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Calculators like this FACIT C113 were a ubiquitous sight in accounting offices around the world in the 1950s and 1960s. My mother, for instance, remembers using one of these during her brief employment at an accounting firm in Budapest sometime in the late 1950s (her boss had an electric version! Of course it was the boss who used her calculator the least often...) I recall spending many an evening playing with one such calculator in the small administrative office of the resort hotel north of Budapest where my stepfather was a manager.
Although much larger, these calculators operate on the same principle as the legendary Curta, using a steppeddrum mechanism. Thanks to the movable carriage, these machines are well suited for multiplication and long division. Although the method is somewhat cumbersome, it is possible to compute square roots on the C113 with reasonable efficiency (I wouldn't want to be doing it all day long, mind you.) The picture to the right shows the result of one such calculation: I just finished computing the square root of π.
These mechanical marvels have all but disappeared in recent decades. The last mechanical FACIT calculator was reportedly made in India in 1982. I was fortunate to have been able to acquire a machine in surprisingly good condition, working flawlessly more than three decades after its manufacture.
As these laborsaving machines are replaced in offices around the world by newer labor saving devices such as accounting software packages and corporate networks, I cannot help but wonder. Cranking the handle can be a pain in the proverbial posterior, but is it really that much worse when compared with frustrating network outages, software bugs, and other mysterious computer problems? Ask your accountant next time what he or she thinks...
Update (September 2013): I wrote the paragraphs above over ten years ago, when I acquired my Facit C113. The machine has been sitting in a cardboard box for many years. Recently, I took it out and discovered much to my horror that it was seized up. At first, I suspected that a mechanical interlock was the culprit. I partially disassembled the calculator (I am not sure I am skilled enough to completely disassemble and reassemble it) but could not find the culprit, not even with the help of an excellent Web site that I found, which provided plenty of useful detail.
Eventually, I realized that the problem was not an interlock after all, but some bearings that seized as the lubricant dried up. Judiciously applying some WD40 did wonders! I had a bit of trouble reassembling the calculator (replacing the carry rotor is not easy because you need to insert it in the correct orientation in order for the carry to work) but it now works as new. Almost as new anyway.
Then, much to my horror, I discovered that I don't remember the square root algorithm anymore! Fortunately, a little further research helped, especially after I found a Germanlanguage description of Töpler's method. I used this exact algorithm in the past when computing square roots on mechanical calculators like the Curta or the C113.
So here is the way it goes. The first step is to mentally divide up the digits of the number into groups of two, starting at the decimal point. Then, starting with the leftmost group of (1 or 2) digits, we successively subtract, 1, 3, 5, etc. If we can no longer subtract the next number from what is remaining, we increase the number by one, and begin the same process with the next digit. We can repeat this process iteratively, computing the square root one digit at a time.
So suppose we wish to take the square root of pi = 3.14159265. Digits are grouped as  314159265. The best way to do this on the C113 is as follows:
0. Clear the calculator.
1. Enter "314159265" into the register. Shift it all the way to the left and then transfer it to the accumulator. The display should now read:
3141592650000 00000001
314159265____
2. Clear the counter and the register. Do not clear the accumlator.
3. Enter "100000" into the register and shift it left 7 places. Subtract it from the accumulator once by turning the crank backwards. The display should now read:
2141592650000 10000000 ()
100000_______
4. Clear the register. Note that you cannot subtract 300000 the same way because it is bigger than 214159. So we move on to the next decimal position. We do that by a) increasing the last digit by 1, and starting the 1, 3, 5 sequence at the next decimal position. I.e., enter "210000" into the register and shift it to the left 6 places. Subtract from the accumulator by turning the crank backwards:
1931592650000 11000000 ()
210000______
5. Clear the register, enter "230000", shift left 6 places and subtract:
1701592650000 12000000 ()
230000______
6. Repeat with "250000", "270000", "290000", "310000" and "330000":
1451592650000 13000000 ()
250000______
1181592650000 14000000 ()
270000______
0891592650000 15000000 ()
290000______
0581592650000 16000000 ()
310000______
0251592650000 17000000 ()
330000______
7. Note that you cannot go further, as 350000 is greater than 251592 so it is time to move on to the next digit. That is, we increase the last digit by one (from 330000 to 340000) and the begin the process anew at the next position. Clear the register, enter "341000", shift left 5 places and subtract:
0217492650000 17100000 ()
341000_____
8. Repeat using "343000", "345000", "347000", "349000", "351000" and "353000":
0183192650000 17200000 ()
343000_____
0148692650000 17300000 ()
345000_____
0113992650000 17400000 ()
347000_____
0079092650000 17500000 ()
349000_____
0043992650000 17600000 ()
351000_____
0008692650000 17700000 ()
353000_____
8. Next digit. Clear the register, start with "354100" shifted left 4 places, and repeat with "354300":
0005151650000 17710000 ()
354100____
0001608650000 17720000 ()
354300____
9. Next digit. Clear the register, start with "354410" shifted left 3 places, then "354430", "354450" and "354470":
0001254240000 17721000 ()
354410___
0000899810000 17722000 ()
354430___
0000545360000 17723000 ()
354450___
0000190890000 17724000 ()
354470___
10. Next digit. Clear the register, start with "354481" shifted left 2 places, followed by "354483", "354485", "354487" and "354489":
0000155441900 17724100 ()
354481__
0000119993600 17724200 ()
354483__
0000084545100 17724300 ()
354485__
0000049096400 17724400 ()
354487__
0000013647500 17724500 ()
354489__
11. The rest is easy, as we can now proceed as with regular division. Do NOT clear the register, just shift the number right by one position and subtract as many times as you can:
0000003012830 17724530 ()
354489_
12. Shift one more time to the right and subtract as many times as you can:
0000000176918 17724538 ()
354489
That's it. The square root of pi is 1.7724538 to 8 significant digits.
There are other methods but this is the best that I can presently recall, which requires no a priori estimate of the square root, no external tools, and relatively little mental arithmetic (you just need to keep track of the number that goes into the register).
One tricky bit is ensuring that you start with the right decimal position; obviously, the digits of the square root of 2 are quite different from the digits of the square root of 20.
The other tricky bit is using the calculator's decimal places with maximum efficiency. This was accomplished by shifting the value all the way to the left before transferring it to the accumulator, and always using 6digit numbers, padded with zeroes, when subtracting.