An astrolabe is an instrument for telling time by sighting the Sun or another known star. The astrolabe is a delightful, surprisingly precise instrument -- provides the time to within a few minutes, except near noon. The astrolabe had to be designed for a specific location and year. They were popular, in western Europe, during the Renaissance, as a status symbol. The wealthy could commission one to be custom-made. It was a rite-of-passage for a scholar -- especially an astronomer -- to design ones own astrolabe. Now, interest is reviving, especially in connection with the reenactment of the Renaissance. A proof is provided for each pertinent theorem, construction, or formula, including a derivation of the equation of time from Newton's law of gravitation.
The name "astrolabe" English comes from the Latin singular "astrolabium", plural "astrolabia". In turn, from the Greek. "Aster" means star. "Lambanein" means take, seize, catch, grasp, apprehend, determine, estimate. Taken together, an astrolabium is a star taker or star finder. We always italicize a Latin or Greek word. As any good scholar, I never use an English word, when a perfectly good Latin -- or, better yet, Greek -- word is available.
The principle of the astrolabium is the stereographic projection, which was studied in classical Greece, c 200BC. Soon thereafter, they discovered that placing the eye-point at the south-pole yields a conformal mapping. Probably, the first astrolabium was constructed in 400 AD; but, possibly as early as 150 BC. The earliest surviving instrument is from c 0900 AD. The astrolabia were popular in Europe, including England, between 1100 and 1700. Traditionally, they were made of brass; but, mine is made of paper, laminated in clear plastic. The astrolabium is a device for telling time, to within a few minutes, at any time -- day or night --, except near noon. It also may be employed to find true north, in the daytime, or at night, even when the polar star is obscured. The astrolabium may be employed to measure the height of a pole or tower. The astrolabium may be employed as a theodelitus (Latin, English is theodolite), in surveying. It was only since 1775, with the availability of the marine chronometer, that the astrolabium could be employed for navigation; but, by that time, the astrolabium had fallen out of fashion.
By 1700, the significantly more precise sextant and theodelitus instruments displaced the astrolabium. However, these specialized instruments lack the ability to perform the required calculations. Instead, one has to have books of tables of the trigonometric functions and ephemerides (Greek singular ephemeris. The positions of the planets.). One also needs a good working knowledge of spherical trigonometry.
Photograph of the Jean Fusoris astrolabium, an exploded view of the astrolabium , and a picture of my astrolabia; each courtesy of Janus. Several other links to photographs of astrolabia: modern, re-creations. [Link rote has set it. Today (S 30-VII-2005), I have removed the broken links.]
A planispheric astrolabium is a sandwich of several plane-surfaces, on a clavus (Latin, pivot or pin), held together with an
equus (Latin, horse) -- a horse-head shaped wedge. On my instrument, it is a small hexagonal plastic (Nylon) nut.
Starting from the middle of this sandwich, and working up to the facies (Latin, face), also known as the obversus (Latin,
front):
The mater (Latin, mother) has an hour-scale on its limbus (Latin, limb) -- the periphery.
The tabula (Latin, table), also known as the tympanum (Latin, plate), has the orthogonal grid of azimuth and altitude (elevation)
inscribed upon it. The tabula has to be designed for a specific parallel of latitude. Thus the tabula is
replaceable.
The aranea (Latin, spider) or the rete (Latin, net), is free to rotate. It has many asters drawn upon it. It
also has either the zodiac or day scale marked on the ecliptic -- the orbit of the Helios (Latin, Sun) around the Earth.
Traditionally, there were two styles employed: The aranea consists of solid brass fingers, each pointing to an individual
aster. The rete is a screen mesh, upon which the asters are mounted. On my astrolabium, the
asters are printed on a clear plastic -- one would call it a rete.
The ostensor (Latin, rule) is employed to point to the hour scale on the limbus. The ostensor has a declination
scale upon it; hence, the name "rule". "Declination" is the astronomical name for the celestial latitude from the celestial equator.
On the dorsum (Latin, back); i.e., reverse-side,
we have the dioptra (Latin), from the Greek, dia- + opsesthai to be going to see. It also is known as the
alhidada (Latin, alidade, rotating radius), perhaps, with sights. It is used to sight the Helios (Latin, Sun) or other
aster (Greek, star). It may have a scale to be employed in computing the tangent or cotangent of an angle.
The dorsum of the mater has a centered circular scale of the Zodiac and an eccentric circular scale of the ecliptic.
Together, these scales convert between a calendar date and the position of the Helios in the zodiac.
The mater may have an unequal-hour scale, as well.
My modern astrolabium has an equation-of-time scale on the dorsum of the mater, to convert between apparent solar-time
and civil-time.
Each scale begins with zero, rather than one. Two noteworthy consequences:
Try to interpolate between the least count, to one-fourth or one-fifth.
The facies of the astrolabium solves the SSS (= side, side, side) problem of spherical trigonometry: Since an astrolabium has to be designed for a specific latitude, the corresponding co-latitude gives us one side. The observed altitude (elevation) of a known celestial object gives us another side. From the declination (latitude from the celestial-equator) of this object we have its co-latitude, which gives us the third side of the spherical triangle. We solve for the polar-angle and the azimuth-angle. We are not concerned about the third angle of this triangle.
For today's date and approximate time, locate the Helios on the ecliptic, on the facies (also known as the obversus-side). On the dorsum (also known as the reverse-side), employing the dioptra (also known as the alhidada), measure the altitude (elevation) of the Helios or other known aster. On the facies, rotate the rete (on a brass astrolabium, usually it would be a aranea) so that the Helios or aster is aligned with the altitude on the tabula (also known as the tympanum). Rotate the ostensor to the Helios (even if you were employing an aster). Read the equal-hour apparent-time on the limbus of the mater.
Elizabethan England still (until 1652) was on the old-style Julian calendar. However, my astrolabium was designed in France, where they already are on the Gregorian calendar, since 1582. Hence, we have to convert the date from Julian to Gregorian. Nominally, we add 13 days. However, we have to subtract one day each for the leap-year or if the time is between sunset and midnight. Astrolabia were being made in France, Belgium, Germany, and Italy; and, of course, in Spain and other Muslin countries. However, apparently, they were not being made in England -- they had to be imported. At least, mine was!
On the dorsum, locate today's date and approximate time on the eccentric-circle. Employing the dioptra, read-off the position of the Helios on the zodiac. Remember this position; you will need it later, on the facies.
Be sure to read the time on the zero to twelve small-scale running from sunset to sunrise and again from sunrise to sunset. Once we have the equal-hour apparent-time, remember it. Read-off the declination of the Helios from the scale on the ostensor. Rotate the ostensor to the proportional-hour scale and see what correction is needed for the current time between the current declination and the equator. The result of this correction is the proportional (variable-hour) Helios-time. Neither time-zones nor daylight-saving time had been invented in Elizabethan England. The equation of time was known, of course; but, you would not be concerned with the equation of time for telling the current (true local solar) time.
Time has become more complicated. When good clocks -- isochronous pendulum clocks -- proliferated around 1680, the mean solar time was introduced; so that the clock would not need to be adjusted to comply with the equation of time. When railroads proliferated, between 1850 and 1900, they did not want to deal with each village having its own time. Thus, they introduced time-zones, each 15 degrees (one hour) wide in longitude, centered on multiples of 15 degrees each direction from Greenwich. There were variations. For instance, Imperial Russia adopted the concept as a single time -- that of the Moscow time-zone -- all the way to the Pacific. Around 1960, the daylight-saving time has become popular.
The beginning of the calculation is easier on the modern astrolabium; because, the days are shown directly on the ecliptic on the rete. Such a scale was conceivable even as early as classical Greece; but, without a computer, it would not have been practical to execute. Besides, they would not have desired to do so for aesthetic reasons.
The termination of the calculation requires three extra steps. First, be sure to read the time on the large-scale running from midnight to noon and again from noon to midnight. Once you have the (equal-hour) apparent-time, remember it. Flip the astrolabium over to its dorsum. Subtract the indicated longitude correction. Line-up the dioptra with the current date on the limbus. Read-off the equation of time correction from the scale on the dioptra and subtract. If daylight-saving time is in effect, add one hour. Now, you have the current civil-time.
In Medieval times, it was a meticulous compass and straight-edge project. Now, a computer program may be written to do the grunt-work.
On the dorsum of the classical astrolabium, the eccentric circle of the dates has its center displaced from that of the circle of the zodiac by twice the product of the eccentricity and major semi-axis of the ellipse of the actual orbit. Actually, the zodiac is an ellipse, with the focal-point at the Earth; but, its eccentricity is so small that at the scale of an astrolabium the distinction would not be apparent. This projection yields the correct speed at each major vertex. Since, by definition, the period along the eccentric circle is the same as of the actual Helios, the average speed also is accurate. The inaccuracy of the speed at the rest of the orbit was less than the observational errors in the classical Greek times. Only the careful observations, in the latter 1500's, by Tycho Brache made the eccentric-circle description not tenable.
The Egyptians devised geometry. The Greeks invented Mathematics and the concepts of axioms and proofs. They axiomatized and systematized Geometry. They discovered the equivalent of Plane Trigonometry and spherical coordinates -- they had to have done so to be able to study the properties of the stereographic projection --; but, did not have Spherical Trigonometry, as yet. However, neither the Egyptians nor the Greeks had a telescope -- Galileo built the first one. The Greeks were prejudiced in favor of circular orbits. It required exquisite observational skill to be able to discern that the orbit of the Helios was centered about a point other than the Earth.
To be precise, the Greeks employed the similar triangles to accomplish the equivalent of Trigonometry. The Muslims introduced the names of the trigonometric functions, as we now know them. Gauss introduced the modern concept and notation of a function.
The modern astrolabium, on its facies, has the dates shown directly on the ecliptic. Since this astrolabium is computer-generated, presumably, the ecliptic is the correct ellipse and the dates follow from Kepler's Three Laws, as shown therein Thus, potentially, the modern astrolabium is exactly accurate, in this regard.
We ignore the parallax of the Helios as seen from the sunrise and sunset positions on the Earth. While the Helios is in the same place on its trajectory of the ecliptic at the same UTC time anywhere on the Earth, since we compute its position from the local-time, there is a built-in conversion from local to UTC time, in the calculation of the position of the Helios. Finally, there is the gradual precession of the equinoxes -- the astrolabium has to be designed for a specific year; but, is acceptably accurate within half-of-a-century. On the other hand, it means that neither any of the surviving astrolabia from the Renaissance nor a precise copy of one would be usable today. Furthermore, the implied position of the precession may be employed to date the construction of an astrolabium, to within a century.
The rete is a stereographic projection of the celestial-sphere onto its equator, with the eye-point at the south-pole, as illustrated. With this specific choice of the eye-point, the projection is conformal. The north-pole projects to the clavus. Regrettably, the equus obscures the vicinity of the north-pole. The asterismos (Greek, asterism) of the prominent constellations are drawn on the rete. The ecliptic maps as approximately an eccentric circle, which is externally-tangent to the circle of the tropic of Cancer and internally-tangent to the circle of the tropic of Capricorn. The rete rotates in synchrony with the sidereal-time. A sidereal (aster) day is approximately 23 hours, 56 minutes, and 4 seconds. Except for the proper motion of the asters, the rete is invariant -- it does not need to be customized. Of course, however, the ecliptic -- especially if it has the dates, instead of the zodiac, shown -- has to be customized, as discussed in the previous paragraph.
In practice, none of the coordinate orthogonal net is shown on the rete. Instead, only the equator, the two tropics, and, optionally, the two artic-circles of the parallels of latitude are drawn on the tympanum. Since they are circles, centered at the north-pole, they are invariant under the rotation of the rete. Also shown on the tympanum are the terrestrial prime-meridian of longitude (which maps to a vertical straight-line) and the meridian at right-angles to it (which maps to a horizontal straight-line). Together, they map into a pair of orthogonal straight-lines, crossing at the north-pole. Since they are terrestrial, they are in a fixed position.
How far south should the rete and tympanum extend? Obviously, their common extent is arbitrary. Since one wants to show the ecliptic, the tropic of Capricorn is required. For use by residents of the northern temperate zone, the tropic of Capricorn is a good compromise. For use by residents of the northern tropics, it would be desirable to provide the southern arctic-circle. The distortion is great any farther south. During the daytime, the astrolabium is useless within the artic-circle, and of dubious value at night. On the other hand, for use in the southern hemi-sphere, one would place the north-pole -- instead of the south-pole -- at the eye-point.
The parallels of latitude migrate to the ostensor, in the form of a scale of the declination -- the latitude from the celestial equator.
The tympanum has several families of curves drawn upon it. Of them, only the azimuth-altitude orthogonal net has to be custom-designed for the specific parallel of latitude of the user.
The azimuth-altitude orthogonal net is centered, at the desired latitude, on the upper portion of the vertical straight-line through the north-pole. The equations for drawing these arcs of circles are provided in the aforementioned discussion of the stereographic projection.
This net is extended to the horizon. Below the horizon, only the 6 (civil), 12 (nautical), and 18 (astronomical) degree crepusculum arcs of the altitude are shown, between the two tropics, because the Helios always is in the tropics.
When does the day begin? Depends upon whom you ask! Astronomers begin the new day at noon, to have the whole night of astral observation uninterrupted by a date-change. For a similar reason, the civil-time begins the new day at midnight, not to interrupt the work-day with a date-change. The classical new day is at sunset -- when the Helios sets, the day is done. For completeness, I propose that the most logical beginning of a new day is at sunrise. Thus, we have a new day beginning on each six-hour. In each case, the dates agree between sunrise and noon. Suitable adjustments have to be made at other times of the day. The actual counting of the hours begins with zero at the beginning of the new day, with 24 hours in the day. Optionally, they may be expressed modulo twelve.
The hour-scale begins at the beginning of the day -- whenever that might be. The degree scale does likewise; but, it may go either counter-clockwise or clockwise. Optionally, it may be expressed modulo 90 or 180. Each of these scales may drawn on the limbus of the mater, instead of on the limbus of the tympanum. There also is a degree-scale on the dorsum.
On the northern otherwise-clear portion of the tympanum, between the tropics, are shown arcs proportional to the horizon. They are numbered from 0 due-west, through 12 due-east, with 6 a straight-line due-north. Thus, they represent the night-time. These arcs are employed to estimate the unequal-hours at a given declination of the Helios. One has to read the negative of the declination of the Helios, to employ these proportional-scales for the day-time unequal-hours.
The azimuth from the northern horizon (intersection with the zenith-meridian) in divided into twelve 30-degree sectors. They are numbered, beginning with I due east, going northward. North is IV, west VII, and south is X. These houses were popularized by the astronomer Regiomantanus (1436-1476). There are other systems of houses.
On its facies, the mater has a replaceable tympanum -- for each parallel of latitude. On my astrolabia, however, the single tympanum is inscribed directly on the facies of the mater.
Each of the correction has to be subtracted from the apparent-time. The longitude correction is
(your-longitude mod(15 degrees)) ((60 minutes of arc) / (1 degree)) ((1 minute of time) / (15 minutes of arc)).
Remember, that west-longitude is negative. The modulo reduction is to yield a result in (- 7.5, 7.5), usually.
The umbra (Latin, shadow) square provides a scale of the tangent and of the cotangent. The umbra is that of a gnomon (Greek, interpreter) bar. The gnomon may be versa (Latin, vertical) or recta (Latin, horizontal).
The correction for the equation-of-time is read from the scale of minutes on the dioptra, when it is lined-up with the current-date on the limbus of the dorsum side of the mater. The equation-of-time is a direct consequence of the second law of Kepler -- it is a differential equation for the conservation of angular momentum We provide a derivation from Newton's laws of motion and gravitation. Since our calendar drifts (remember the precession of the equinoxes) with respect to the perihelion, the equation of time has to be adjusted to the year for which the astrolabium is designed. Historical note: The Greeks already could have approximated the equation of time from their eccentric-circle representation of the motion of the Helios in the ecliptic. However, the mean solar-time only became of interest when pendulum clocks proliferated
For sighting the Helios or other objects with the dioptra, one has to have the astrolabium in a plumb vertical position. To this end, the top of the astrolabium is provided with an armilla (Greek, bracelet, ring) fixa (Greek, fixed; together, they mean throne), to which is attached an armilla reflexa (intermediate ring), which is hung from the armilla suspensoria (suspension ring). This armilla suspensoria is supported by the thumb of the left-hand of the observer.
It is a missed opportunity. There should be a circular slide-rule inscribed on the dorsum and another rete. At the very least, the A, B, C, CI, D, S, and T scales should have been provided. It would be even nicer if the family of L scales had been included. Since logarithms were invented by Napier, these slide-rule scales would be out of context on the classical astrolabium; but, they would be a most welcome addition to the modern astrolabium.
Since the astrolabium must be custom-designed for the user, neither the original Medieval or Renaissance nor exact copies of them would be usable, at present. These Medieval or Renaissance astrolabia were meticulously-inscribed precision instruments. Modern brass astrolabia tend to be just decorative or stage-props. I have obtained a pair of delightful, precise astrolabia from Janus. They are custom-designed, printed by a computer program on paper, and laminated in clear plastic. They are very reasonably priced. Also, on this site, there are articles and references.
| Greek | Latin | English |
| alhidada | alidade, rotating radius [on the back] | |
| aphelion | farthest from the Sun | |
| aranea | spider [whose legs indicate the stars] | |
| armilla | bracelet, ring | |
| armilla fixa | fixed portion of the suspension | |
| armilla reflexa | intermediate ring | |
| armilla suspensoria | suspension ring | |
| aster | star | |
| asterismos | asterism, prominent stars of a constellation | |
| astrolabium | astrolabe | |
| clavus | pivot, pin | |
| crepusculum | crepuscular, twilight | |
| dia- + opsesthai | dioptra | to be going to see, [alidade, on the back] |
| dorsum | back, reverse-side | |
| ephemeris, plural ephemeredes | --, [position of moon & planets] | |
| equus | horse, [horse-head shaped nut] | |
| facies | face | |
| gnomon | gnomon | interpreter |
| gnomon, umbra versa | a vertical bar, which casts a shadow | |
| gnomon, umbra recta | a horizontal bar, which casts a shadow | |
| Helios | Sun | |
| limbus | limb, periphery | |
| mater | mother | |
| obversus | front [front side of a coin or other flat object] | |
| ostensor | rule [on the front] | |
| perihelion | nearest to the Sun | |
| rete | net [which supports the stars] | |
| reversus | reverse [back side of a coin or other flat object] | |
| tabula | table | |
| theodelitus | theodolite | |
| tympanum | plate | |
| umbra | shadow |
Copyright (c) 2000, 2,3 by R. I. 'Scibor-Marchocki. last modified on Thursday 13-th June 2002. An external link added on Wednesday 02-nd July 2003. Several broken links deleted on S 30-VII-2005. A link corrected on Monday 29-VIII-2005.