Calculators like this FACIT C1-13 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 stepped-drum 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 C1-13 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 labor-saving 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 C1-13. 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 WD-40 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 German-language 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 C1-13.

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 | 3|14|15|92|65|. The best way to do this on the C1-13 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 6-digit numbers, padded with zeroes, when subtracting.