Keuffel & Esser 4071 Duplex

The model 4071 duplex was fabricated in the United States from 1901 to 1917.  This twelve inches long rule - ten-inches long scales  mahogany duplex linear slide rule is coated with celluloid on the front and back only, the edges are bare. Scales are engraved on the celluloid. This model was a new version of the 4070 duplex model fabricated since 1891, where the main different characteristic is the cursor, a chisel cursor in the original 4070, and a clam shell glass indicator in the 4071.

The model 4071 was first announced in the K&E 1901 Product Catalog with the description: "like No. 4070 but with glass indicator", and a $ 8.00 sale price (about $238 in 2023 USD). This model with the  clam shell cursor first appears in the 1909 Catalog, with a $ 5.00 sale price (equivalent to $165 in 2023) .

The 4071 model has eight scales in total but two of them: A and B, appear on both sides of the rule, so this model really has only six different scales.  Another interesting characteristic is the two reversed scales: the single logarithmic CI and a double logarithmic BI, on one side of the rule slider. Keuffel & Esser used this double feature in several duplex models produced in the period 1899-1920, but never used this again in later models.

The scales on the 4071 are:  A [B,  C] D on the front face, and A [BI, CI] D  on the back face. The specimen in this collection has no serial number, only the patents dates: Jun 5, '00  Dec 22, '08, so it can be assumed that this specimen was fabricated between 1909 and 1915, being this last the year when the frame-less cursor was patented, according with the specimen displayed by the ISRM.

The 4071 model had several modifications during the productions years: different indicators like the frameless (introduced in 1915) and column cursor (introduced in 1913), and additional L, S, and T scales. There were also models with the wire framed cursor, as the rule shown below, that can be assumed were produced after 1916. The 4071 model is not pictured or mentioned in any K&E catalog printed in the period 1921-1938 (McCoys KE Catalogs). 

Years later, in 1939, Keuffel & Esser introduced the new model 4071-3 Polyphase Duplex Decitrig, with a much more robust scale set.

The number of numerical problems that could be solved with the original 4071 model small scale set, may be seen as very limited. No log-10 scale? No trigonometric scales? Not even a cubic scale? What operations beyond multiplications-divisions, squares and square roots, or their combinations could be performed? Well, the skills of the user should no be discarded...  Calculating cubes or cubic roots without  a K scale? Well, certainly this cannot be done directly but it can be done with one slide rule setting. 

• Calculating Cubes. To calculate the cube of a number without a K-scale but using the scales A, B, and D of the rule's front face,  consider that the cube of a number is the product of its square times the number

(*) Important: In many cases when using the double scales A and B for multiplication or division, you will find the result on any of the two portions of the double log scales, so you may select to read the answer in any of the portions, but don't forget the right placement of the decimal point after you multiply. 

• Calculating Cube Roots: To calculate a cubic root without a K-scale, using the scales A, BI and D on back face, a couple of things will have to be considered. 

(1). Assume the number "x"  to be cube rooted is the cube of another number "a", then algebraically we have: 

(2). The number  x  to be cube rooted has to be located in the correct portion of the double scale A. The next figure can be used as reference: 

Cubic Root of 64:  Set right index on scale BI to 64 on scale A (on right section), move the cursor to the left section of scale A until hairline be over the same number on scales A and CI: 4

Cube root of 640:  Set left index on BI to 640 on A (left section), and on the same left section, move the cursor until the hairline is over the same number on scales A and CI: ~ 8.62

Cube root of 6400:  Set rigth index on BI to 6400 on A (right section), and on the same right section, move the cursor until the hairline is over the same number on scales A and CI: ~ 18.6

Compositions of squares and cubic roots or cubes and square roots can be calculated with the 4071 combining the above steps with the regular procedures to find square and square roots using scales A-B and C-D.

References:

• McCoy, Clark: Keuffel and Esser Catalog Site. K&E Catalogs and Price Lists for Slide Rules.  http://www.mccoys-kecatalogs.com/KEmain.htm 

This website is the most extensive digitla file of Keuffel and Esser product catalogs and price lists (prices  shown here were taken from these lists)  from 1881 until 1972. You can find also user manuals for many slide rule models, as well as other interesting brochures and materials.

• Shawlee, Walter: Keuffel & Esser Co. Slide Rules. https://www.sphere.bc.ca/test/ke-sliderule.html 

This website gives a good overview of models, and other interesting information and links