The Mathematica Journal
Departments
Feature Articles
Columns
New Products
New Publications
Classifieds
Calendar
News Bulletins
Mailbox
Letters
Write Us
About the Journal
Staff and Contributors
Submissions
Subscriptions
Advertising
Back Issues
Home
Download this Issue
gehrig.nb

Investigating the Astronomical Clock of Prague

David Gehrig
Wolfram Research, Inc.
davidg@wolfram.com

Introduction

For use on Mathematica 4.0 promotional materials, Wolfram Research created a sample notebook briefly sketching the geometry behind the Orloj, an astronomical clock, which since the year 1410, has tracked the motion of the sun and moon in the sky over Prague. This article will describe the clock in more detail, including a description of the mathematical model behind it.

The clock adorns the tower of the Old Town Hall in Staromėstské námestí, or Old Town Square, not far from the crypt of the astronomer Tycho Brahe. It has two large dials, the lower one a calendar dial listing the cycle of feast days and the upper one the astronomical dial pictured here. Over the centuries the clock has also acquired flanking allegorical figures such as Vanity and Death, a mechanical procession of the Apostles, and a golden cockerel which crows on the hour. The internal mechanism of the astronomical dial is essentially unchanged from its earliest incarnation, despite the effects of time and a firebombing by the retreating Nazis in the last days of the Second World War.

[Graphics:Images/index.new_gr_4.gif]

Figure 1. Astronomical dial of the Orloj. (Photo: PhotoDisc © 1998)

Modeling the Motion of the Sun

The clock's 24-hour dial keeps track of two distinct motions, adding them together to calculate the position of the sun at any given time. The first motion is the slow annual procession of the sun around the ecliptic; this is modeled by having the sun disk travel exactly once per year around the circular dial marked with signs of the zodiac. But this dial itself also moves, spinning eccentrically once per sidereal day to model the second motion, the daily turning of the celestial sphere with respect to the sky of Prague.

The Prague sky is shown on the hindmost dial, the horizon dial. This dial--the one marked with Roman numerals--corresponds to horizon coordinates, altitude, and azimuth over Prague. The ecliptic dial then corresponds to celestial coordinates, the right ascension, and declination for a given point on the celestial sphere. (A small star attached to the ecliptic dial--visible almost directly below the center in the photo above--represents the first point of Aries, or 0 hours right ascension.)

The Horizon Dial

The horizon dial depicts the local sky. Here is a schematic diagram of the main details.

[Graphics:Images/index.new_gr_5.gif]

Figure 2. Schematic diagram of the horizon dial.

The horizon line corresponds with the horizon of Prague; when the sun disk crosses this line, it marks the transition from day to night or vice versa.  A second system of time measurement is also shown on this dial, one in which the daylight is divided into twelve equal parts. This system is usually referred to as unequal hours, since in this system a summer hour is longer than a winter one. The daylight part of the horizon dial is divided, roughly radially, into twelve zones, each corresponding to an unequal hour; the time is read by noting which zone contains the sun disk. Unequal hours are numbered from sunrise; we still speak of something being done "at the eleventh hour," meaning the last possible moment before the end of the day, because of this ancient system of timekeeping.

Three concentric circles on the horizon dial mark the Tropic of Cancer (the outer rim of the dial), the celestial equator, and the Tropic of Capricorn. We can recreate the horizon dial as follows, ignoring unequal hours. We can use Mathematica's knowledge of three-dimensional shapes, as found in the standard Graphics`Shapes` package, to help with our construction.

[Graphics:Images/index.new_gr_6.gif]

The function circleSegs creates a set of line segments approximating a circle parallel to the [Graphics:Images/index.new_gr_7.gif] plane.

[Graphics:Images/index.new_gr_8.gif]

Calculating the positions of the tropics requires knowing the obliquity or tilt of the earth's axis,e.

[Graphics:Images/index.new_gr_9.gif]
[Graphics:Images/index.new_gr_10.gif]

Here we build a little globe, tilt it to compensate for Prague's latitude, add circles for the tropics and celestial equator, and then plot the stereographic projection onto the plane [Graphics:Images/index.new_gr_11.gif] using [Graphics:Images/index.new_gr_12.gif] as the point of projection.

[Graphics:Images/index.new_gr_13.gif]

[Graphics:Images/index.new_gr_14.gif]

[Graphics:Images/index.new_gr_15.gif]

[Graphics:Images/index.new_gr_16.gif]

The formula for the stereographic projection in this case was found by solving the two-point formula for a line, substituting [Graphics:Images/index.new_gr_17.gif] as the point of projection, and setting the [Graphics:Images/index.new_gr_18.gif]-coordinate of the result to -1; this is the calculation shown on the Mathematica 4.0 box.

[Graphics:Images/index.new_gr_19.gif]
[Graphics:Images/index.new_gr_20.gif]

Final Notes

The mathematics of the Orloj is closely related to that of a medieval device called the astrolabe. One of the best contemporary presentations of the astrolabe--what we would now call a user's manual--was written by Geoffrey Chaucer, author of the Canterbury Tales.

Astronomical clocks continue to be built even today; a well-known example is the Clock of the Long Now, a device which is being designed to work accurately for the next ten thousand years. (See www.longnow.org for details; my thanks to Stewart Dickson for drawing this to my attention.)

Finally, old Bohemian time is a compromise between fixed hours and unequal hours; it divides the time from sunset to sunset into twenty-four equal parts. Old Bohemian time can be read by the sun disk's position against the outermost ring of Arabic numerals on the Orloj; this ring oscillates back and forth once a year depending on the time of sunset, which always corresponds to the start of the first old Bohemian hour.

References

P. Duffett-Smith, Practical Astronomy with Your Calculator, 3rd Edition,  Cambridge University Press, 1988.

G. Chaucer, "A Treatise on the Astrolabie," The Complete Poetry and Prose of Geoffrey Chaucer (John H. Fisher, ed.), Harcourt Brace, 1990.

J. D. North, "The astrolabe,"  Scientific American, 230:(1)(1974), pp. 96-106 (January,1974).

"Prague astronomical clock." http://otokar.troja.mff.cuni.cz/RELATGRP/Orloj.htm.

About the Author

David Gehrig is a technical marketing writer at Wolfram Research, Inc. He holds an undergraduate degree in computer science from the University of Illinois at Urbana/Champaign and a master's degree in English from the University of Illinois at Springfield.


Converted by Mathematica      April 14, 2000