**Last Update: Sept 18, 2015 -- THE CURTA REFERENCE**

- How to Calculate with a Curta - by Nicholas Bodley
- The Amazing CURTA! - by Bruce Flamm
- CURTA 2000 - by George E Heath
- Algorithms for calculating Natural Log and e^x - by Steven Alford
- Calculating Curta square roots - by Daniel F F Ford
- Curta Observations - by Chris Writt
- "Crunching Numbers" - by Cliff Stoll
- "The Curious History of the First Pocket Calculator" - by Cliff Stoll
- When was your Curta made? - by Daniel F F Ford
- An algorithm for long divisions - by Silvio Henin
- Curta-NYTimes-110652.pdf - from Jim Chen
- Curta-NYTimes-011058.pdf - from Jim Chen
- Curta-NYTimes-050460.pdf - from Jim Chen
- Curta-NYTimes-010961.pdf - from Jim Chen

One of the most delightful small mechanisms ever sold is the Curta brand of handheld mechanical calculators. These are a radical rethinking of the layout of a calculator, and were first put on the market roughly about 1953. They were very popular with sports-car rallyists because of their architecture; the number entered could be retained very easily (it had to be cleared out manually, one or a few digits at once).

I know of two models, the larger one being able to work with more digits. I used to own one of the smaller ones; it would accept an eight-digit number, and had an eleven-place accumulator with six positions. In one calculation, you couldn't see all 6 + 8 = 14 digits of a product.

Apparently, the inventor was Curt Herzstark, if memory serves me right. They were originally made in the Principality of Liechtenstein, which is generally east of Switzerland, on the border.

The Curta is basically a cylinder with a handcrank on the end. It has an amusing resemblance to a pepper mill. The mechanism that does the job of the carriage in desktop machines is like a large knob, on the crank end of the cylinder. The dials are tiny cylinders (thick disks) on shafts with their axes extending radially from the center. You shift the "knob carriage" (let's call it a "display") by pulling it against spring tension away from the main body of the machine and rotating it to its new position, then gently letting go.

You hold the calculator in your left hand, with thumb and forefinger on the display. Your right hand operates the handcrank. You always turn the crank clockwise, the "easy" and natural direction. (Sorry, as so often happens, this machine is distinctly right-handed!) You should NEVER turn the crank backwards! On my Curta, the anti-reversal ratchet is quite delicate, and easily overridden.

One turn enters the number into the accumulator once; multiplication requires several turns, typically, and shifts of the "knob". To subtract, you pull out on the handcrank, which snaps happily into place; a shiny collar shows that the handcrank is out. You turn the crank the same way, and the number is subtracted from the accumulator, but by adding the complement.

The display, in addition to the accumulator, contains a cycle counter that shows the number of machine cycles in each of its positions. There is a complete carry, and the sense of the count can be reversed for division and square root (use the "fives method") to show a correct quotient/root.

There is a finger ring that swivels from a stowed position and locks into place; this ring is attached to a rotating piece that clears either the accumulator, the cycle counter, or both. It normally remains in either of two positions. You need to lift the "knob" away from the main body to clear. Incidentally, everything is interlocked; you can't lift the "knob" while the machine is in mid-cycle, you can't clear the dials if the "knob" is in its normal position, and you can't turn the crank if the "knob" is pulled away from the body.

You enter numbers by positioning tiny sliding knobs that stick out through slots in the side of the housing. They detent nicely as you change settings; tiny dials show the number you've entered. These knobs position tiny 5-tooth gears to engage with the proper level of a big, central "stepped reckoner" which has interspersed layers of complement-count teeth for subtraction; if you carefully misposition a digit-entry slider, you can engage these teeth.

Pulling out the handcrank moves the stepped reckoner by (I think) one and one-half "digit spaces", referring to the digit-entry sliders.

Subtracting all zeros rotates the least-significant accumulator dial (that is engaged with the body, that is) by exactly one full turn; this is guaranteed to cause a carry. All other columns advance their dials by 9 places, and the carries propagate to return all to their original positions.

This is a really beautiful piece of machinery, the equal of the innards of a fine camera. Some idea of its relative sophistication can be gained by considering what is on the shaft for each display dial. First, of course, is the dial itself. Then, there's a 10-lobe detent cam that ensures that a dial is always positioned correctly, never between digits. There's a tiny spring-loaded ball that bears against this cam.

There's also a little pin or such that sets the carry slide when the dial moves between 9 and 0. There is, quite likely, also a Geneva-drive locking disk that ensures that a dial will rotate only when the main mechanism unlocks it; a dial is probably locked at all other times. This mechanism ensures that the dial doesn't overshoot if someone develops a really-fast handcrank technique. (All calculators except the newer Marchants have such a lock.)

There is still more; there's a 10-tooth gear to advance the dial when the shaft in the main frame turns a tiny 5-tooth crown gear. This gear, most likely, serves to clear the dial when you clear the display. To keep the dial from going past zero when clearing, a part of one tooth is machined away so that the dial will stop when it reaches zero.

All of these various mechanical elements are mounted on a tiny shaft that is probably shorter than 1/2 inch.

If you are a careful person, you can see part of the mechanism by removing the screws on the end cover opposite the crank. You won't see much relating directly to the display, but you can watch the carry slides set and reset. The stepped reckoner in the middle is dramatically apparent, and it's quite easy to see how the setting slides and the stepped reckoner work.

In my calculator, I removed the setting shafts and knobs; I was astonished to find that the tiny porous-bronze bearings at the ends of the shafts were individually hand-fitted; I'd assumed they were interchangeable. They weren't; I did my best, having a big, long, trial-and-error session of swapping!

If you know what you're doing, this is a good opportunity to "clean house" and relubricate judiciously (use clock oil?) inside. It's a sacrilege to use a cheap oil! Reinstalling the cover is rather easy; there is a small "key" that positions it precisely. Rotate it back and forth a wee bit to make this "key" drop into its slot.

I would not recommend trying to unpin the handcrank unless you're an experienced machinist or mechanical technician; to begin with, on my Curta, at least, it's probably a taper pin, and it isn't obvious which end is bigger. The crank absolutely MUST be "backed up" with a large mass, such as a kilogram or so of lead. (I have not seriously tried, on mine.) Don't even think of laying the Curta on its side and banging away on a pin punch.

To a person who appreciates small, precision mechanisms, a Curta is a very special device, a real treasure.

Perhaps no calculator of any type has generated more discussion than the Curta. Although a book could be written on this incredible device and its inventor, I will limit this article to a very brief summary. In the future I'll discuss more details and explain how this wonderful machine actually works.

What is it? The Curta looks like a small metal pepper-mill or coffee grinder. It is, in fact, a precision instrument which performs calculations mechanically using no electric or electronic parts. I can best describe the sensation of turning the crank on a Curta as being similar to winding a fine 35mm camera. To the best of my knowledge it is the smallest mechanical calculating machine ever built.

Who invented it? Mr. Curt Herzstark of Austria.

How was it invented? Over the years I've heard rumors that Mr. Herzstark secretly developed the Curta while imprisoned in a German concentration camp. Apparently this is almost true. Herzstark was a prisoner at Buchenwald but the camp leaders were aware of his work and encouraged it. They apparently wanted to give the invention to the Fuehrer as a victory gift at the end of the war! Herzstark was given a drawing board and worked on the design day and night. The camp was liberated in April, 1945 by the Americans. Herzstark survived as did his revolutionary concept for a miniature calculator. (More details about how Curt Herzstark ended up in a concentration camp in a future issue.)

When were the Curtas made? Although several prototypes were made, the first production began in April, 1947. The last Curta was made in November, 1970 but they were still sold until early 1973. By then, pocket electronic calculators were selling for under $100 and a precision mechanical instrument like the Curta could no longer compete.

Are there different Curta models? The Curta II is slightly larger than the original Curta. It was first produced in 1954 and has a larger numeric capacity than the Curta I.

How many Curtas were produced? In 1949 only about 300 Curtas could be produced each month. By 1952, production had increased to about 1,000 units per month. Over the course of about 20 years approximately 80,000 of the Curta I and 60,000 of the Curta II were constructed.

Are there any Curta prototypes? Mr. Herzstark kept three prototypes in his home in Nendeln, Liechtenstein. Upon his death in 1988 they were sold to a private collector.

Are there different protective cases? Since the Curta is a precision instrument, it was sold with a protective capsule or case. These are screw-top cylinders often with internal padding. Some plastic capsules were made but these were found not to be suitable so production was stopped and the metal cases again prevailed. Apparently Prince Heinrich von Liechtenstein found the metal case to be to rigid for a precision instrument so he had a special leather case crafted for his Curta.

Why does the cap of the Curta's protective capsule screw on backwards? I had always assumed that this was to prevent someone from hastily unscrewing the cap and dropping the fine instrument on the floor. I recently learned that this was actually a design change made to prevent the accidental turning of the Curta's operational crank when the case was screwed shut. This accident would leave the Curta not ready for calculation when the case was opened. Early metal cases having a clockwise closing cap are apparently quite rare.

Special thanks to member Bob Otnes in Palo Alto, CA and Peter Kradolfer in Germany for providing information used in this article. Please send any interesting information that my be helpful for future articles on the Curta to Bruce Flamm, 10445 Victoria Avenue, Riverside, CA, 92503.

Curt Herzstark envisioned a world free of unnecessary thinking when his Liechtenstein jewels first appeared in 1947. The original design was so nearly perfect that very few changes were ever introduced during its 23 years in production. Back then, Y2K was a part number for a vacuum tube. But time and progress do not stand still and if Curt were alive today, he would be producing the Curta 2000 model. Alas, Curt is not alive to carry out this significant product improvement, so I have taken up the challenge and developed a modification for your type I and II Curta calculators.

Curt eliminated unnecessary thinking - now the drudgery of manual cranking is going to be a thing of the past. This article provides the details to convert your low-tech pepper grinder into the motor-driven Curta 2000. It's like upgrading your 286 computer to a Pentium III - and nearly as easy.

For the stage one upgrade, you'll need the following tools and materials;

- A Nikon MD4 or similar motor-drive

- A hacksaw (18 tooth blade -Curt says it cuts faster)
- A flat bastard file (medium or coarse)
- 3 feet of duct tape (the handyman's friend)
- Black modeler's paint (optional)
- A hot glue gun
- A screwdriver or pry bar.

- Take a 2-inch piece of the duct tape and place it on the top of the calculator, under the crank handle. This will protect the fine finish from being marred by the subsequent sawing and filing.
- Locate a line 4mm from the crank axis and cut off the crank with the hacksaw.

*Curt's Tip #1 - Save time and labour! Just saw 3/4 of the way through and pry up on the crank with the screwdriver - it'll break right off. You've just saved about 15 saw strokes and 20 seconds of time! - Use the flat bastard file to clean up the cut surface of the crank - remember appearance is important

*Curt's Tip #2 - Try to prevent too many of the metal particles generated from the preceding steps from entering the calculator - they can cause rough operation and binding of the gears. If your unit does seize - just force the crank to work the chunks through. Use vice-grips if the crank has already been removed. - Apply a bit of black paint to the exposed metal on the crank (and any plier marks) - it gives a nice finished appearance.
- If you have one of the early calculators with a rounded dome at the crank axis, flatten it with the file.
- Apply a generous gob of hot glue to the crank stub, and quickly press the drive end of the Nikon motor drive into the soft glue.
- Wrap about 4 turns of duct tape around the now joined calculator and motor drive to permanently connect them together.

*Curt's Tip #3 - Position the motor drive so that it does not interfere with the movement of the setting knobs or the counting dials.

Curt say's "if some's good, more's better, and too much is just enough!" And who can argue with that? You've already made a big improvement to your calculator with the Curta 2000 modification- but why stop there? You can get more - much more - from this transformation.

The Nikon MD4 is powered by 4 AA cells, which provides 6 volts DC at fairly low amperage. In stock trim, the MD4 will cycle in about 3/4 seconds - faster than hand cranking, but still not fast enough. Tool Time's Tim Allen would tell ya' - this baby needs more power! Using the booster cables from your trunk, connect your car battery to the motor drive. You've doubled the voltage and hooked into virtual limitless amperage.

*Curt's Tip #4 - Watch the polarity! If you've reversed the connection, you'll drive the calculator backwards. This setup has the power to overcome the ratcheting mechanism that normally prevents this condition. If the calculator makes a crunching sound and metal bits fall out when you cycle the motor drive - it's reversed. Also, you may want to wear oven mitts when using the 12 volt connection - it tends to get hot. If thick smoke or flames appear, discontinue use momentarily.

You've completed the mechanical and electrical mods to your Curta, or should I say - your Curta 2000 - now its time give the ascetics a kick into the third millennium. The possibilities are endless but here are few tasteful changes that are real head turners. American car and motorcycle manufacturers learned long ago that you can't beat chrome plating (the more the better) to give your product the right look. The drab satin black finish on the Type I can be completely concealed under a luxurious layer of bright chrome. Just take the whole unit to an automotive bumper shop and have them throw it in their plating tank. Presto - like magic you have a gleaming icon of fine industrial design. Note: This process can obliterate all the numbers on the unit. If this happens to yours, just reestablish them with an ink marker.

In addition to, or as a separate option, what about speed stripes. You choose the colour and size - express your individuality. Stripping tape is available at auto parts stores.

Interior lighting, engraved initials, digital interface, drilled lightning holes, anti-theft alarm, shot peening, …you name it. I'll bet some of you creative types are thinking of mods right now that you could make to improve the performance and appearance of your Curta.

Machine ready!

Editor's Note: Mr. Heath is currently working on a mechanical/electronic hybrid version of the Curta 2000. The output digits will be replaced with LED displays and the input sliders will be covered by a numeric keypad! The unit will retain the gear based mechanical CPU. It may be powered by a small wind generator, but early experiments have proved disappointing. The minimum 11 mile per hour air flow is seldom present in indoor settings. One of the possible solutions is to increase the diameter of the turbine from 6 inches to 4 feet, but this would add an extra 45 pounds and considerable bulk to the once pocket sized calculator.

Mr. Heath is interested in obtaining type I and II Curtas (by donation) to further his research. Your unit will be returned as a prototype when completed.

**Note 1.** Do not carryout any of these modifications on your calculator!

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Using the Curta calculator to calculate the Natural Logarithm and Exponential functions without using tables.

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Natural Logarithm

Preliminary requirements: memorize three numbers.

Ln(2) = 0.69315 (remember as 6-9-3-15) Ln(5) = 1.60944 (remember as 160-944)

Bottom of range = 0.77

Step 1.

Eliminate tens

For example, if you want to find ln(187.2), you would find ln(100*1.872), which is equal to ln(100) + ln(1.872)

Step 2.

Move into the "range" by eliminating twos or fives.

The "range" is from 0.77 to 2*0.77, or from 0.77 to 1.54.

In this case, we would say that ln(1.872) = ln(2*0.936) = ln(2) + ln(0.936).

Note that 0.936 is in the range [0.77,1.54].

Step 3.

Use formula for part that is in range. Make sure to subtract 1 first!

Note that the Taylor series expansion for the natural log is as follows:

Ln(1+x) = (x)/1 (x^2)/2 + (x^3)/3 (x^4)/4 + (x^5)/5 … for -1 < x < 1 < 1

Now a way to show successively improving approximations of this series is as follows:

(x)/1 (x^2)/(1*2)

(x)/1 (x^2)/2 + (x^3)/(2*2)

(x)/1 (x^2)/2 + (x^3)/3 (x^4)/(3*2)

(x)/1 (x^2)/2 + (x^3)/3 (x^4)/4 + (x^5)/(4*2)

We will only need to go as far as the second approximation. It can be rewritten as:

x + (x/2)(x/2)(x 2)

3A: find x.

Since we're using 0.936 = 1+x, we have x = -0.064.

3B: square half of x.

So to start to do this on the Curta, we will manually enter |x|/2 on the SR. In this case, 0.064 / 2 = 0.032. Square this to get 0.001024. Now enter this in the SR and clear the RR.

3C: multiply by (x 2) Multiply this by (-0.064 2) on the Curta. Since all the additions in 3C and 3D are negative, you can just do them as positives and deal with the negative sign later.

3D: add in x.

Then clear the SR and enter x (-0.064 in this case), and add it in once. Again, since all the additions are negative, you can just do them as positives and deal with the negative sign later.

Now you have ln(0.936) = -0.066113536. This will be accurate to within + or - 0.002. The true value is ln(0.936) = -0.0661398.

Step 4: Add in your twos and fives

We have calculated ln(0.936) = -0.06611. Clear the Curta and put this in the SR, and use one negative turn to subtract it.

Now add in ln(2) = 0.69315, memorized above. Now we get ln(0.936)+ln(2)=0.67204, which means that ln(0.936*2) = ln(1.872) = 0.67204.

Step 5: Add in your tens.

Since we're looking for ln(187.2), we need to further add in ln(2) twice and ln(5) twice. Since we already have ln(2) in the SR, two more positive turns, then two positive turns with ln(5)=1.60944 in the SR.

Final result: ln(187.2) on the Curta is 5.23222, + or 0.002. Actual ln(187.2) is 5.23218. Error is 0.00004.

Final note on error:

Most of the error of + or 0.002 is in the ends of the [0.77,1.54] range. If you subtract 0.001 if you are very close to the ends, in the [0.77,0.82] and [1.49,1.54] ranges, your error will decrease to + or 0.001.

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Recap of steps:

Step 1. Eliminate tens by division

Step 2. Move into the "range" [0.77,1.54] by eliminating twos or fives by division.

Step 3. Use formula for part that is in range: ln(1+x) ~= x +
(x/2)(x/2)(x 2)

3A: find x.

3B: square half of x.

3C: multiply by (x 2)

3D: add in x.

Step 4: Add in your twos and fives

Step 5: Add in your tens.

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Exponential Function (e^x)

Preliminary requirements: memorize one number

e^0.5 = 1.64872 (remember as 16-48-72)

(assuming you have already memorized e=2.7182818.)

Step 1: separate out wholes and halves from remaining fraction.

Exp(1.74) = Exp(1.5) * Exp(0.24)

Step 2: use formula for fraction

Note that the Taylor series expansion for e^x is

e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + (x^4)/4! + (x^5)/5! … A good approximation for e^x over the range [0,0.5] is:

e^x ~= 1 + (x)/1 + (x^2)/2 + (x^3)/5.

This can be restated as

0.1 * [(10 + 10x) + (x^2) * (5 + 2x) ]

2A: find square of x

0.24^2 = 0.0576. Now clear the Curta and put this in the SR.

2B: multiply by (5 + 2x)

Multiply the 0.0576 by 0.24 twice and by 5 once. You should get 0.315648

2C: add in (10 + 10x)

Clear the SR, and add in 2.4 once and 10 once. You should get 12.715648.

2D: Divide by 10 (just move the decimal point).

You will now have 1.2715648

Step 3: Multiply by whole and half powers of e. Exp(1.74) = Exp(1) * Exp(0.5) * Exp(0.24) = 2.71828 * 1.64872 * 1.27156 = 5.69874

Actual value is 5.69734, error is 0.00140

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Recap of steps:

Step 1: separate out wholes and halves from remaining fraction.

Step 2: use formula for fraction: e^x ~= 0.1 * [(10 + 10x) + (x^2) * (5 + 2x)]
2A: find square of x

2B: multiply by (5 + 2x)

2C: add in (10 + 10x)

2D: Divide by 10 (just move the decimal point).

Step 3: Multiply by whole and half powers of e.

The Calculator Reference by Rick Furr (rfurr@vcalc.net)

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