Archimedes, Badenia, Brunsviga, Diehl, Euclid, Thales, Triumphator, Walther: These were machines from the golden age of mechanical computing. They had no printing mechanism and were primarily used for adding, multiplying, and dividing. In the late 1950s, the engineer Willi Faber created a machine in the East Westphalian town of Neesen that could also square numbers and extract roots.
Squaring isn’t difficult - it’s just 2 multiplies, scale as required.
Square root… on the other hand…
I have a few mechanical calculators and I remembered the manual for my Curta has a method- but right now I can’t find the manual, however
This would readily adapt for the Brunsviga/Nippon/Busicom pinwheel types. The first one is classic Newton - involves a lot of remembering numbers along the way, the other they call the “Dibble Dabble” method.
That really in a “Frankencalc” above though - going to need good arm muscles to use it regularly…
No idea how my Sharp Compet 361 does it, in all its discrete components, magnetic core and nixie tube wonder. It doesn’t even advertise that it can do roots, but it can.
I’m not super keen to dig about in it while it’s running, as those display power lines run at 185 V. A little too bitey.
It has some weird things in there, like the red = key for subtraction. Its spec shows 0.31 seconds for a 16 digit square root: not bad for a 50 kHz machine, and possibly faster than later Texas handhelds like the SR-50/TI-30
It may be noteworthy to mention that the problem almost disappears in binary: a candidate either fits or does not, and what’s taking guesses and backtracking with larger radix systems becomes a simple binary condition with an immediate answer. Doing this in base-10 and mechanically is really “on the other hand”!
I wish I could remember the algorithm the HP35 calculator used; it was probably a cordic variation. I do remember starting with 1, then adding 3, then 5, then 7 (producing the square) until it got bigger than the number whose swaure root was being taken. The number of iterations was an approximation of the square root. It then did something like adding .1, .3, .5 etc , then .01, .03 , .05, successively refining the answer until it got to the precision it needed. The HP35 was a marvel - all those scientific functions in 767 words of 10bit ROM (one word was not possible to use for very obscure reasons having to do with the keyboard…)
What I was thinking of was something along the line of this [1], where I’m trying to make sense of some early 1960s code for the DEC PDP-1.
[1] https://masswerk.at/spacewar/inside/insidespacewar-pt6-gravity.html#square_roots
(This is part of an exursus covering the math routines found in Spacewar!. I’ve no idea where this particular routine came from: multiply and divide routines came from BBN, the sine/cosine routine came from Adams Associates, who were working with Itek on a CAD system, but there’s no info or comment on this one.)
The lesson of this is that some math problems are much easier in binary, since choices become simple conditions with definitive answers, thanks to the small radix. Conversely, algorithmic complexity increases with the radix size – and then doing this in mechanics…
When there was a Friden in use in the room, everyone there knew it. The room lights dimmed, the motor whirred, the entire desktop shook, and the wheels spun with a sound somewhere between a threshing machine and a car in dire need of Mr. Transmission. It sounded as though the machine couldn’t possibly last out the day, but in fact, the Friden was quite a reliable machine. The basic Friden was strictly a four-function calculator, with no memory except the little wheels on the carriage. However, a square root version was available at considerable extra cost. Our office had only one square root Friden, so we all had to take turns with it when we needed square roots. Or so I thought.
One day I was busily thinking deep-space thoughts, while my officemate was banging away on our common, non-square-root Friden. I heard a strange sound that went something like “punch-punch-cachunk, punch-punch-cachunk, punch-punch-cachunk-DING, punch-punch-DING-clang-clang” in a repeated rhythm. I thought my mate had either lost his marbles, or was creating some new kind of computer game.
I asked him what the heck he was doing. He said, “Finding a square root.”
“But, um…,” I said, “this isn’t a square root Friden.”
“I know,” he replied. “That’s why I have to do it this way.”
