Slide Rule Wristwatches

Slide Rule Wristwatches

SLIDE RULE WRISTWATCHES

Robert Adams

Robert graduated with a Electro‐Technology Diploma from the SA Institute of Technology in 1970, using a slide rule (Thornton P221) and with a Engineering Degree in 1979 using a electronic calculator (HP25). He is currently the Principal Strategist for ElectraNet, an electricity transmission company. He started collecting slide rules approximately 10 years ago and has currently approx. 400 rules in the collection. A main focus of the collection is “Electro” slide rules and rules that have hyperbolic functions. Robert currently resides in Enfield, South Australia.

A slide rule watch can be defined as any watch which combines the normal chronological

functions with a calculating device that uses logarithmic scales. A small number of watches have only one logarithmic scale that rotates around a fixed time scale or have only one logarithmic scale for decoration, I do not consider these to be slide rule watches.

IntroductionThe first slide rule watch was probably a pocket watch designed by Meyrat & Perdrizet in

France near the turn of the 19th century.

The slide rule wristwatch has a relatively recent origin, appearing from 1940 onwards. There is some dispute about who manufactured the first slide rule wristwatch, but it was certainly a Swiss firm. The first three slide rule wristwatches came from the firms MIMO, Juvenia and Breitling. The Mimo‐Loga was possibly the first, with its patent application appearing on July 27, 1940, some weeks before Breitlingʹs patent for the

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Chronomat which was submitted on August 26, 1940. The Juvenia Arithmo is usually dated around 1945 when it became commercially available. The images of the Mimo‐Loga, Juvenia and the Breitling below are taken from Art Simon’s Slide Rule watch site.

However In 1952, Breitling also introduced a pilotʹs wristwatch with an integrated circular slide rule that incorporated scales specialized for flight calculations: the Navitimer. This watch which was referred to by Breitling as a ʺnavigation computerʺ, featured airspeed, rate/time of climb/descent, flight time, distance, and fuel consumption functions, as well as kilometre–nautical mile and gallon–litre fuel amount conversion functions. This watch, available in larger commercial quantities, came to epitomise the slide rule watch.

With few exceptions no other watch manufacturers introduced slide rule models until the 1960’s. This may be attributed to the fact that Swiss patents have a term of 20 years, and so after the 1940 patents expired, other watch manufacturers felt free to incorporate logarithmic scales on their products.

The Japanese (Seiko and Citizen) introduced models around the 70’s. And now even budget brands such as Casio have introduced slide rule models.

Types Slide rule wristwatches can be broadly categorized into two classes based upon their scale

types and the number of scales included. The categories are described as follows.

Calculating Watches Aviator (or Navigator) Watches

Calculating watches These wristwatches usually have only two scales usually a C and D scale arranged in

similar fashion to the Mimo‐Loga patent. (Note: in this paper I will refer to the C scale as being the outer most scale on the watch). Examples of this genre are:

The Girard – Perregaux The Ventura Loga

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Aviator watches An aviator or navigator slide rule wristwatch by

definition should assist the navigator in calculations required by their profession. To do this the slide rule wristwatch needed to emulate the main functions of the E6‐B computer, the pilot’s manual calculating device.

The Breitling Navitimer was the first to do this by

combining the C and D scales on a rotating Bezel and inner face and with a direct 9‐hour speed scale on the clock face in a similar layout to the E6‐B. This arrangement enabled pilots to calculate airspeed, rate/time of climb or descent, flight time, distance, and fuel consumption functions, as well as kilometre–nautical mile and gallon–litre fuel amount conversion functions.

An E6‐B Flight Computer

Examples of this type are:

Breitling Navitimer with 9‐hour speed scale. (Note: not all Navitimers have this scale)

The Pulsar (probably 100 times cheaper than the Breitling!)

Use

The uses of the slide rule scales on a slide rule wristwatch are as per the normal methods used on any circular slide rule and any instructions for a circular slide rule would be able to be used for the slide rule watch. To illustrate the general calculations and aviation calculations I have condensed instructions from the Casio Watch Company. The full and unabridged instructions can be found on their website http://world.casio.com/. The examples use an aviator watch which has a outer and inner logarithmic scale and also a inner time scale. In these explanations I will use the terms inner and outer to refer to the logarithmic scales and the term inner (time scale) to refer to the time scale.

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http://world.casio.com/


Math Calculations Multiplication Example 12 x 15

Align 12 on the outer scale with 10 on the inner cale. Then, 15 on the inner scale corresponds to 18 on

xample 300 / 15

e outer scale with 15 on the inner ale. Then, 10 on the inner scale corresponds to 20

on

square roots can e obtained.

istance ert 45 miles into nautical miles and

5 on the outer scale with STAT on the ner scale. Then, NAUT on the inner scale

co

Example: Convert 16.4 oil lbs. into U.S. gallons and and litres.

scale with OIL LBS on e outer scale. Then, U.S. GAL on the outer scale

co

a outer

xample: Convert 13.1 fuel lbs. into U.S. gallons and and litres.

ner scale with FUEL LBS on e outer scale. Then, U.S. GAL on the outer scale

co

s the outer scale, taking into account the position

of the decimal point to obtain the answer of 180. Division E

Align 30 on thsc the outer scale, taking into account the position

of the decimal point to obtain 20.

With other methods ratios andb

Conversions DExample: Convkilometres

Align 4inrresponds to about 39 nautical miles on the outer

scale, and KM on the inner scale corresponds to about 72 km on the outer scale

Weight

IMP gallons

Align 16.4 on the innerthrresponds to about 2.2 U.S. gallons on the inner

scale, and IMP. GAL on the outer scale corresponds to about 1.8 IMP gallons on the inner scale, nd LITERS on the scale corresponds to about 8.3 litres on the inner scale Volume EIMP. gallons

Align 13.1 on the inthrresponds to about 2.2 U.S. gallons on the inner

scale, and IMP. GAL on the outer scale corresponds to about 1.8 IMP. gallons on the inner

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scale, and LITERS on the outer scale corresponds to about 8.3 litres on the inner scale Aviation Travel Time required

me required for the flight of an aircraft at 160 knots for 240

Align 16 on the outer scale with the speed in

r

Speed to distance e knots (air speed) for 250 nautical miles with a flight time of 1 hour and 40 minutes

Align 25 on the outer scale with ʺ1:40ʺ on th

Scales y all current slide rule wristwatches use a same scale layout with the two C and D

sc

wever some notable exceptions. he CI Scale

verse of the C scale. Invariably positioned as the outermost scale it increases in m

Example: Obtain the tinautical miles

dex (in this case the MPH line) on the inner scale. Then, 24 on the outer scale corresponds to ʺ1:30ʺ on the inner scale (time scale). Thus, the time required for the flight is 1 hours and 30 minutes. Note this only works for the same dimensions i.e. in this case knots and nautical m160 kilometres per hour and 240 nautical miles without conversion of one of the units.

iles. It would not give the correct answer fo

Example: Obtain th

e inner scale (time scale).Then; the speed index on the inner scale corresponds to 15 on the outer scale. Thus, the air speed for the flight is 150 knots. Again consistent units are required.

Nearlales running left to right. The other scales from the standard slide rule which are used for

roots, trigonometry, logarithms or other mathematical operations are rarely seen on wristwatches.

There are, ho

TThis is the inagnitude in an anticlockwise manner. It normally replaced the C scale on the watch as can be

seen in the following image.

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The Breitling for Bentley

Only the Juvenia Arithmo, early Breitling manual‐wind Chronomat, the current Breitling for Bentley models and a Chinese replica of the Breitling for Bentley have this scale arrangement. Although you can still perform the usual multiplication and division, the reason for this inverse scale is a little difficult to establish, it is really only convenient if multiplication by reciprocals (i.e. division) is your usual calculation. Trigonometric Scales

Very rarely have trigonometric scales been included on slide rule watches. The only example that I have seen is the magnificent Seiko 6138‐7000,

Seiko 6138 - 7000 Scale designations

The images above show that the outermost scale is an inverse sine scale (SI). The scale provides the answer to 1/sin(x) which would be of advantage in any calculation involving the law of sines. It is interesting to note that the scale also has the converse to the sine scale i.e. the cosine scale indicated in orange, which would also allow the calculation of problems involving 1/cos (x). This watch is also unique for another reason which will be elaborated in a later section.

Index Mark The index mark is placed at the 12 o’clock position usually on the inner scale. In the

majority of the aviator and other slide rule wristwatches the value on the D scale (the normally fixed scale) at the index position is 60. In other cases the slide rule watches mimic a normal circular slide rule and the value at the index position is 10 or 1.

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Slide Rule Wristwatches Examples

Index Mark = 60 Index Mark = 10

The reason for the placement of the index mark in these cases seems to be arbitrary. In most

cases Aviator type watches tend to have the index mark at 60. Where as most calculator wristwatches tend to have the mark at 10. Although this convention is not consistently applied.

Gauge Marks Many wristwatches, particularly the Aviator type, include gauge marks for many of the

calculations. Most calculations involve a fixed relationship and are used for the conversion of one quantity to another e.g. nautical mile to kilometres or litres to gallons. As such the placement of the gauge marks could have been arbitrarily distributed along the scale. I.e. taking the statute mile to kilometre gauge marks for example, as long as the relationship of approximately 1 to 1.61 is maintained between the marks, any values could be used. But nearly all watches conformed to the gauge mark values contained in the following table. Category Gauge Mark Value

Distance Nautical Mile 660 or 327

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Category Gauge Mark Value Statute Mile 760 or 380 Kilometre 1222 or 611

Fuel Litres 485 Fuel Lbs 766 Oil Lbs 960 Imperial Gallon 1065 US Gallon 1280

Mathematical Pi 3.14 Seconds 360 Lbs 3630 Kgm 1623 Feet 1430 Metres 4360

Cursors

Cursors on slide rule wristwatches are indeed rare; the only real example would again be the unique and magnificent Seiko 6138‐7000.

Rotating Cursor

Cursor Parts

Cursors are indeed a rare inclusion for a number of reasons. Not least is the fact that normally only two scales are usually involved and therefore an index mark is all that is required. Another reason could be that the implementation of a cursor on a watch means that a protuberance would be required and this would be prone to catching on pockets, garment edges etc.

Another form of cursor is that used in the Mondia illustrated below. The Mondia had two movable scales and had an engraved red hairline on the watch class. This provided a “cursor” function.

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The Mondia

Accuracy Sufficient accuracy and precision was and is the difficulty faced by all small length slide

rules and slide rule wristwatches being some of the shortest length scales suffered the most from problems with accuracy and precision.

The “accuracy” of a calculation system is the degree of proximity of the calculated result to its actual value. The “precision” of a calculation is the degree to which repeated calculations show the same results.

As the scales on a slide rule wristwatch are concentric the repeatability of the calculation would normally be guaranteed if there is no flex in the mounting of the scale rings. The placement of the scales had a direct bearing on the readability of the scales and hence the repeatability (precision) of the calculation. This is best demonstrated by the following images

Placement of the scales as close as possible to each

other made reading straightforward Whereas placement of the scales in this example

makes accurate reading of the result difficult to repeat.

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In most linear slide rules, even, the cheapest rule had scales that “tick” marks that aligned along the entire scale. The early and high end slide rule wristwatches, such as the Juvenia and Breitling also showed the same manufacturing accuracy. But some modern slide rule wristwatches demonstrate somewhat shoddy manufacturing techniques and the alignment of the tick marks can be so far out of alignment that errors in calculation can easily approach 10%.

Citizen Wingman Index aligned Tick Mark at 30 aligned

GE Ollech and Wjas

Index aligned

Tick mark at 30 out of alignment. 30 on the inner scale aligns with

29.8 on the outer scale

Evolution

As electronics drove to ever decreasing sizes it was inevitable that calculators could be produced in the size aspect of wristwatches. There are many examples of “4 function” calculators produced in wrist watch form but very few in scientific styles. Perhaps the most complicated watch is the Casio CFX‐200 shown in the following image.

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Like the HP35 to the slide rule this would seem to be the death knell for the slide rule wristwatches. But no, the Casio CFX‐200 has been and gone and the slide rule wristwatch is more prevalent than ever. Why? I would like to think that the slide rule, has at last found a position in the modern world, but alas I think it is a reflection of fashion or mode for the retro feel. Acknowledgements Firstly I would like to thank Art Simon for allowing me to use many of the images from his excellent Slide Rule watch site. http://sliderulewatches.googlepages.com/history.html. The use of these images greatly assisted me in the preparation of this paper. And I relied upon his site for the introductory historical information. Secondly, I would also like to thank David Rance for again editing this paper and his many suggestions that significantly improved it.

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http://sliderulewatches.googlepages.com/history.html


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Slide Rule Wristwatches