There are at least three aspects of this problem: accuracy, ease of use, and elegance. We might be able to agree that dials with a built-in analemma are conceptually the most elegant because they utilize to the maximum extent the information from the sun and require a minimum of external input. With ease of use, I have my problems with the analemma. And if I have problems, how is a citizen going to react who is not a scientist and possibly has never seen a sundial before? Arrows on the analemma may designate unambiguously which branch should be used, but take too much thought. Labeling with the seasons is better in this respect. This is well and good for a noon mark, but an accurate sundial should have marks every five minutes or so. Analemmas already start to overlap at half-hour spacing. A sundial with a hundred analemmas may be a work of art and an accurate instrument, but it will be impossible to read, especially if the branches have to be labeled.
And now another word or two about accuracy. If we could read the position of the shadow *exactly* (and always knew for sure which branch we needed), then a dial with analemmas would be perfectly accurate (though perhaps impossible to read in practice). The fundamental disadvantage of a dial without dates becomes evident when the shadow cannot be read exactly. During most of the year a dial with analemmas will be about as accurate as one with date lines because the analemma lines are nearly "vertical". The problems arise at the solstices where they are nearly "horizontal". Since the shadow is used to determine the declination as well as the hour angle, the accuracy decreases with the slope of the analemma. The more accurately the shadow can be read, the greater the disadvantage of pure analemma solutions, because a reading will come closer to trying to use a purely horizontal mark. We might ask, for example, at how accurate the shadow would have to be read before the leap year problem of a dial with 365 day-lines becomes greater than the solstice problem of a dial with unmarked analemmas. With knowledge of the date but not the year, the first type of dial cannot be made more accurate than +/-(1/2) day, which in late December means +/-15 sec (EoT changes by 30 sec/day). As explained above, a dial with analemmas will generally be less accurate than one with dates, but if the accuracy of reading the shadow becomes better than +/-15 sec, then the former will continue to improve, but the latter will be stuck with this uncertainty. But to reach this accuracy near the solstice, the analemma would also have to be able to determine the time of the solstice within +/-(1/2) day, which in turn would require an angular accuracy of +/-1.6 arc sec [(23 deg)*(2pi*(1/2)/365)^2/4]. If you could determine the position of the sun to this accuracy, then you could also read your sundial to well better than 0.1 msec! So much for the leap year problem. One way to combine the best of both approaches is to label the analemma not just with the seasons but with the dates. The problem remaining is the ease of use, which can be solved by employing moving parts. I have not been able to think of any purely passive design that simultaneously does not require knowledge of the date, but permits use of the date to increase accuracy, and is intuitive to use. As long as we are talking about the limits of accuracy at special times, I would also like to mention the refraction of the atmosphere. This can introduce an error of a minute or so for most designs. The only design which is not affected by refraction is an azimuthal dial (in the strict definition). If the terms of John Davis' "design challenge" were accuracy from sunrise to sunset, rather than from 9am to 3pm, this would be another argument against a dial using analemmas. --Art Carlson
