheim >> Modern Mannheim.  "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 (as differentiated from Polyphase Mannheim).   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, in our chapter covering Double-Sided/Duplex-Type, "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 these two chapter labels, which becomes the way I think about these general mathematics types of slide rules.  

The chapter covering Specialty Rules can be thought of rules that are "everything else" if you'd like, but this would be inaccurate for two reasons.   First, that would not be fair to some important families of rules in the specialty class.  The Merchant, Stadia, Radio/Electrical, and Demonstration families are every bit as important throughout our history here and are a fundamental part of the eco-system from the very beginning of the company.  Secondly, there are many specialty rules that are not associated thematically with a broader family of rules, and so those should be talked about separately.  Either way, rule in this chapter, within these families, didn't always care about using the same construction or form-factor.  For example, there are both singled-sided Mannheim-style (the Model 4100) and doubled-sized duplex (the Model 4102) "Stadia Family" slide rules.  So, for this chapter, it's important to note that the Specialty rules are not miscellaneous K&E rules, but rather those non-general-math rules that still reside within broader family types. 

There remains many specialty rules that are not specific to a "family" of rules, and thus these rules get their own "Miscellaneous" chapter.  These are will be those items listed in a K&E Product Catalog to be discussed individually, out of the context of a more broad categorization.   Some of these rules are historically significant and are some of the more exciting products in a K&E collection.   As such, the dismissive "miscellaneous" moniker should in no way be interpreted by the reader as being slide rules that are not as worth of discussion as everything else. 

And finally, we have a chapter of rules, which here we call the "Out of Catalog/Custom Rules," that consists of items not be found within a K&E Catalog, but yet exists "in the wild."   They could have been custom rules built for industry or they might have been a short-term slide rule that K&E hoped would catch on, yet was introduced and then discontinued between two issues of their product catalog.  We will see that such catalogs, and other K&E price lists and brochures, often went many years between publication.  These represent the most mysterious of K&E products with less of a story historically shared, yet lends itself to a plethora of historical questions.   Such questions are the subject matter of many advanced collectors of K&E rules and provides some of the more exciting discussions about K&E rules in the present day. 

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

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Chapter 2: 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.   

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 "single-sided" section of slide rules.  But because those are actually duplex in construction, we will talk about them in the next chapter.  

​So, what follows will be those historical K&E slide rules that are foundationally  "Mannheim" in its scale set.  These rules fall into three broad categories.  The first is the traditional Mannheim scale set of 7 scales (A, B, C, D, S, L, & T) and as such many K&E rules regardless of construction type are considered to be in the "Mannheim Family" of rules.   Secondly, we will look at the "Polyphase Mannheim Family" that is an evolution of this same scale set, yet also formatted with same type of non-duplex type of construction.   And thirdly, we'll look that the "Modern Polyphase Family" of rules, that were all-plastic rules of the non-duplex type, yet still founded on the traditional Mannheim scale set of slide rules. 
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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 side only with A, B, C, and D scales, and 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.  Pre-1900 models in this family were named "Engineers Slide Rules" within catalogs.  But at the turn of the century, it's clear that K&E shifted the marketing of these rules towards the everyman, showing that anybody could benefit from doing basic arithmetic and trig operations made possible with the Mannheim rules.  In 1901, K&E revamped their entire product line-up and marketing campaigns, pushing toward broader markets, including the in-house production of all slide rules.  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 revamped series of the 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.

Earlier models in the Mannheim family of rules pose some degree of difficulty to describe.  Mostly, we can thank the lack of surviving samples of these rules as well as the absence of good descriptions for this.  But there will be a recurring theme until we get to the "flagship" Mannheim model in this family, the Model 4041 Series...up until the turn of the century, K&E seemed content to provide slide rules to customers using whatever was readily available to them.  As we will see, the development of a standard celluloid-covered mahogany Mannheim side rule with the ever-so-familiar "K&E look" was most definitely a process!
  
Historically, the major slide rule models in the Mannheim family include:

Model 479 

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

(Note: Renown collector Bob Otnes conjectures that this rule could have been produced by John Rabone and Sons in England. In Otnes' article, "Keuffel & Esser - 1880 to 1899" in the Journal of the Oughtred Society, Spring 2001, p. 18, Otnes bases this thought on similarities between a Rabone 8" rule in his collection and the 10" Model 479.  This is possible, though the same can be said of Tavernier-Gravet, which is historically connected with the Model 479, as can be seen on rules of this period with both Tavernier-Gravet and Keuffel & Esser maker's marks.  Until otherwise conclusive, I will credit Taverner-Gravet with the production of the Model 479, though it is not without reason that Rabone could have made the earliest version of this rule prior to T-G.)  

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 an old K&E rule with this model number) was introduced at 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.  These were true Mannheim rules and predecessors 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 Model 1745 and Model 1746 rules, both 10 inches.  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.  Their catalog description in 1890, and as printed below in the 1895 catalog, states...

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Sidebar: Mahogany vs. Boxwood

As early as 1900, we see the Keuffel & Esser shift the majority of its slide rule construction from boxwood to mahogany wood.  This, following on the heels of their major supplier, Dennert & Pape, having done the same.  Likewise, American competitor Dietzgen also followed suit.

The exception to this great "mahogany migration" were K&E's budget and beginner's rules which remained boxwood.  Even so, by the end of the slide rule era, every wooden slide rule manufactured by K&E would be made of mahogany.  

So, clearly, there is an aspect of mahogany that makes it the preferred wood for making slide rules.  Yet, boxwood isn't entirely cast to the side.  I believe if we hope to understand K&E's mind on this, a deeper dive into the nature of these woods is required.   As a woodworker myself, I think that perhaps I can bring some clarity.  

 

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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.  

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).  

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.   

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).

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.  Thus, there is variance in wood use between models as well as model years.  Exact dates of the use of wood within models are unknown. 

The Transitional Models 4028, 4030, 4032, and 4040

During the last 5 years of the 19th century, it is obvious that K&E was experimenting with what they wanted their Mannheim Family to look like.   Some rules had obviously been made in-house, but not all, showing that K&E was ramping up their in-house production of rules during this time.  A few slide rules seem to have an identity crisis, as they would not resemble the rules of the product line a decade later.  Some of these rules shifted from boxwood to mahogany construction prior to 1901, while in others boxwood lingered beyond. Others held on to the metal pointing indicator when it was clear that glass was largely preferred.  Many of these rules could be classified in the Model 4041 Series (see next); however, these rules I've classified as "transitional" are different enough despite their spiritual heritage and similar model numbers.  Therefore, here are some models made between 1901 and 1905 that held onto life as remnants from the 17XX model line or remain different enough to separated from the rules talked about in the next section.   

​These rules are as follows...

• Model 4028 Gunter - This 1901 model succeeded the 479, 479-1, and 1745 "Gunter" rules of previous years.  Remaining cursor-less, this model was discontinued after 1906, and would have likely never been offered for sale in the 20th century if K&E didn't still have a back-stock of slide rules in which to dispose.  Their rarity today is likely an indication of sales; I know of no known samples of this Gunter rule. 
• Model 4030 - This model is the 5" Model 1747 described above, both in form and function.  It is non-adjustable boxwood with the basic Mannheim scale set.  Like all models for the 1901 years receiving a new 4XXX designation, the 1747 was christened the Model 4030 and sold for the next couple of years.   I would speculate that the sole reason it sold after 1901 was because of back-inventory of the 1747 rule, and as such is a Dennert & Pape imported rule made prior to 1900.  
• The Early Model 4031 - Continuing with 5" slide rules, this rule is essentially a Model 4030 made to be adjustable like the mahogany Model 4041 Series.   But because it is boxwood, it brings into question if the early 5" rules of the Model 4041 series were either prototypes or an actual production rule?   If the latter, then it would suggest either that K&E wanted to use up some boxwood blanks still in supply or that Dennert & Pape provided the parts to the rules, leaving final glue up to K&E.   I find that more likely these early models are indeed prototypes of the new found adjustable slide rule.  Noting that this is a 5" slide rule, I would surmise that K&E had the most back-stock of this size since, as we will see with most all early wooden slide rules, K&E couldn't sell shorter slide rules at a discount to the longer ones.  As such, it is doubtful that the 5" rules would have been in demand at their given price point - they'd most certainly opt for the 10" variations instead for that same price.  So with essentially an over-supply of 4030 slide rules laying around, then adapting them to the new technology makes sense to me. 

Model 4041 Series

Once K&E replaced the 1746 and 1748 rules, as well as moving past their Transition period of various and sundry slide rules mentioned above, the Mannheim product lineup solidified and standardized around the Model 4041 series, both in form and function.   These rules, and thus the model number change, were manufactured in-house by K&E, as all their rules would be from 1901 forward.  The Model 4041 Mannheim rule ($4.50 in 1901) was a lovely 10" mahogany rule covered in celluloid with engine-divided scales and glass cursor (evolving over time).  This rule was also produced between 1906 to 1935 with a finely-divided scale option known as the N4041F (a $3.50 upgrade). These 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 Model 4031 in a 5" length, the Model 4035 in an 8" length (the first 8" rule since the original Model 479-5), the Model 4045 in a 16" length, and the Model 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.  

The most important aspect of these rules to differentiate them from those I mentioned in the previous section is that all rules of this series (and all Mannheim framed rules thereafter) are made to be adjustable, based on Keuffel's patent of the "Mannheim Adjustable Slide Rule Frame" in June, 1900.  K&E understood that weather could change the moisture content of the wood, causing seasonal swelling - remember their comment above regarding the tendency of the 20" Model 1748 to warp?  So any rule without the patented adjustability, namely those made prior to 1901, would have been hard to use on occasion.   As such, after 1901, the "flagship" models with the Mannheim frame, including the Model 4053 "Polyphase Mannheim" rules in the next section, would boast "adjustability" as a significant improvement, whereas "budget" rules would not include the extra feature. 

As I mention in the Collector's Outlook (see right), the 10" Model 4041 is quite ubiquitous; however, the length variations of the 4041 are quite hard to find, as is the N4041F, "finely-divided" model of this series.   I would conjecture fewer of the variations of the 4041 were sold due to the pricing structure.  $4.50 could have bought the consumer any of the 5", 8", or 10" models, as they were all equally priced.   One might question why this is the case, but it makes sense when we consider that production time and cost would have been similar regardless of the rules' length.  And most consumers of the day would have only been able to choose one such slide rule due to the high cost, not feeling compelled to have, for example, a pocket model to go with their full-scale daily driver.    Similarly, the 16" Model 4045 and the 20" Model 4051 were priced $10 and $12.50 respectively.  Those prices represent a rather steep requirement for slide rules that wouldn't have been significant functional upgrades to the 10" rule.   

Modern day consumerism, whereby products providing 10% performance improvements at double the price is an acceptable trade-off today, would not have been the nature of K&E's market in the earlier part of the century.   Thus, to the consumer, it's easy to see how the 10" Model 4041 rule would have represented the very best in value, and as such would have been K&E primary seller in this model series.  100 years later, it has made tracking down the entire series of rules quite a challenge!  ​

As I mention in the Collector's Outlook (see right), the 10" Model 4041 is quite ubiquitous; however, the length variations of the 4041 are quite hard to find, as is the N4041F, "finely-divided" model of this series.   I would conjecture fewer of the variations of the 4041 were sold due to the pricing structure.  $4.50 could have bought the consumer any of the 5", 8", or 10" models, as they were all equally priced.   One might question why this is the case, but it makes sense when we consider that production time and cost would have been similar regardless of the rules' length.  And most consumers of the day would have only been able to choose one such slide rule due to the high cost, not feeling compelled to have, for example, a pocket model to go with their full-scale daily driver.    Similarly, the 16" Model 4045 and the 20" Model 4051 were priced $10 and $12.50 respectively.  Those prices represent a rather steep requirement for slide rules that wouldn't have been significant functional upgrades to the 10" rule.   

Modern day consumerism, whereby products providing 10% performance improvements at double the price is an acceptable trade-off today, would not have been the nature of K&E's market in the earlier part of the century.   Thus, to the consumer, it's easy to see how the 10" Model 4041 rule would have represented the very best in value, and as such would have been K&E primary seller in this model series.  100 years later, it has made tracking down the entire series of rules quite a challenge!  ​

Models 4054, 4055, and 4056 - "Favorites"

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). 

I should note that the original boxwood versions of the 4054 and 4056 Favorite rules are very hard to find.  I suspect this is because they weren't as long-lasting as the adjustable mahogany rules that litter eBay.   These rules, if found, are easy to mistake as "ordinary," so it's likely that a collector might not have to pay a lot if they stumble across one. 

​Model 4058

A "student" model for any budget, made of a less expensive boxwood, glued-on paper-scales, and a metal-bracketed, glass cursor, the new Model 4058 was the carry-over of the previous model 1749-1 introduced a few years earlier.  Reintroduced in 1901, the rule cost a rock-bottom price of $1.  These rules did not include the inch/centimeter scales on the sides, but otherwise offer the Mannheim scale set.   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). 

While these served the purpose of a budget rule, these were not intended to last long, though earlier models are reputed to have been quite enduring. It is thought that the boxwood supply in earlier years was well-seasoned (as stated in the product catalogs) and likely maintained stability, making adjustability mostly unnecessary - that feature was priced out of a budget type of slide rule.  But by the 40s and 50s, judging by several samples in my own collection, it is obvious that the source of boxwood was not as well-seasoned, or in the very least was not finished in a way to prevent wood-movement, and therefore would no longer function well in humid conditions.  As such, many they were prone to warping over time.  While boxwood is considered a stable wood, it required a good selection of it be endure as long as the more expense mahogany.  Of course, you get what you pay for in that regard.  Please see SIDEBAR:  Mahogany vs. Boxwood earlier in this writing for more perspective on those woods choices in slide rules.  

I will say that of the warped samples in my collection, they are the 4058W version of the rule with the painted-white faces.  Yet I have two 4058C samples from 1930 to 1935 that are actually quite lovely, as they also came in nice boxed cases.  These "C" versions are bare wood, with scales painted directly on the boxwood, yet finished nicely in what is very likely shellac or varnish.  90 years later, they still function very well.   

Some form of these wooden rules ran from 1900 to 1961, but they are always consistently underwhelming from a collector's perspective.   They are neither rare nor are they expensive to acquire today.  

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, which are found easily on eBay for maybe $20.  The exception is the original boxwood version of the 4054 and 4056 rules, which are very difficult to find, as most in the wild are the later mahogany versions.  Very few have come up on eBay over the last couple of decades, though prices are less than $100 for the 4054 and $20 for the 4056 when they do. 

Any of the Ever-There Pocket rules, as well as most of the more modern pocket rules made of the better plastics 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 rule between 1911 and 1943, but is rather rare.  Perhaps this is because the upgrade to finely-divided scales came with a $3.50 premium!

In 1954, all 4053 versions to follow would employ semi-plastic construction - the back of the rule 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.   K&E also produced a variant of the rule for special use by the U.S. military known as the Model D4053-3 beginning in 1954.  This rule, stamped "U.S. Government," came with manual, magnifying cursor, and custom leather case to also accommodate the higher-profile cursor.   

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 (with the military version called the D68-1617) 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.
​
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 and lettering kits, with Doric indicating their most basic kits.  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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Chapter 3: ​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. 

At anytime throughout K&E history, the "flagship" slide rule for Keuffel and Esser would always be of a duplex design.   As a general principle, duplex rules would always be more costly than their single-sided, Mannheim-type relatives, likely due in part to more surface area being covered by the celluloid laminations, as well as more scales required to be finely etched.  In the modern era, most duplex rules would shift to all-plastic construction, but they would always produce their flagship, classic rules of mahogany.   And these rules would always be more expensive than their plastic counterparts.   

The duplex rule would evolve over the 70+ years of its manufacture, which I will describe more fully within the historical context of each model itself, but other than cursor materials and design, font usage, and edge laminations, the celluloid-covered mahogany rules would remain remarkably similar throughout these build histories, a testimony to the long-lasting quality of these rules. 

From p. 276 of the 1954 K&E Product Catalog, this detailed drawing of the Model 4070-3 Polyphase Duplex slide rule shows the detail that went into duplex slide rule construction. This rule, and many others, will be discussed within this chapter.

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.

​K&E rules of this type will be easy to compare to those from other manufacturers, particularly the bamboo-constructed rules made by Hemmi in Japan.   As collectors, we love them all, but in general the K&E rules are absolutely ubiquitous in the United States, so without too many exceptions these slide rules are easily acquired on this side of the world.   As such, not much of what we will talk about in this chapter are necessarily rare, or with extraordinary value, with only one notable exception in the Modern Duplex era.    

It, and other K&E duplex rules will not be fully discussed, 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.   ​

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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 an early appearance - but not the first - 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 Family 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 glass cursor with metal rails in 1915, followed by a switch to plastic rails in 1916. 

(Note of general amusement: This version of the cursor was patented in August, 1915, as number 1,150,771 by "Willie" L. E. Keuffel.)

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. 

Sidebar: The European Influence on K&E

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 companies 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 producer 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, especially once William Keuffel hired his second cousin, W.L.E. Keuffel, in 1884.   Also named William (or sometimes "Willie), he would eventually become the Vice President of Manufacturing.  He should likely receive more credit for K&E slide rules than any other single individual, as he was the guiding force for pushing the company in the direction of in-house production of all rules, including the writing of seven patents from 1898 to 1916, as well as the development of tooling to make slide rule production happen.   His own son, A.W. Keuffel, would eventually write 11 slide rule patents himself. 

As a company, K&E was always evolving; pushing diversity; 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. 

But in the early days of slide rules, 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.  

But 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 obvious.  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.  

Celluloid rules were 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, importantly, 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.

However, this business model was not unique.  Perhaps no other company was more influential to K&E's business model on the whole as the W.F. Stanley and Co. in London.  The Stanley Company was enormous in England by the turn of the 20th century, having long broken ground in the arena of drafting and survey equipment themselves.  Not only would K&E sell many Stanley-made products over time, K&E likely purchased most of their own tooling from Stanley as well, particularly their dividing engines that Stanley invented in 1861.  Stanley laid the blue-print for such a business some 15 years prior to Keuffel & Esser, having more than 3000 products in their own catalog by 1881.  

This idea of borrowing ideas, designs, tooling, and supply from other companies shouldn't be a surprise, nor does it make K&E any less innovative.  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, instead farming out their manufacturing to Sun Hemmi in Japan.  Likewise, Chicago-based Dietzgen was in no way particular about from whom they would acquire licensing rights, especially from the Charles Bruning Co. 

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.   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.  

And borrowing from European markets did not stop when K&E started producing all their slide rules in-house.   It is a fact that K&E still borrowed or licensed certain technologies to produce a majority of their own rules.  After all, K&E was not a think-tank of invention.  Whereas they had smart guys behind smart products, they relied heavily on inventors and consultants outside company walls to come up with many ideas.  We see this on a variety of rules, like their Log Log Duplex Vector Family and Model 4110 Power Trig rules, as well, perhaps, on their mainline of rules.  For example, because the Model 4092 Log Log Duplex rule, first made by K&E in 1909, did not identify an author on the manual, it's thought by many that the design of the rule was likely brought over from Dennert & Pape, who was producing the "Yokota"-designed rule the year prior.  The layout of the Model 4092 is very much of similar design.   

But K&E wasn't a reactive company where innovation was concerned.  They understood market trends and product demand.  So even when they licensed the "duplex" copyright from William Cox, they were always at the forefront of bringing technologies to market.  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.  And we also know that their innovation extended to all their other products as well.   

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, which allowed for the computation of arbitrary exponentials and roots.  This was 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.   

Unlike previous Log Log rules that used a single scale, it is a certainty that more makers wouldn't have thought production of a Log Log rule to be useful unless multiple continuous scales could be utilized, providing worthwhile resolution. Such a design already existed known as the Yokota layout, which required three continuous log log scales on a stator rail.   Enter European slide rule maker (and industry leader) Dennert & Pape, who developed their first "Yokota" designed Log Log rule in 1908, one year before K&E produced their first Log Log Duplex rule.

Note:  As previously discussed, dividing machines up until the mid-1900s could only function at the physical edge of a stator rail, so only 2 scales on the rail were possible. This seems odd to me, as companies could put 3 scales on a slide for many years previous, but nevertheless there was a technological limitation to that point in time. 

K&E likely retooled with these new capabilities simultaneously with Dennert & Pape, but it wouldn't be until a year after the Dennert & Pape effort that they could roll-out a product to market.  But it was obvious that K&E's first order of business was to make such a slide rule, likely in what I feel was a partnership effort (see Sidebar: The European Influence on K&E" ​for more). 

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 latitudinal 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 manufactured.  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, rising 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! 

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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.  

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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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Chapter 4: 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.  Beginning in 1901 with the introduction of the 4xxx naming convention, K&E started using 41xx series numbers to denote many of these specialized types of slide rules, particularly the linear rules. The notable exceptions to this naming scheme are many of the cylindrical or circular types of rules, as well as the Merchant's Family of rules, all of which carry 40XX model designations.

Logarithmic rules can take on a variety of forms, like this Thacher Calculator in my collection. Here we discuss those K&E items that do not follow traditional form or general-mathematics functions.

As we discuss the Merchant's Family of rules first, 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, mostly because there are so few models to describe.
​
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, yet solid enough to be worked with one hand (freeing a hand to write down computations).  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. 
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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

Model 4140/4141 Hudson's Horsepower Computing Scale

If I am K&E and I want to be known as THE source for slide rules in the USA, then I will not care so much if I made the rule or not.  In the case of the Model 4140/4141 Hudson's Horsepower Computing Rule, K&E became licensed to sell the product from its maker, the W.F. Stanley & Co., Ltd. in England.  We will talk more about Stanley later in the context of the Fuller Calculator, but there is no mistake that K&E followed this company closely in terms of their business model, especially working together with them in many areas.  One of these is to license for sale in the US the Model 4141 Hudson's Horsepower Computing Scale.  

Origins of this slide rule can be traced back as early as 1877 in the W.F. Stanley product line.  These early versions were made of pure ivory, whereas later versions might incorporate celluloid covered boxwood, as does the version marketed by K&E known as the Model 4140, or simple cardboard construction, as with the K&E Model 4141 version of the rule.  Either version is only 4.5" long.  Price of the two rules in the 1913 catalog is $3 for the cardboard Model 4140 and $6.50 for the wooden Model 4141 version.    K&E sold these rules between 1901 and 1916.   Both version came with a leather-covered sheath.

The rule has two slides which move independently between two fixed stator rails, bracketed on the ends.  As such, it is fully duplex, as there are indeed scales on the back side.   There is no cursor.   Scales are as follows...

Front Side:  Indicated Horsepower [RPM] [Stroke Mean Pressure]  Cylinder Diameter
Back Side:  Mean Pressure Scale

The slide rule works on the principle that if you know any two of an engine's piston speed, its displacement size, or its power, then you can compute the third quantity.   Displacement size would be a product of the stroke length and its cylinder diameter, giving 4 total inputs, which makes the dual slides a necessity in a cursor-less design.    700 RPMs is shown as the maximum measurement, as such would be for slow revving engines of the day, most usually steam engines, but not necessarily. 

J.G. Hudson, designer of the rule, invented 3 other rules similar to this horsepower rule, each of which was produced by Stanley.   These include a Pump Duty Computing Scale;  the Shaft, Beam, and Girder Scale; and one known as the Photo Exposure Scale. 

Why K&E chose only to sell the Horsepower rule is unknown, but all such rules are similar in style and function.   The horsepower rule is the most common; however, as well as the longest selling, sold by Stanley (and AG Thornton) until approximately 1931.    Even so, it's still quite rare, as only two Model 4141 rules have come up on eBay since 1999, averaging around $460.  

Model 4135 Power Computer

A likely reason for K&E dropping the Hudson's Horsepower Scale in 1916 is that the company began selling their own version of very similar slide rule 3 years previously.  The Model 4135 Power Computer rule, first appearing in the 1913 Product Catalog, allows for similar horsepower computations of steam, gasoline, and oil engines.   Two versions of the rule would appear over the product's history.   The first rule, made by K&E from 1913 to 1921, was based on the same design as the cursor-less, Hudson rule, with identical celluloid-covered boxwood and dual independent slides.  It was longer, however, at 7 inches.   The K&E version also makes better use of both sides of the duplex rule, with a general-purpose Mannheim scale-set on the front side of the rule (A, B, C, and D scales only) and the specialty scales on the back.   Cost for this rule in 1913 was $7.00, only slightly more expensive than the Hudson version despite the much larger size and increased utility. 

Like the Hudson rule, the Model 4135 is used to compute the formula:  

H. P. (horsepower) =  (P * L * A * N) / 33000, where P is the pressure (mean effective) , L is length of stroke, A is area of one piston in square inches, and N is the number of power strokes per minute.   In this case, 33000 ft.lbs/min. is James Watt's equivalent for 1 horsepower (hp).  

As such, scales on the back side of the rule are like the Hudson rule:

Indicated Horsepower [RPM] [Stroke Mean Pressure]  Cylinder Diameter

​In 1922, K&E completely changed the format of this slide rule, adding the "N" prefix.   The new model N4135 was converted to the 4088-1 model body, also introduced that same year.   Thus, it was a normal 5" pocket duplex style using typical celluloid-covered mahogany. This rule added a glass cursor   Scales were as follows: 

Front Side:  A [B CI C] D
Back Side:  P [R W] D, where P = horsepower, R = Stroke, W = RPM, and D = Cylinder Diameter  

Essentially and functionally, the new format worked the same as the old, except stroke length and RPM are switched on the slide relative to the earlier version, and the cursor replaces the need for dual slides.   This rule would last appear in the 1937 catalog and the 1938 price list at the price of $10.  Of note, 1938 catalog pricing for the 4088-1 polyphase duplex pocket rule, from which the N4135 was derived, was $9.50.   

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Description from my 1921 K&E Catalog. Click on the image for a close-up of the scales for this rule. Note front and back scales are denoted A B [C D E] F G, adjusted for steam pipe calculations on the front and water pipe calculations on the rear.

4142 Allan Friction Head Duplex

This slide rule might as well be a unicorn - I am unaware of a single known sample of this slide rule anywhere.  But it is described in every product catalog from 1915 to 1927 at a cost of $18.  

Keeping to the theme of "steam" power, the Model 4142 Allan Friction Head Slide Rule is a 20" duplex formatted slide rule designed to do steam and water pipe computations.   According to the catalog, the rule is setup to compute one of 5 variable inputs as follows:  volume, friction, diameter, velocity, and pressure.   Those relating to steam computations are handled on the front side of the rule, with scales for water calculations on the back.  

The detailed illustrations in the 1921 catalog (at left) show a very clearly labeled and well thought-out slide rule.   I would certainly love to see a scan or image of this rule, or better yet, the real deal in person.  Alas, there is simply no internet record of an actual sample anywhere.   More searching (and research) is necessary. 

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The Radio/Electrical Family

As a slide rule collector, those specialty rules designed to help with electrical applications are among my favorites.  Most slide rule makers have a variety of slide rules dedicated to this purpose, especially since specifics electronic computations are easily derived from special logarithmic scales and because of the need to do these computations in the field. 

Radio communications is merely an extension of electronics, and traditionally a radio engineer needed the capabilities that an electronics rule would provide.  Typically, such radio specific rules will add a scale or three specific to an aspect of radio engineering to those of the electronics and the general-math abilities they provide.   

4133 Roylance Electrical

The focus of this slide rule is on electrical wiring, including amperage limits for a particular gauge and type of copper wiring, as well as wire temperatures at particular power draws.   It is based on the 4035 Mannheim rule in its format, 

Front Scales: in || A [ B C ] D || B&S Gauge  
Back Scales:  [ S L T ]

Of note are extra markings, in red, on various scales for extra purposes: 

• Between 9.9 and 20 on the B scale, there is a scale for wire temperature in Celsius for doing resistance calculations.
• On the right hand side of the A scale is a "W" gauge marking to represent the constant (.003027) for calculating weight in pounds per 1000 feet of bare copper wire.
• The C scale is marked in red at 746 to assist with conversions of watts to HP (and vice versa). 

The front edge of the rule has what's known as the B & S (Brown and Sharp) standard wire gauge.  These numbers are the equivalent measure to the better known AWG (American Wire Gauge) commonly used today.   On the rule, it's a three line continuous scale running from 0000 to 40, representing common wire gauges from large to small.   Most electrical calculations on the slide rule begin with this setting. 

Additionally, there is a scale beneath the slide on the main body with columns of numbers in 4 rows.   Those numbers are used with the B & S Gauge scale to give the following information: 

• Top row: Rubber-covered wire
• 2nd row: Weather-proof wire
• 3rd row: The amp carrying capacity for the rubber-covered wire
• 4th row:  The amp carrying capacity for the weather-proof wire

Perhaps the most unique aspect of this slide rule, among other K&E rules, is the use of two additional hairlines on the cursor.    This is distinctly a European slide rule feature, yet K&E implements it here on the Roylance Electrical rule.   In this case, if the diameter of a circle is set on the right hair-line, then its area can be read from the left hair-line.   A very interesting feature! 

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4138 Morrison Radio Engineering

4139 Cooke Radio Rule

4082-3 Radio Special

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The Sector Family

4175 Ivory Sector

4176 Boxwood Sector

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 Demonstration Rules

Most slide rules makers produced larger, full scale versions of their popular slide rules for demonstration use.  These rules are typically measured in feet, not inches, and are usually large enough for a room full of people to see.   Such rules were often supplied to schools with the purchase of classroom sets of the regular sized rules.  They also have a dual-purpose as marketing displays.  

Such demonstration rules were surprisingly inexpensive, or least much less that you might otherwise imagine.   An example is the 1958 price of the Polyphase N4053-3 at $13.50 and the 100 Mannheim Demonstration rule at $22.   This is mostly because production techniques for the demonstration rules were not as involved as the real item.   For example, the Deci-Lon Demonstration Rule in my collection is made of redwood, milled down to two rails and a slide in a 6.5 foot length.  The rails are grooved and the slide is rabbeted, allowing them to slide within each other, with easy to fabricate metal brackets placed on the ends.  They are painted white and marked with painted on scales and other labeling to simulate the real rule.  The cursors are typically two sheets of plexiglass, held together with wooden cursor rails and marked with a center-line.  Gravity serves as the cursor spring. 

As such, K&E demonstration rules were not intended to be exact-scale models, but merely functional duplicates of the real thing.  No engine-divided celluloid facings or expensive wooden stock.  

Another form of demonstration rule were those made for overhead projection.  Thus, slide rules models could be made completely from a transparent or translucent material, marked with opaque scales, and placed on a 10 x10 inch "overhead projector," as commonly found in most classrooms of the era.  

For the collector, these rules are obviously display pieces.  They are rare to find in the wild unless posted for local/regional sale on Craigslist or other online market-place sites.    They also pop up with some frequency on eBay, but buyers will always pay upwards of $100 extra for shipping, which reduces the number of actual sales of the large rules unless people finally change the price to something that makes more sense to buyer.   Typical pricing of such rules can be between $50 to $2000, plus any relevant shipping charges, depending on the condition of the rule and demand.  

100 Mannheim Demonstration Rule

68-1944 Mannheim Demonstrator

101 Polyphase Duplex Demonstration Rule

N105 Log Log Duplex Decitrig Demonstration Rule

68-1923 Log Log Duplex Decitrig Demonstrator

68-1929 Deci-Lon Classroom/Demonstrator

Found on Craigslist in Austin, Texas, I am proud to have this massive slide rule in my collection.   Most demonstrator rules are indeed not built of the same material as those from which they are modeled, so this giant Deci-Lon is not made of Ivorite plastic, awesome as that would be.   Instead, this Model 68-1929 Deci-Lon Demonstrator is made of big chunk of redwood about 6.5 feet long, with faces painted in white, and then screen-painted with scales.  The cursor is two plates of plexiglass attached to wooden cursor rails, with a black cursor line affixed on both sides.   Gravity keeps the large cursor in place, so no spring is required.   The scales are painted black, with all inverse scales colored in red, and with red maker's marks, Deci-Lon 10 logo, and model/copyright numbers also printed in red.   All scales from the actual Deci-Lon 10 slide rule are present on the demonstrator. 

To hang this fully duplex slide rule on a wall, as I do in my classroom, it must be hung in a fixture that allows use of both sides.    As such, K&E originally sold an optional trunnion (metal brackets) for mounting the slide rule to the wall, allowing it to pivot to front and back positions.   I mostly use mine as a display within my classroom, but I sometimes use it to demonstrate its use to my students.   I also allow some of my students to use it, as I have extra credit for students who make a paper slide rule and are willing to demonstrate how a slide rule works.  They enjoy using the enormous slide rule on my wall to demonstrate knowledge!  

Perhaps one of the most satisfying aspects of having a demonstration rule comes from understanding its historical provenance.   My sample came from Mineola ISD, as known by the metal ID asset inventory tag still attached to the rule.   I find it exciting to know that a very small school district approximately 120 miles east of my residence in the DFW metroplex had K&E's most powerful demonstration rule hanging in a classroom.   The implication is that their high school likely had an advanced math class, similar to my own, which not only used such a slide rule, but would have had a classroom set of Deci-Lon 10s for their students to use.   Certainly this meant that some serious math computation was happening in Mineola, Texas, whereas a more simple demonstrator just wouldn't have sufficed!

Originally introduced in 1962 with the actual Deci-Lon family of slide rules, the 68-1929 demonstrator would have set you back $60, a very reasonable price considering its size.  Remember, the actual Deci-Lon 10 slide rule was $25 itself.   The 68-1929 would appear one last time in the 1967 catalog, disappearing after that.  

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68-1960 Log Log Duplex Decitrig Overhead Projection Rule

68-1955 Deci-Long Overhead Projection Rule

Beseler Projection Slide Rule

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Chapter 5:  Miscellaneous K&E Rules

There exists a number of products over the history of K&E that were highly specialized devices distributed by the company.  All of these are known items with descriptions listed in their product catalogs.  I deem them as "miscellaneous" because they are unattached to a "family" or theme like we saw in the previous section.  Some of these items were made in healthy quantities and aren't necessarily rare, but still have value due to age and uniqueness.  Others were more expensive, produced in short supply, or existed for a very short duration of time.   Many such devices 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 in good condition today, and all of them have excellent collector value.  

As you will note, many of these are not linear slide rules, but rather cylindrical, circular, or pocket-watch style.  In fact, one is completely mechanical.   The term "computing device" is probably a more apt title for rules in this chapter. 

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.  

This is the Model 4012 Thacher Calculator in my collection, serial number 2758 which places its production date right around 1910 to 1912. It's a really good sample, with only a light amount of bleeding/spotting on the triangular scales covering the brass rails. This version was purchased through the F. Weber Co. in Philadelphia, which often distributed K&E products, as well as produced the occasional slide rule of their own. Note that the Weber badge on the box uses the "Thatcher" spelling as well.

Thacher Calculator (Models 1740/41 and 4012/13)

We start with this device because of its importance to history - and not just K&E history.   The Thacher Calculator was first patented by Edwin Thacher in 1881 and manufactured by Stanley Ltd. in England.  K&E arranged to have it distributed in the USA beginning around 1884, which is the copyright date of the first instruction manual produced by K&E.  The device would first appear in the 1887 Catalog, known as the Model 1740.  K&E likely imported the first Thacher Calculators, and shortly thereafter worked in partnership with Stanley to build them, with Stanley producing the scales and K&E building the rest, including the lovely box-jointed carry cases in which they came.  There are slight variations in the construction of the earliest samples, enough to make me believe that there was indeed a transition between the earliest Stanley-made Thachers and those that would be made almost entirely by K&E.  

Regardless, I believe the scales, indicated by the "divided by Stanley" text on the device itself, were always supplied by Stanley regardless of who made the remainder of the device. 

​At this time the main competition for such a high precision calculating machine came from another Stanley cylindrical rule, known as the Fuller Calculator (later carried by K&E as well, as discussed below) and the monstrous, heavy mechanical calculators of the day.  Incidentally, the scales of the Fuller (made entirely by Stanley) look like they could have been divided by a similar machine to the Thacher, lending credence to the idea that K&E always sourced the scales from Stanley.  

The Thacher features a 4" diameter drum 18" in length.  The drum (also known as the slide) is covered with a single continuous "A" scale amounting to the equivalent of an 80" linear slide rule, rotating in a brass cage, with a secondary "B" and "C" paper scale on each of the triangular rail covered with shellac.  Resolution is 4 to 5 significant figures, depending on the computation.  Instructions for operating the rule were always affixed to the front of the Thacher's base, with serial number and maker-marks engraved on the back of the base.  Likewise, the end of the drum contained patent information and the name of Edwin "Thatcher" curiously misspelled - note that the actual patent name differed from the name on the instruction plate as well as that in K&E Catalogs.  "Thacher" is the correct name.  

To use the Thacher, numbers are computed similarly to a normal slide rule.  For example, to multiply something like 4.56 x 3.24, the number 456 on the drum (A scale) is aligned with the index of the B scale (there are indices at both the edge and the center of a triangular rail for convenience).  Spin or slide the drum to the appropriate scale.  Then, by spinning the cage to find the number 324 on the B scale, the product can be read off of the A scale.  Like any slide rule, supply your own decimal!   For division, bring the divisor to the dividend instead, and then read the quotient opposite the index on the B scale.  Other computations, like cubes, can be computed by reversing the drum.  Again, instructions for doing any such computations are affixed to the front of the device's base.  These instructions, like the scales on the triangular rails, will often be deteriorated on many of these devices due to age.  

By the turn of the century, it is estimated that around 1000 Thacher's were produced, with a rate that would increase greatly over the next couple of decades.  These Thacher's are known as the Type I today, which includes the Model 1740 as well as a Model 1741 that came with a brass rail with attached 3" magnifying glass.    These first appeared in the 1887 Product Catalog, priced at $30 for the Model 1740 and $40 for the Model 1741.   The Model 1740 (equating to $970 in today's money) was mostly purchased by research and science facilities within university or government settings, as well as within industry for heavier number-crunching in engineering fields.   It is widely known that Lowell Observatory used (and still displays) a Thacher Calculator for computing ephemeris data for celestial bodies, including Pluto, as discovered in 1930 by Clyde Tombaugh at that very observatory.  

In 1901, with the introduction of the new production line of slide rules built in-house, K&E changed the model number to the 4012, all of which are known today as the Type II version of the machine.  It would remain known as the Model 4012 until the end of its production.  The magnifying version was called the Model 4013 and is definitely considered the more rare of the two Thachers.  These versions represented a significant change from the original models.  The changes included a more open design of the ends of the brass cage where it is mounted to the base, as well as a possible change to the 4" drum from a metal to wood.  This newer version would use only 2 screws on each end to mount the cage, whereby the Type I model used 3 screws per side.  ​ Bob Otnes (J.O.S, Fall 2001, p. 39), who owns both Type I and II Thachers, is of the opinion that K&E took over production entirely at this point, making the numerous changes because Thacher's original copyright ended in 1898 and was not renewed.  This would have given K&E the liberty to make the device themselves without royalties to Thacher (or Stanley) and could build the units as they see fit.  

The Type II would exist from 1901 to 1927 with an annual production rate increase to around 164 units.  According to the September 1994 Journal of the Oughtred Society (Feely and Schure),  the last known Model 4012 has a serial number of 5432.   I own a Type II variant myself, serial number 2758, which places it maybe a little later than 1910 for a production date.  It still uses scales with the "divided by Stanley" text on the drum, so it's clear to me that Stanley still had something to do with the Type II, even if Thacher himself might have been kicked to the curb, so to speak. 

In 1927, along with many aesthetic changes that K&E was making to their slide rules, a new Thacher Type III variant began, changed enough to take on a "N-" prefix.   As such, the new Model N4012 and Model N4013 added a single pedestal attachment to the cage ends, as well as knurled rings and white number bands at the ends of the cage.  The Model N4013 sales ended right at the start of World War II in the early 1940s, with the Model N4012 lasting until the early 1950s, appearing in the 1952 catalog one last time.  Priced at $130, it's easy to see why the Thacher would no longer be a viable option during that "golden-age" of nice, less expensive, yet powerful slide rules, not to mention the ingenious Curta Calculator (see sidebar below) introduced in the 1950s.  And traditional usage of the Thacher could be replaced entirely with vacuum tube and transistorized mainframe computers into the 50s and 60s.   

This scaling off of production was precipitous after the war, with the last known serial number ending just below 7000.   As such, in all, there are less than 7000 of these devices out there despite the 70 year production run.   Most are in rather bad shape today, coming up in auctions and on eBay typically priced between $600 to $1000 in a condition where the triangular scales will be only partially intact.   Great samples, especially earlier Type I versions, are exceedingly rare and would carry a much higher premium.   Mine is in quite good condition, especially the wooden case, and could sell for $2000 or more.  

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Fuller Calculator (Models 1742 and 4015)

This device was never made by K&E.  Not at all and not in part.  But it was distributed by them.  While only one model of the Fuller Calculator would be sold by K&E over a relatively short history of the device, I will discuss it on the whole, including the models never sold by K&E, if only to give historical perspective.  

The product was patented by George Fuller on September 2, 1879, slightly more than two years prior to the Thacher, and produced by Stanley Ltd. in England - also known as the "W.F. Stanley & Co."   Not to be confused by USA-based hand-tool maker Stanley Toolworks, the English company's founder, William F. Stanley, was a kind-hearted genius - his story is quite remarkable - who began his company in 1853 with drafting and surveying equipment.   The company was self-proclaimed as the "largest business of its type" by the turn of the century.  His 1881 catalog states that his company employed 80 people and produced over 3000 products.   Stanley invented the dividing engine in 1861, which of course is an important invention for every product mentioned in this article.  Stanley would also produce many products that would be distributed by K&E in the States.  If Keuffel & Esser needed a model to follow in the development of their own company, then they needed to look no further than Stanley.

NOTE:  Why William Stanley's company produced a limited number of linear slide rules is worthy of further investigation.  But suffice it to say, Stanley seemed content to license technology to other companies like K&E and Thornton, among others, as opposed becoming a world-wide leader in any and all products that involved the use of his own dividing engine technology.  And whereas they needed to fill out their catalog with a product, then historically they would distribute from other makers.  Stanley marketed slide rules from Faber, Aristo, and Thornton/PIC, among many others. 

Concerning his own products in the late 19th century, Stanley saw a great need for higher resolution "computers."   As such, he would develop the Thacher and Fuller Calculators at roughly the same time, as well as the long-scale, pocket-watch style Boucher Calculator.   To differentiate, the Thacher was a larger desktop model, capable of being used single-handed.  The Fuller required two-handed use, but was certainly more portable.  Stanley would have viewed the two devices for differing applications and, therefore, would not consider them to compete in the same markets. 

In other words, it's conceivable that customers wealthy enough to afford them could have used BOTH.  And it's this same reason that K&E, more than a decade later, included the Fuller in their catalog as well.  Beginning in 1895 and listed as the Model 1742 for a price of $28, the "Fuller's Spiral Slide Rule" featured a single scale on a movable cylindrical "sleeve," spiraled around at a linear length of 42 feet long (500 inches), and numbered from 100 to 1000.  At this length, up to 5 significant figures in resolution would have been expected for many computations.   

Two brass pointers serve as the indices for the computations, one fixed to the body of the rule near the handle and the other to a rotating top cap of the rule.  The latter is marked with 4 indices labeled B1, B2, B3, and B4, any of which can be used for convenience.  The inner cylinder, on which the outer sleeve slides could have been left blank to be functional, had a simple scale for finding logarithms as well as a sine table for reference.   As such, the device provided simple, but nicely resolved multiplication and division of numbers.   Included with the calculator is a box-jointed case made of mahogany, similar to that of the Thatcher, only slightly smaller.  The case also functioned as a stand for the device, with a brass attachment affixed to the end of the box on which a cantilevered-Fuller could rest.  Early Fullers, made of mahogany, had a removable, threaded handle with a smaller case than those that would come later when the device evolved to a permanently affixed handle.   The outer sleeve was constructed of papier-mâché with paper scales hardened with shellac.   The inner stationary cylinder was wood, wrapped in papier-mâché, and then covered in scales.  

In 1901, with the shifting of all K&E model numbers to the 4XXX series, the Fuller Calculator became the Model 4015.  But it should be noted that this is a catalog designation only, as the device was always produced entirely by Stanley for the duration of its product life, never taking on a K&E label or maker's mark on the device itself.   I find it interesting, as pointed out by Clark McCoy at his "everything" website about K&E slide rules, that K&E changed the catalog description for the Fuller rule in a seemly negative manner, pointing out that its spiral design only provided a single computation while not showing all the other simultaneous proportions compared to the Thacher and other linear slide rules. This bit of seemingly unnecessary "marketing" couldn't help sell more devices, so it remains a curiosity why this would be noted from 1901 until the 1927 catalog when the Fuller Calculator would make its last appearance.  However, the idea of pushing consumers toward the Thacher would make sense if K&E did take over all production rights as mentioned earlier, so perhaps this is the case?

For Stanley, this model was known as "Professor Fuller's Calculating Slide Scale," which would see a change to fully-molded Bakelite construction around 1928 (perhaps not coincidently when it was discontinued by K&E).   Stanley would continue to sell it abroad until the 70s when the hand-held calculator finally ended the run.  Collectors, in retrospect, have coined this device the "Model 1." 

Sidebar:  The Curta Calculator

If you lived in the 1950s and 60s and wanted a mechanical calculator that could be held in your hand, then you could have purchased a Curta Calculator.  Introduced in 1949 after the most amazing early history of development imaginable (more about that later), Austrian inventor Curt Herzstark produced his Curta Calculator.   

It would not be hyperbolic to state that Herzstark's pocket-sized device is one of the 20th centuries most incredible inventions.   Originally built its Type I incarnation, the Curta, with the size and metallic feel of a hand grenade, is mechanical calculator capable of multiplying numbers of astonishing size, able to add, subtract, multiply, and divide numbers with up to 11 significant digits using a stepped drum, cogs, carriage, and hand crank.   

An exploded view of the Type 1 Curta demonstrates the level of genius behind the device. Click for a larger view.

Built in Liechtenstein until the early 1970s, the Curta cost $125 when it first appeared in 1949 (around $1500 in 2022 money), but was guaranteed a life-long warrantee.   3% of all Curta calculators would need service over the life of production, a figure that Herzstark anticipated due to its amazing complexity - the Type 1 used over 600 parts - which is why he adjusted the sale price accordingly.    In 1954, the Curta II was sold, raising the capability to 15 significant figures of precision.  Slightly larger in size, but still capable of fitting in a pocket, introductory price was $180.  

Demand for the device was extraordinarily high.  Millions wanted the Curta Calculators, but production could never supply them.  From 1949 to 1972, an estimated 80,000 Type I and 60,000 Type II Curtas were sold, or around 140,000 in total. Contina AG (the name of the company producing the devices with Herzstark as the technical director) produced the two Curta models around the clock for 23 years.  

Despite the rate at which the devices would need to be serviced - likely because curious people would take them apart and be incapable of putting them back together - both models of the Curta were actually remarkably durable and reliable, so much so that they remained in use by Rally Car drivers and their navigators throughout the 80s and the 90s. 

But perhaps the most inspiring part of the Curta story is the circumstances in which it was first developed.  

Curt Herzstark was imprisoned during the Nazi occupation of Austria during World War 2 until they discovered who he was.   At that point, German leadership approached Herzstark to produce his device, giving him the resources and freedom to draw up the plans and produce prototypes.  It is said that it would have been a gift for their Führer once the Nazis had won the war, though I suspect they also understood the need for such technology.   

Once the end of the war came, Herzstark had produced the plans while also nearing completion of a prototype. At this point, Herzstark was discovered by the Prince Josef II of Liechtenstein, who provided him with the factory, Contina AG, to produce the Curta.  

While the story doesn't redeem the atrocities of the war, it's certainly good that there was a happy ending for many, like Herzstark. 

The "Model 2," which Stanley called simply "Fuller's Rule" in their 1912 catalog, added a scale on the fixed cylinder for computing the sine of angles.  Three other, very rare models are also known to have been produced.  A "Fuller-Bakewell" calculator, which adds sin^2 and sin x cos scales for stadia/tacheometric measurements, is shown in the 1912 Stanley catalog, as well as the "Barnard's Co-ordinate Calculating Rule," which was "modified to calculate co-ordinates."  

In the early 1960s, the "Stanley Whythe Complex Calculator" was produced.  This was" designed to do conversions from rectangular to polar coordinates, as well as to do operations with complex numbers.  

For the collector, prices at auction and eBay will typically be $300 to well over $1000 for the Models 1 and 2, depending on condition and age.  I have both models in my collection.   The other models listed are rare, with very little information with regard to price, though a single Whythe Complex Calculator did come up on eBay within the last 23 years and failed to sell at its $1200 price tag.  ​

The Fuller Calculator "Model 1" from the collection. Serial number 11376. Made in 1954. Shown with Sine Table on the fixed, inside cylinder. Resolution sufficient for all angles in 10 minute increments.

The Fuller Calculator "Model 2" from the collection. Serial number 12165. Made in 1957. Shown with Sine Scale on the fixed, inside cylinder.

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Reckoning Machine

Charpentier Calculator

Boucher Calculator

Sperry Calculator

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This sample of the Model 4090 Universal rule came up on eBay in April of 2019, selling for $685, which would be a bargain for slide rule collectors. Note the dual slides and triple rails and the way they are connected. The idea is simply to give more "edges" to the rule to accommodate more scales. Images courtesy of eBay.

Model 4090/4091 Universal Rule

This is a unique and important slide rule based on W.L.E. Keuffel's March 1899 "dual slide" patent.   The patent was founded on the desire to have more scales on a duplex rule, whereas dividing engines of the day did not have the ability to etch scales on the center of the stator rails.   As such, this patent called for a rule with three rails and two slides to allow for more total scales.  Note that these slides and rails do not move independently; the 10" rails and slides move collectively like a typical slide rule as they are bound together.   The result was a duplex rule with more edges for the dividing engine.

Scale arrangement was, of course, unique due to the dual slides and triple rails.  But they were also innovative, constituting the earliest example of "folded" scales on a K&E rule.  

Front Side:  A [B C] D CF [CF S] A
Back Side:  A [BI CI] D  CF [CI L T] D

Note that the CF scales were not labeled such on the rule.  They also were labeled "C," but unlike the normal C found on the upper slide, these were indeed folded over sqrt(10) without the CF label.  This is a decade earlier than the folded scales rolled out on the 4088 Polyphase Duplex model.  Of course, as we saw with that model, as well as the Merchant's Family of rule that used folded scales, these are included to assure operations are always found on the rule due to the single index in the middle of the scale. 

The rule itself was put into production in 1901, built in-house and rolled out with all the other new products that year.   Materials for the rule are the same as other duplex rules of the time, mahogany wood covered in celluloid with a full metal dual-sided pointing indicator.  In 1906, as we've seen with most K&E rules, the cursor was changed to a glass indicator.   With this change came the "N" designation, now catalogued as the 4090N model.  In the same year, the company produced a 16" version of the same slide rule known as the Model 4091N.

This "Universal Rule" was expensive; $20 for the Model 4090 (and 4090N) and $30 for the longer model.  As a comparison, the normal 10" Model 4070 duplex rule was priced at $6.50 (or $8.00 for glass cursor).  As such, the Universal rules would be short lived, remaining in production for 5 more years until 1911 when it would be discontinued.  By that point, K&E had gained the ability to etch scales at the centers of the stator rails, making the Universal rule (and its patent) unnecessary.  And the rule couldn't have been a great seller at that cost anyway.  After discontinuation, the model numbers of 4090 and 4091 would be recycled by K&E at a later date within their Log Log Duplex family of rules. 

Needless to say, these "Universal" rules are rare, but they do come up at auction on occasion.   Five such rules have come up on eBay over the last 23 years, ranging in cost from $700 to $4000.   Another, in not-so-good condition, sold at a major online auction in January, 2023, for $4550.   All samples were the 10" model; I have yet to witness an actual 16" version of the rule in "the wild."   Needless to say, due to price, this rule is NOT part of my collection; though as with many rare rules, there's always a chance of a fortuitous meeting in any collector's journey.  

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Model 4095 Triangular Metal Rule

Model 4096/4098 Metal Mannheim Rule

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Model 4180 Fraction Adder

Model 4185 Brunton Slope Chart

Model 4108 Military Rule

Model 4019 Goodchild Mathematical Chart

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Chapter 6: Out of Catalog, Custom Rules

At this point, we turn to short-lived slide rules that never appeared in a K&E catalog.  Most such rules are custom-made in small quantities, therefore rare, and likely never meant to be listed in a catalog.  But K&E was often known to produce rules between catalog years that simply failed to sell for whatever reason.   What follows are descriptions of such slide rules.  Some of the descriptions are rather short due to the lack of information available.   Others are widely known, yet simply weren't worthy of a long production run.  

D4053-3 Military Rule 

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Model 4104 20" Longscale

Model 4110 Power Trig

This interesting slide rule is built from a 1942 patient, licensed by K&E, and produced for a limited time in the early 1940s.  There is some indication that production of the slide rule predated the patent as the 2,283,273 patent number is nowhere indicated on the actual slide rule.  The rule does indicate "patent pending," however.  Likewise, the instruction manual of the Model 4110 Power Trig rule has a 1941 copyright date.  

This slide rule was never described in a K&E catalog, so it becomes difficult to know under what arrangements the rule was licensed or for how long, but most certainly its production by K&E was very brief.  A 1941 to 1942 production date seems likely.  

Scales for the slide rule are front side only:  P F A [B T S CI C] D.    There is nothing on the back of the rule, not even a conversion table, nor are there any scales on the back of the slide.  The trig scales are in degrees and minutes, so this is not a "decitrig" rule. 

This Mannheim rule used the same oversized stock as the 4096M Merchant's rule - the handheld version - which was produced simultaneously.   As such, it's also longer than a typical slide rule, with a 13" scale length.   The extra length comes from using two decades worth of C and D scale.  This is a strange feature, as the rule yields reduced resolution for multiplication and division, the equivalent of a 6.5" slide rule, in fact.  However, this sacrifice is also the reason for both the "Power Trig" name and the patent, since the sine and tangent trigonometric functions now have greater range, capable of much finer outputs for small input angles, while doubling the effectiveness of the single S and T scales as compared to rules keyed to typical C and D scales.   As such, it largely replaces the S, T, and ST scales on a traditional 10" slide rule, trading away some resolution for the simplicity of having the entire input range on single scales.  

There is another noteworthy feature of the rule that you might have seen and that's the addition of F and P scales on the top stator.  An F scale has been used before, denoting "frequency" in K&E's 4082 "Radio Special" rule, and you no doubt notice that P scales ("Pythagorean") can be found on many European slide rules.   But in this implementation, F and P is neither.  Instead, these are log log scales, whereas P is for computing positive bases of exponentials and F is for negative bases.    These are keyed off of the four-decade A and B scales.  Doing so replaces the eight scales of a typical Log Log design.  It's a tremendously space-saving and convenient feature; albeit, the reduction in resolution is significant.   Quite elegant though!   And an extra benefit was that inverses could be read directly off the F and P scales. 

However, I find this slide rule to be perplexing.  And, ultimately, I can see why the Model 4110 was so short-lived.  

There are clearly two advantages to the rule's design, those being its ease of use, as well as its wonderful space efficiency.   In the Model 4110, ease of use is certainly the primary reason for it's existence.  But make no mistake, all of the typical computations normally performed on a 10" slide rule are significantly reduced in resolution.  I feel that if this rule were implemented across 20" of real-estate, and maybe finely-divided to recover even more resolution, then the slide rule makes some sense, other than the undoubtedly exorbitant cost of such a rule.   

As for space efficiency, the whole back of the slide was left blank.  (!)   While folded scales were made obsolete by the double-decade main scales, and typical L and K scales are unnecessary on a rule where base 10 logs and cubes are performed more easily on the scales it has, then it wouldn't have made much sense to include them on the reverse of the slide.   But I feel that K&E missed an opportunity here.  In keeping with the "Power Trig" name, why not add extra trig functionality?  Maybe add an actual Pythagorean scale to the rule (which they never did on any rule) or maybe provide decimal trig scales on the back?   A "Power Decitrig"? 

This real factor though...and ultimately it's downfall?   Price.   We simply do not know how much this slide rule cost; however, I do feel strongly that we can derive at a good estimate! 

This was built materially on the 20" 4096M Merchant's rule platform, which carried a $15 price tag in 1939.   We know K&E never discounted their earlier wooden rules just because they were shorter.  Production costs would have still been the same, as would the overall price; that is, if pricing history of the K&E Mannheim platform rules have taught us anything.  Likewise, we know that certain rules that carried higher licensing costs (see the Log Log Duplex Vector) ultimately passed on that expense to the customer.  The patent for this rule was owned by four individuals, all of which would have likely received a cut of the profits.  As such, I would estimate the cost of the 13" Model 4110 Power Trig rule to be no less than $15, and very likely closer to $20.   And in my mind, this prices the rule out of its intended market. 

Thus, I view the rule as a contradiction...designed for people who want a simple, easy to use slide rule, yet costing more money than the vast majority of rules K&E made.  I believe this is why the Model 4110 slide rule appeared briefly, probably sold a few samples, but never enough to make it to the next published catalog. 

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K&E Lumber/Construction Rule

Reymond Leather Calculator

Belleville Spring Washer Computor

Beatley I-Q Slide Rule

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Brick-O-Meter

Here is a custom rule, commissioned by the Mackensen C. Co. Inc., with the purpose of allowing quick computations of the amount of a construction material needed for a job depending on square footage.   Called the Brick-O-Meter, this 10" slide rule was built using the same white painted boxwood and black painted scales as the Model 4058W "Beginner's Rule."   Cursor design and overall look corresponds to a 1938 to 1942 production date.   It has no K&E model number, no serial number, and appears in no catalog.   However, originally, it did come with 5 or 6 stapled pages worth of instructions wrapped around the rule in a black boxed case, identical to the 4058W rules of the time.  

Scales on the rule are front side only:  Q-Quantity [M-Material  H-Height] L-Length.    The L scale, which is essentially a D scale, and the H scale (inverted D scale), sets the size of what is to be covered, be it with bricks, tile, gypsum board, or any other material.  This material is then lined up under the cursor on the M scale.   The quantity required for that material would be read off the Q scale.  The M or Material scale is interesting, with 7 marks for standard material sizes, which likely was customized to Mackensen needs.   They are as follows:

C.B. - For any material in one square foot size, like a 12"x12" or 8"x16" tile 
B-2 - For a 2 brick thick wall (around 9") where there would amount to 13 standard sized bricks to a square foot
B-3 - For a 3 brick thick wall (around 12") where there would amount to 21 standard sized bricks to a square foot
T - Stands specifically for "tile" sized for 2.2 units per square foot, such as a 5"x12" rectangular sized tile 
G - Smaller "Gypsum" tiles of the era, with 4 units within a 10 square foot area, with no waste allowed
F.L. - For a standard sized brick laid flat; or approximately 4.5 bricks per square foot area, regardless of design and with no waste allowed
B-1 - For a 1 brick thick wall (around 4") where there would amount to 6.5 standard sized bricks to a square foot

The amount of a material read from the quantity scale accounts for the typical 3/8" to 1/2" mortar spacing and the approximate 20% overage/waste, which is typical in the industry.  For the B-2 and B-3 readings, the value would be multiplied by the number of bricks used per square foot, which would need to be estimated if the brick's size is non-standard. 

This is a very flexible rule in that there is plenty of clean white areas for the user to mark their own lines on the material scale.  The rule would be useful in the modern era as well, especially for today's do-it-yourself home renovations, where DIYers might need help computing flooring materials. 

The Brick-O-Meter is quite rare, coming up on eBay at a frequency of maybe once per year, usually with an average price approaching $100.  This seems expensive, as materially it's the same rule as the 4058W rule which collectors wouldn't pay more than $5 for.  But it is a unique rule and a rare K&E item with a interesting story.​​

The unique Brick-O-Meter in my collection.

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National Union Radio Reactance Computer

M4 and M16 Graphical Firing Table

E-6B Aerial Dead Reckoning Computer

Model 8858 Special War Time Issue

Syrup Rule

Short Base Triangulation Rule

Model 4010 Stima Calculator

​K&E Chemco
link

K&E Decimal Keeping Mannheim
​link

American Optical Company
link

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Chapter 7: 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

Brick-O-Meter

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

​

 Demonstration Rules

68-1929 Deci-lon 10 Demonstration rule 6.5 ft