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(2a)  The Sundial


1. Stargazers

1a. Celest. Sphere

1b. Pole Star

2. The Ecliptic
2a. The Sundial

3. The Seasons

3a. Angle
        of sunlight

4. The Moon (1)

 4a. The Moon (2)

 4b. Moon Libration

5.Latitude and

5a. Navigation

5b. Cross-Staff
        "Some people can tell what time it is by looking at the sun. But I have never been able to make out the numbers."
                (Attributed to an essay by a student
                 in elementary school.)

The simplest sundial is a vertical stick rising from a flat horizontal surface.

[IMAGE: A Simple Sundial]

As the Sun rises, passes the highest point in its path (at noon and to the south, in the northern hemisphere) and sets, the shadow rotates around the stick in a clockwise direction, and its position can be used to mark time. Indeed, it has been claimed that the "clockwise" direction in which the hands on a clock rotate was chosen for this reason.

A sundial with a vertical pointer ("gnomon") will indicate noon correctly when its shadow points north or south. [North in northern middle latitudes, south in southern ones, while near the equator it can be either way, depending on season.] However, the direction of the shadow at some other time of the day may depend on the season--its value in summer, when the Sun's path is high, may differ from what it is in winter, with Sun low above the horizon.

Such a sundial will however work equally well at all times if the pointer is slanted, to point towards the pole of the celestial sphere (click here for an explanation--but be warned, it is a bit complicated!). The angle between it and the base then equals the geographic latitude of the user.

A Paper Sundial

Ornamental sundials are often found in parks and gardens, with the pointer widened into a triangular fin, which must point northwards. A sundial of this type can be constructed from folded cardboard or stiff paper: click here to see the basic design used around latitude 38 North of the equator, here for a corresponding one in the southern hemisphere. Either can be printed and then photo-copied onto suitable sheets of stiff paper or cardboard [You may want to use the "option" menu to reduce size to 90% before printing--but make sure to return the setting to 100% afterwards!]. It is meant to be used at a latitude of 38 degrees and should work adequately in most of the continental US.


  1. Cut the paper along the marked line: one half will serve as base, the other will be used to construct the gnomon.

  2. In the gnomon part, cut away the two marked corners.

  3. Fold the sheet in its middle, in a way that the two secondary printed lines (leading to the cut-off corners) remain visible. The line of the fold is the gnomon.
      Note: In stiff paper, straight folds are helped by first scoring the paper, by drawing a line along them with a black ballpoint, guided by a ruler and pressed down hard.

  4. With the page folded in its middle, cut out along the curved line, cutting a double thickness of paper in one cut. The cut begins near the top of the gnomon-fold and ends on the secondary line. Do not cut along the secondary line. No pieces come off.

  5. Score the other two secondary lines, then fold the gnomon sheet along them. The fold is opposite to that of the fold in the middle. These two folds should form 90-degree angles, so that the two pieces with the corners not cut in step 2 can be placed flat on the table, and the triangular gnomon rises above them.

  6. In cut (4), the fin of the gnomon was separated from two pieces with curved outlines. Fold those pieces so that they, too, are flat with the table. One goes above the other, and the slots they form near the secondary lines create a place for the fin to fit into.

  7. You are almost done. Take the base sheet, and note the apex where the hour-lines all meet (that is where the bottom corner of the fin will go). Carefully cut the sheet from this point along its middle line, up to the small cross-line marked on it. Do not cut any further!

  8. Slide the fin into the cut you made, so that all horizontal parts of the first sheet are below the base sheet; only the fin sticks out. Its bottom corner should be at the apex.
      The sundial is now ready, but you might use tape on the bottom of the base-sheet to hold the two pieces together firmly. For further stability, and to prevent the sundial from being blown away, you may attach its base with thumbtacks to a section of a wooden board or a piece of plywood.

  9. Finally, orient the fin to point north. You may use a magnetic compass; before pocket watches were available, folding pocket sundials were used in Europe, with small magnetic compasses embedded in their bases. If clear sunlight is available, the shadow of the tip of the fin now tells the time.

If you want to make a sundial of more durable materials, draw the pre-noon hour lines at the angles to the fin (given in degrees) given below. These lines are meant for a latitude of 38 degrees; if your latitude is markedly different, see note at the end.

6 -- 90°    9 -- 31.6°
7 -- 66.5°   10 -- 19.6°
8 -- 46.8°   11 -- 9.4°


   The sundial will obviously be one hour off during daylight saving time in the summer, when clocks are reset.

   In addition, "clock time" (or "standard time") will differ from sundial time, because it is usually kept uniform across "time zones"; each time zone differs from its neighbors by one full hour (more in China and Alaska). In each such zone, sundial time matches clock time at only one geographical longitude: elsewhere a correction must be added, proportional to the difference in longitude from the locations where sundial time is exact.

    (Up to the second half of the 19th century, local time and sundial time were generally the same, and each city kept its own local time, as is still the case in Saudi Arabia. In the US standard time was introduced by the railroads, to help set up uniform timetables across the nation.)
   Finally, a small periodic variation exists ("equation of time") amounting at most to about 15 minutes and contributed by two factors. First, the Earth's motion around the sun is an ellipse, not a circle, with slightly variable speed in accordance with Kepler's 2nd law (see here as well as the section preceding that page). Secondly, the ecliptic is inclined by 23.5 degrees to the equator, which means the projection of the Sun's apparent motion on it (which determines solar time) is slowed down near the crossing points of the two.

Note on Latitude

The angles listed above are intended for a latitude of 38 degrees. If your latitude is L, denotes "square root of" and K (=cotg2L) is

K = cos2L/ sin2L

then the angle between the fin and the line corresponding to the hour N+6 (N going from 0 to 6) satisfies

sin A = cos(15N) / (1 + Ksin215N)

Here 15N (=15 times N) is an angle in degrees, ranging from 0 to 90, and of course, the afternoon angles are mirror reflections of the morning ones. If your calculator has a button
(sin-1), if you enter (sin A) and press it, you will get the angle A. For an explanation of sines and cosines, look up the math refresher. And don't forget to adjust the angle of your fin to L, too!

And by the way...

The sundial described here, with a gnomon pointing to the celestial pole, is a relatively recent invention, probably of the last 1000 years. Yet sundials were used long before, often with unequal hours at different times of the day. The bible--2nd book of Kings, chapter 20, verses 9-11 (also Isaiah, ch. 38, v. 8) tells of an "accidental" sundial, in which the number of steps covered by the Sun's shadow on a staircase was used to measure the passage of time. In that story, the shadow miraculously retreated ten steps on the staircase built by King Ahaz.

Exploring Further

    The "Sundial Bridge," with a unique design which may well make it the largest sundial anywhere, opened July 4, 2004 in Turtle Bay Park in Redding, California, at the foot of Mt. Shasta. Designed by the innovative Spanish architect Santiago Calatrava, it resembles his stunning 1992 bridge erected in Seville, Spain. It is a pedestrian bridge, connecting two parts of Turtle Bay Park, and it also operates as a sundial, using plaques set in a semicircular upper plaza.

[IMAGE: Sundial bridge]

For a more detailed article about this bridge, see Sundial Bridge at Turtle Bay

    Before the days of affordable wristwatches, people often carried a folding sundial in their pocket ("poke" below), with a small magnetic compass embedded, to show the north direction. In "As You Like It" by William Shakespeare (act 2, scene 7) one of the characters tells of meeting in the forest a fool (= witty court entertainer) carrying such a "dial":

"Good morrow, fool," quoth I. "No, sir," quoth he,
"Call me not fool till heaven sent me fortune:"
And then he drew a dial from his poke,
And, loking on it with lack-lustre eye
Says very wisely, "It is ten o'clock:
Thus we may see how the world wags:
'T is but an hour ago since it was nine;
And after one hour more 't will be eleven;
                              (and continues)

   You may also be interested to know that a North American Sundial Society (NASS) exists, with its home page at http://sundials.org. From the main page, the visitor can click on "About NASS", and/or access the many other features the site offers. And in case you wonder about the creatures drawn at the bottom of the "About NASS" page, they are toves, the whimsical invention of Lewis Caroll in his poem Jabberwocky. Concerning what toves are, see Humpty Dumpty's explanation, also reachable by clicking the winking sun icon on the top of the NASS home page. The reference is from the 6th chapter of Lewis Carrol's Through the Looking Glass.

The British Sundial Society also has its sundial page.

  A sundial was included as part of the Mars lander mission and is shown in "Astronomy Picture of the Day" for 28 April 1999. It has a thick vertical gnomon, so that its readings may need some extra corrections

.    For those with serious interest in history (and access to a good library): "The Material Culture of Astronomy in Daily Life: Sundials, Science and Social Change" by Sara Schechner (History of Sci. Dept., Harvard) Journal for the History of Astronomy Vol. 32, part 3, August 2001, p. 189-222, with many illustrations.

From the 1.1.2000 book catalog of Willman-Bell in Richmond, Virginia (www.willbell.com):

    Easy-to-make Wooden Sundials, by Stoneman, 38 pp., $4.95
    Sundials: History, Theory and Practice by Rohr, 230 pp, $12.95.
    Sundials: Their Theory and Construction by Waugh, 19 chapt., $8.95

Questions from Users:   What is the Analemma?
                  ***         The equation of time
                  ***         The "Sundial Bridge" in Redding
  ***         Latest Sunrise, Earliest Sunset

Next Stop: #3 Seasons of the Year

            Timeline                     Glossary                     Back to the Master List

Author and Curator:   Dr. David P. Stern
     Mail to Dr.Stern:   stargaze("at" symbol)phy6.org .

Last updated: 3-1-2007

Above is background material for archival reference only.