Having drawn you in with an absolute statement, please allow me qualify it. For most sundials, why does the time as indicated by the shadow, nearly always differ from the time indicated on your watch or cell phone? There are a number of reasons for the discrepancy that range from the comical to the astronomical. This discussion may take a while, so grab a beverage of choice, get comfortable and let us unravel and correct the errors as best we can.
When describing the Sun’s daily or seasonal movements across the sky, I will frequently use the Earth-centric view. We understand that this interpretation is erroneous. The Solar System is heliocentric, the Sun is fixed and its observed movements across our sky are in fact the result of the Earth’s various motions in three dimensions. For example, the Earth rotates once a day relative to the Sun at a rate of 360° in 24 hours, 15° per hour. From the ground, we perceive the Sun moving from east to west at the same rate, 15° of longitude per hour, 1 degree every 4 minutes. However, with respect to the movement of the Sun’s shadow on a sundial, the two perspectives are equivalent.
I will reference this discussion to a flat plate sundial, as shown above, set up in the northern hemisphere. First, a few definitions. The dial plate is the horizontal base of the sundial, inscribed with the hour lines and their corresponding times. The Sun’s shadow is cast on the hour lines by the clean top edge of the gnomon (pronounced no-mon, to rhyme with go-john). The word comes from the Ancient Greek, and etymologically is related to “know” and “knowledge.” Gnomon can be translated as “the one who knows,” and was also the name of the carpenter’s/builder’s square. The vertical plane of the gnomon is aligned with the north-south meridian and the angle the gnomon makes with the horizontal dial plate is the same as the latitude of its location. The gnomon points directly to the north celestial pole, conveniently marked for us by the star Polaris.
Now, to borrow a concept from Terry Pratchett, change the perspective, draw back, view the orb of the Earth from space. Imagine several very large sundials set up on the visible circumference. At the North Pole, latitude 90° north, the gnomon points vertically up to Polaris. On the equator, latitude 0°, the gnomon is horizontal, but the north end still points towards Polaris. Similarly, for all sundials in between, their gnomons, set at their respective latitudes, always point to Polaris. Thus, we can conclude: all gnomons are parallel, they just have different addresses.
With our sundial correctly positioned, and viewing it from south to north, the Sun rising in the east casts the shadow of the gnomon to the left, indicating say 6:00 a.m. As the Sun then moves up and across the sky, the shadow proceeds clockwise around the dial, 6:00 a.m. to noon, then from noon to 6:00 p.m. in the afternoon. Let me recast that last idea. The hour hand of a clock moves sundial-wise around the clock face. Sundials came first, early clock designers retained the template.
Just as the angle of the gnomon changes with the latitude, the angles the hour lines make with the gnomon are also latitude dependent. The 6:00 a.m. hour line is always 90° from the gnomon and noon is 0°, but the angles of the hour lines in between vary according to the time and the latitude. The equation to calculate the values involves the tangents of a couple of angles and the sine of the latitude, though, rest assured, we are not going to use it here. Just be aware that the latitude influences the angle of the hour lines. For those who are interested, the equation can be found online and the following site also includes a handy calculator (1). It should also be noted that flat plate sundials are most accurate at mid latitudes, say from 25°- 65° north.
Now we appreciate that an accurate sundial is customized for its given latitude, both with respect to the angle of its gnomon and the angles of the different hour lines, we can identify a major source of errors affecting sundial time. Commercially-available, mass-produced sundials frequently found in home gardens have their gnomons set at about 40° degrees, one size fits all. This angle works well for those on that latitude, e.g., Philadelphia to Denver to Eureka, CA., but not so much for Bismarck, ND at 47° north nor Miami, FL at 26° north. Similarly, the locations of the hour lines on mass-produced dials have more fealty to artistic design than scientific accuracy. Finally, there is no guarantee that home garden dials have been set up correctly with the dial plate horizontal and the long axis of the gnomon aligned to the north - south meridian. In my neighborhood there are sundials where the orientation of the gnomon seems to be either random or aesthetically pleasing. Many years ago, I saw one dial where, to my horror and bewilderment, the gnomon had been assembled back to front! Instead of the gnomon pointing to Polaris, it was oriented to the south, in the general direction of the noontime sun. Therefore, until demonstrated otherwise, please regard the time indicated by a garden sundial with considered suspicion.
For a variety of reasons, we have daylight saving time in the summer. Since one cannot “spring forward and fall back” the shadow on a sundial, DST instantly introduces a one-hour error. The correction is easy. We do not need to change the hour lines, only to provide each one with an additional number for DST. For example, 12 noon standard time is also labelled 1:00 p.m. DST, with the numbers say in two concentric rings. One can also make the correction mentally.
We have considered the effects of latitude on our dials, what about longitude? The answer relates to our time zones. For convenience, and in accord with the 1884 International Prime Meridian Conference (2), the Earth’s surface is divided into 24 one-hour time zones, each centered 15° degrees of longitude apart girding the planet. Since the Sun travels 15° in 1 hour, it takes this long for it to pass from being overhead at the eastern edge of a time zone to being overhead at its western edge.
To determine the effect of longitude on our sundials, let us examine, at 12 noon central standard time on April 15th, the time shown by the shadow on dials in three cities in the central time zone: Panama City, FL. (86° West), New Orleans, LA. (90° W) and Austin, Tx. (98° W). On its journey from the Atlantic to the Pacific, the Sun will shine from due south sequentially over each of these three cities. At noon in New Orleans, the Sun is due south and its shadow aligns with the long axis of the gnomon, indicating 12:00 noon. However, on its traverse from east to west, the Sun passed due south of Panama City at 16 minutes before noon. Panama City is 4° to the east of New Orleans, every degree equals 4 minutes, hence the Sun was due south of Panama City at 16 minutes before noon. Consequently, when the dial in New Orleans indicates 12:00, the shadow on the dial in Panama City reads 12:16 p.m. In contrast, Austin is 8 degrees to the west of New Orleans, therefore at noon in New Orleans, the shadow on the dial in Austin indicates 11:28 a.m. (8° x 4 minutes each = 32 minutes subtracted). The Sun still has a long way to go to be due south of Austin.
There are two approaches to resolve this longitude effect. A small plaque can be added to the dial to indicate the number of minutes to be added or subtracted from dial time to give standard time. Alternatively, the sundials in Panama City and Austin can be redesigned from scratch making allowances for the permanent offsets. For the dial in Panama City, the 12:16 p.m. hour line becomes the new noon, with all other hour lines adjusted around the dial accordingly. For the dial in Austin, the 11:28 a.m. hour line becomes the new noon. These reworked sundials will then show the standard time for the central time zone.
Dear Reader, I never claimed that this process would be easy.
Let us summarize. We have designed a sundial for a specific location, taking into account its latitude for the gnomon angle and hour lines, its longitude for the appropriate time zone offset and two sets of hour numbers for standard and DST, respectively. Congratulations! Set your dial out in the garden with the gnomon aligned to the north – south meridian. Stand back and admire. Unfortunately, despite our best efforts, this dial will show the correct time on only 4 days of the year: April 15, June 15, September 01 and December 25. There is yet another complication to consider.
To understand the cause, we must again consider the music of the spheres. Once more, change your perspective, but this time to envision the Earth orbiting the Sun. The Earth orbits the Sun not in a circle, but in a slight ellipse. Over the course of a year, the Earth traces out the plane of the elliptic around the Sun. This ellipse is slightly off-centered from the Sun with respect to its long axis, therefore the Earth is closest to the Sun in December and furthest away in June. Additionally, the Earth’s axis of rotation is not at right angles to the plane of the elliptic, but is canted over by 23 1/2°, the axial tilt. In the short term (astronomically), the axis of rotation is fixed, with Earth’s northern end pointing constantly at Polaris. Each year as the Earth orbits the Sun, the northern hemisphere points alternately towards and then away from the Sun. This variation gives Earth its seasons with their consequent effects on terrestrial life.
Now, return to Earth and look up at the sky. Because of the axial tilt, the Sun is higher in the sky in the summer, lower in the winter. Does the rise and fall affect the functioning of our dial? Minimally; it makes the shadow of the gnomon shorter or longer, but that’s about all. However, the ellipticity of the Earth’s orbit causes the Sun to track left and right in the sky twice during the course of the year.
The combination of these celestial motions, up and down, left and right, results in the Sun tracing out an analemma in the sky each year. To visualize the analemma, take multiple photographic exposures of the Sun, in a fixed direction, at the same time (UTC) every few days for a year. The result is the beautiful elongated figure of 8 shaped path. (See Photo).
Our final set of corrections will unwind the effects of the analemma on our sundials.
The figure below displays the annual track of the Sun across the sky more precisely. Just after the southern summer solstice, say Christmas day, the Sun is due south of Greenwich, U.K. at noon, the 180° azimuth line. Standard time and dial time agree. As January stretches into February, at noon standard time each day, the Sun will be further and further east in the sky, reaching a maximum around February 12th. This eastward track causes the shadow on the dial to be further to the left, to be slow relative to standard time by approximately 14 minutes on February 12th.
The Sun then swings back to the west, causing sundial time to catch up with standard time on April 15th, then be ahead by about 4 minutes on May 15th. Continuing on its S-shaped track, the Sun causes sundial time be 6 minutes slow on July 25, then 16 minutes fast at Halloween, before returning to be in agreement again at Christmas. The complex cycle then starts over again and continues, year by year, the Sun tracing out the analemma, journeying on to infinity.
The differences between standard and sundial time are also known as The Equation of Time. As you can appreciate from the figure, except for around the solstices, the corrections to be applied to bring dial time into agreement with standard time vary every day. To maintain our sanity, let us approximate. Some dials have a small table inscribed on the dial plate indicating the average number of minutes to be added to or subtracted from dial time each month to bring it into close agreement with standard time. For instance, in February: Add 14 minutes, in May: Subtract 4 minutes, etc. Some sundials have a small graph indicating The Equation of Time, which again indicates the correction to be applied depending on the date. There are tables available online which give the analemma/Equation of Time correction for every day of the year (3).
For an uncorrected sundial, all three adjustments described above must be totaled to convert sundial time to the prevailing standard time. The table below illustrates such a calculation for dials in two cities, Panama City, on the last day of central standard time and Austin on the first day of central daylight time in 2021.
City, Longitude: Panama City, FL. 86° West Austin, TX. 98° West
Time and Date: Noon, CST, March 13, 2021 Noon, CDT, March 14, 2021
DST Correction: 0 (Not in effect) Add 60 minutes
Longitude Correction: Subtract 16 minutes Add 32 minutes
Equation of Time Correction: Add 5 minutes Add 5 minutes
Total Correction : Subtract 11 minutes Add 1 hr. 37 minutes
Time shown on sundial: 12:11 p.m. 10:23 a.m.
At this juncture, I am reminded of the words of Cassius in William Shakespeare’s play, Julius Caesar:
“The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings.”
With respect to sundials, this quote is 180 degrees reversed. Our measurements, our math and our manufacturing are first rate; we are not underlings. The fault, dear Reader, is not in ourselves, but in the stars. The revealed truth is that the Sun and the Earth are very imprecise timekeepers. We have traveled a long, tortuous journey to reach this, perhaps disappointing, conclusion.
However, with our journey comes a greater understanding of a very practical, intimately related application. A sundial at a known latitude and longitude uses the angle of the Sun, the meridian and the date to determine the time. Rearrange those variables to determine a different unknown. Navigators use a sextant to measure the angle of the Sun with respect to the horizon at a specific time, then knowing the meridian and the date they can calculate their latitude and longitude. With our knowledge of the analemma and the daily movements of the Sun, we can now appreciate the complexity of the navigational tables required to reduce the basic observation of the sun angle to precise coordinates.
Returning to our sundials and applying all the appropriate corrections for location and date, they will give us the time to an accuracy of about plus or minus two minutes. For most of us in our daily lives, eating, sleeping, working, heading to the ball game, etc., this accuracy is more than sufficient. Yes, everything we do with and through our cell phones and computers requires split second timing, many professions also need precise timing, but for everyday living, not so much.
Apart from the scientific and timekeeping aspect of sundials, there are its many artistic qualities to be admired. I find pleasure and also a certain poetry in deriving the time of day from the motions of the Sun and the Earth. Sundials make pleasing public art, be it in a garden, in a city square, at a religious institution or a university. An online search will reveal sundials, some of which incorporate the analemma into their three-dimensional designs, that synthesize art, architecture, mathematical rigor and functionality into beautiful creations.
To conclude, there is one crowning jewel of the sundial that we have not yet examined. Despite its apparent imperfections, when the watches have all stopped, the cell phones and computers run out of juice, the broadcasters are silent and the central timekeepers are long since defunct, when you have no other way to tell the time, step outside, there in the sunshine, your sundial will find the time anew.
The basis for my understanding of sundials came long ago from a wonderful little book that is thus a deep and wide reference for this essay: Sundials: Their theory and construction by Albert E. Waugh. June 1973, Dover Publications, Incorporated. Used copies are available online for a few dollars.
1) https://sundials.org/teachers-corner/sundial-mathematics.html
2) https://en.wikipedia.org/wiki/International_Meridian_Conference
3) http://www.ppowers.com/EoT.htm