(2a)  The Sundial

Lesson Plan for (2a) The Sundial

        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.

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. 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, which 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.

Instructions:

  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°

Accuracy

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") due to the Earth's uneven motion around the Sun (an ellipse, not a circle), amounting at most to about 15 minutes.

Note on Latitude

The angles listed above are intended for a latitude of 38 degrees. If your latitude is L, SQRT 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) / SQRT(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.

...and believe it or not, a British Sundial Society exists, and it has put on the web its sundial page.


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Author and curator: David P. Stern,
Last updated 3 April 1999


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