## Analog Thinking

About five years ago, I found an old book, probably now an introuvable, Korn and Korn’s Electronic Analog Computers (D-c Analog Computers), 1956 [1].

Well, that mostly unrelated to today’s post, except that in some cases, relatively crude analog methods may give surprisingly good results in numerical problems (the topic came up while I was discussing the golden ratio and its place in art with friends, trying to make a point that it was mostly heuristic, bordering on the numerological). Take the pyramids, for example. The Ancient Egyptians did not have GPSes and laser guided telemetry. Yet, they managed to get some of their buildings incredibly well aligned with the celestial north.

So what do you need to find the celestial north/south axis?

Very little, I think: you need a plain where you can see the horizon on a clear night, a rope of good length (several tens of meters), three pickets ($A$, $B$, and $C$), and some strong tea (optional).

1. Find a plain where you can see the horizon on a clear night. Plant picket $A$ somewhere, while vaguely facing north (it doesn’t have to be very precise at that point). Choose a bright star vaguely close to where you think north is.

2. As the star rises above the horizon, plant picket $B$ at distance $L$ (that’s why we have the rope) of picket $A$ so that picket $A$, picket $B$, and the star are in line.

3. Savor your tea by this lovely starry night (optional). An aged oolong shared with a good friend would be appropriate.

4. As the star is about to set, plant picket $C$ at distance $L$ of picket $A$ so that picket $A$, picket $C$, and the star are in line.
5. Now that picket $C$ is installed, you have that picket $A$ is the center of a circle of radius $L$ and that B and C lie on its circumference.
6. We now have to find the bisectrix of angle $CAB$. If you fold your rope to have some length $L' \leqslant L$ and draw two circles with pickets $B$ and $C$ as centers ($L'$ has to be long enough for circles to intersect (not necessarily in two distinct points), you find two intersections.
7. The small circles intersection(s) and picket $A$ are necessarily lie on a line, which points to the celestial north.

So the Ancient Egyptian didn’t need any sophisticated hardware to find the celestial north. What remains to be seen is how sensitive the method is to errors, but I would guess that a few centimeters off here and there when $L\approx 10\mathrm{m}$ will lead to very precise estimations, well within a tenth of a degree.

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I did figure this myself at one point a long time ago but it turns out that it’s a favorite theory on how the Ancient Egyptian found the celestial north. I saw the method in a documentary on PBS a few years ago (or maybe it was Discovery?), but I can’t seem to find a solid reference now—most of what Google turns up is pseudo-scientific garbage; probably I am not using the right query. A page that gives an idea of how to do so (with nice pictures but in less details than what I gave here) can be found here.

[1] Granino A. Korn, Theresa M. Korn — Electronic Analog Computers (D-c Analog Computers) — McGraw-Hill Book Company, inc. 1956