If you’ve ever spent time thumbing through back issues of magazines like Scientific American or New Scientist, you may have seen adverts for the Curta – a strange little device that resembles a pepper mill. It cost a shocking amount of money and was claimed to perform all sorts of arithmetic functions purely mechanically and with incredible precision. Rather than being a scam in the order of upmarket X-ray specs, the Curta lives up to the claims and the story behind its creation has its roots in a Nazi death camp.

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For most people today, a calculator is just another phone app. It’s one of those devices that we take for less than granted, yet for those of us of a certain age it was nothing less than a technological liberator when it arrived, freeing scientists, engineers, and anyone who routinely used math from hours or days of tedious calculations. Indeed, it’s sobering to think of how many years everyone from Johannes Kepler to unknown bank tellers lost due to number crunching prior to its arrival.

Some relief from tedious manual number crunching came in the 17th century with the invention of logarithms followed by mechanical aids like the slide rule and the first adding machines. Unfortunately, they weren’t that much of a help. Slide rules couldn’t handle numbers to more than two or three decimal places and, for centuries, calculators were little more than curiosities that were about as practical as a toy automaton.

The Curta was sold mainly through mail order

By the late 19th century, commercial desktop calculators began to appear, which were a definite improvement, but these crank-powered monsters were so expensive that only larger businesses could afford them. They were also about as portable as a fishing anchor, with even the “lightweight” machines clocking in at around 34 lb (15 kg). As the new century dawned, many improvements were added, like multi-key systems, a motorized mechanism, and the ability to automatically multiply and divide. But they were still huge and expensive, which limited their appeal.

Then along came Curt Herzstark, a young man who in the 1920s regularly traveled through the former Austrian Empire selling mechanical calculators to banks and other businesses. It was on these travels that he heard the same complaints from his customers.

“And again and again, wherever one went, competitors came with wonderful, big machines, which were ever more expensive and electric, but something was missing in the world market,” said Herzstark in an extensive interview (PDF) conducted by the Charles Babbage Institute in 1987. “‘I would like to have a machine that fits into my pocket and can calculate. I am a building foreman. I am an architect. I am a customs officer. I have to be able to pick something up. I cannot go 10 kilometers to use a calculator in the office. The slide rules are not useful for my purpose. Slide rules cannot add or subtract. And aside from that you can only read three values from the markings on them, not more. For an invoice I have to know exactly.’ So, I continually found interest in a pocket calculating machine. Of course, the whole world seemed to be interested in solving this problem.”

Examples of mechanical calculators

This would simply have been a keen observation by a perceptive salesman, except that Herzstark was the heir to the firm of Rechenmaschinefabrik der Austria Erstanden Compagnie in Vienna, which was one of the first Austrian calculator manufacturers and sold improved versions of American machines to the local market. Born in 1902, Herzstark was being groomed to one day take over the business started by his father and, in addition to sales, he had already received extensive training in how to design and build intricate mechanical devices. So it was no small wonder that customer complaints started him thinking along practical lines.

For the next 10 years, Herzstark mulled over the problem of how to radically reduce the size of calculators, but it was far from a simple task. There were handheld calculators of a sort on the market, but these were little more than cheap, crude toys that worked along the lines of an abacus and could do little more than add and subtract – if even that.

Real calculators were so large and heavy because they were enormously complex devices that had to be built out of solid metal parts if they were to work accurately without jamming. Also, it was common practice to have a complete set of number keys for each column of digits, so the user was faced with a solid slab of keys, including special ones for accounting numbers.

What’s worse was that inside the device each digit of a number was set on a separate register with its own mechanism that had to be repeated up to eight to 10 times. Then to compound the problem, subtraction meant duplicating all these in reverse, plus a special mechanism to handle carrying the number when necessary. No wonder a handheld calculator seemed as unobtainable as easy-open blister packages.

Hertzstark’s answer was to forget about the inside of his tiny calculator and concentrate on designing it from the outside in.

“I started to concentrate on possible solutions and at first, naturally, didn’t get any further,” said Hertzstark. “Later, I had an idea that I should look at everything backwards. I thought to myself, I’ll pretend that I have already invented everything. What does this kind of machine really have to look like, so that someone could use it? It cannot be a cube, or a ruler; it has to be a cylinder so that it can be held in one hand. And if one can hold it in one hand, then if is miniaturized, you could adjust it with the other hand. And you could work either its sides or top and bottom. You can make the answer appear on top.”

Diagram of the Curta calculator

Then in 1937, Hertzstark had a breakthrough. Instead of making a machine that could add and subtract, make one that did nothing but add, but in such a way that it also subtracted.

“I can remember. I sat in a compartment alone and looked out and thought at that moment, ‘Good Grief! One can get the result of a subtraction figuratively by adding the complementary number to it.’ This has long been seen with the Burroughs machine which only added. When someone entered in something and it was wrong, one could correct it by adding a number, which when added to the wrong number yielded zeros and hence the unwanted number was out again. Then I thought that works exactly the same way as subtraction registers. So if I enlarge the second step register, the result can be achieved through pure addition … that was that … “

What Hertzstark did was apply what’s called the method of complements to the problem. Without getting tangled up in the math, this method relies

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