That’s impressive, but, at some point… a 5-place printed table of logarithms takes up fewer than 15 pages, for example.
Most circular slide riles are relatively small in diameter, and don’t give you any advantage in acuracy. In fact, since the scales get smaller as they get closer to the center of rotation, many of the scales are usually pretty hard to read.
I have a pencil cup in the form of a cylindrical slide rule, with rotating (instead of sliding) scales, so they’re all the same accuracy. It’s the only cylindrical slide rule of this type (as opposed to the high-accuracy one i mentioned above), and it’s pretty neat. Except that it’s very obviously a novelty item and isn’t made to a high degree of accuracy.
It looks like this
As I mentioned in that other discussion:
Franklin Reck’s book The Dilworth Story (available on archive.com) mentions a BIG slide rule Mr. Dilworth had : “A fabulous spiral 105-inch job that carried logarithms to seven decimal places.”
If you wanted more significant digits, you used your tables instead of your slide rule. Your cheap basic table was 4 digit with 1 extra digit of interpolation: your bound copy was 5 digits giving 6: your library copy in 10~15 large volumes was ??? 10 digit ???
If you wanted fewer digits — my dad was required to memorize 1 digit logarithms for his professional certification.
Only one digit? That does not seem so onerous:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|
| 0 | 3 | 5 | 6 | 7 | 8 | 8 | 9 | 10 |
The legend, of course, is that Gauss had logarithms memorised out to however many digits, to speed up his calculations.
When I was an undergrad, we had a guy living on our floor who could recite three-place logarithms from his head. He claimed that he could visualize the curve of log x.
What’s amazing is that he could do it – arguably faster – when he was drunk.
Like many art projects, slide rule layout templates were drawn large, then optically reduced to the desired size. Minor errors in the layout are therefore minimized.
Another example: matchbook covers were drawn way oversize, then reduced. I say “were”, because today’s computerized art production might be different.
Same with cartoons, comics, and printed circuit boards – draw oversize, then reduce to print. I have a reproduction lithograph in a frame on my wall of one of Walt Kelly’s Pogo episodes. The original size was about 14" x 17".
My sources: Besides my personal involvement in the art world, I knew a matchbook cover artist once; a friend lived next to a slide rule factory and they gave me boxes of parts and discards; and the aforementioned Walt Kelly drawing.
I was trying to figure out a way of being correct, but I figured it didn’t matter. He had to know the log of 1 digit numbers.
You’ve got a PM. Dunno if Discourse notifies of those by default or not.
Got it. Thanks.
I’m surprised that nobody has mentioned the slide rule’s distant cousin, the Vernier caliper, yet. The construction methods are pretty similar, though they serve very different functions. Of course, I’m thinking of a real Vernier and not one of the dial or digital models. I had to learn to read Vernier scales when I was in shop class in junior high and I’m actually a bit shocked that so few people can use one accurately.
“Let’s see, two times three is five point nine… five point nine eight… ah, I’ll just call it six.”
I had a slide rule custom manufactured twenty four years ago, for a special purpose. It was printed on stiff plastic sheets, an outer one that was bent to fit around an inner one, without a sliding indicator around the whole thing. There were two edges where scales were against one another. The company I found had a mathematician that would work with me, and I described the underlying functions. I did already know how to create the scales and had folded paper examples to show them.
I could only find special purpose new slide rules for sale now. There are several on Amazon, for the trades – masonry, paving, electricians, that sort of application. They’re real slide rules with some log scales and maybe some more obscure scales.
There was also a higher accuracy slide rule called Thacher’s Calculating Instrument, made by K&E. It had a cylinder that rotated and translated inside a cage of stationary scales. I think it bought the user an entire extra digit of accuracy. I’ve seen them on display occasionally but have never used one.
I graduated high school in 1984 and even then, slide rules were obsolete. But my high school physics teacher required us to obtain and learn to use a slide rule. Later we learned the reason was that doing so forces you to read a scale properly, estimate and think about orders of magnitude (is that result 100 or 1000). So kids were running around town asking stationery stores if they sold slide rules and of course almost none did. My brother was two years older and already had physics under the same teacher, so I inherited his. I think I still have it, though I don’t remember how to use it.
(If I ever decide to build a collection, I was thinking one of slide rules and other scientific instruments might be fun.)
Yeah, as a practical tool, for doing real-world calculations, slide rules are far eclipsed by electronic devices. But as a teaching aid, they’re still wonderful.
I remember my father had a big cylindrical rule like that… no idea where he got it. I wish I had kept better track of it: would probably be quite a collectors item nowadays.
I took a course on computer calculations in university. the very first versions of 8-bit BASIC would do things like 2+2=3.999999 due to issues with the binary-to-decimal conversion.
The one I remember most was the demonstration of finding the intersection point when two lines crossed. Given the line parameters and error ranges, the prof pointed out - “OK, where do these lines cross?” You get an answer to 8 digits. Now calculate the error. Since the lines were very close to parallel, the overlap of the error/uncertainty zones meant the answer would be something like 2.1734866 +/- 4.5 - i.e. the uncertainly was high enough the answer was effectively meaningless.
Similarly, we calculate an answer for a quadratic equation, where the X^2 coefficient was something like 1.8 +/-2.0; the prof pointed out - you’ve got a solution where can’t be sure the answer is a parabola opening upward or downward.
In the real world as opposed to math, the error is a significant part of the calculations. The story goes that the first attempt at using a computer to calculate required thickness for jet aircraft wing spars, the answer came out to be “11 feet thick”. It’s important to understand whether iterative calculation increase or decrease the concurrent range of uncertainty.
I did some Googling and found this:
They apparently have brand new ones still in boxes that were manufactured but never sold.
A real example: Back in my college days, in my orbital dynamics class, we had a big project where we were given a prior estimate of the six orbital elements for an orbit, along with their 1-sigma uncertainties, and then we were also given four highly precise measurements of distance from the satellite to known fixed points, and tasked with determining the new best-fit orbital elements.
One of the given orbital elements was the eccentricity, which was very close to zero. So close, in fact, that the uncertainty in the eccentricity was larger than the eccentricity itself. Which leads to seemingly-nonsensical results, because eccentricity can’t be negative. My code managed to deal with that (IIRC, by transforming to a different set of variables that could go positive or negative), but not everyone’s did. The “negative eccentricity” portion of the solution space ends up mapping to “positive eccentricity, but with perigee and apogee swapped”, and the best-fit solution turned out to be in that portion of the solution space.
Of course, that’s not just an issue with the uncertainty being greater than the measurement. The same issues could, in principle, show up with any measurement constrained to be nonnegative and any nonzero error, because a 2-sigma or 3-sigma or 10-sigma error is also possible (though increasingly unlikely).
…which is why the professor was teaching you about the condition number of a matrix and how to deal with it. All that error analysis stuff would apply to slide rules as well, for that matter.
If the parameter is constrained like that, you are supposed to quote a (correctly calculated) confidence interval, which will naturally not include any negative numbers, not merely an uncertainty.
Example: your variable is Poisson-distributed and you observe 0 events. Then the 95% confidence interval for the mean number of events is [0.00, 3.09]
Exactly. That’s why, for example, the Chi-squared confidence intervals are not symmetric.