- Almagest Ephemeris Calculator
This web page is organized as follows:
- Introduction
- How to Use the Almagest Ephemeris Calculator
- Calendar Date Module
- The Precession in Longitude and the Obliquity of the Ecliptic
- The Position of the Sun
- The Position of the Moon
- The Position of Mercury
- The Position of Venus
- The Position of Mars
- The Position of Jupiter
- The Position of Saturn
- The Luni-Solar and Planetary Aspects
- Bibliography
Introduction
This web page provides a set of
JavaScript calendar and ephemeris modules for calculating geocentric luni-solar
and planetary positions for an arbitrary calendar date and time according to the
kinematical models of the Sun, the Moon and the planets described in the
Almagest (also known as the Mathematike Syntaxis) of Claudius
Ptolemy (c. 150 AD).
The mean motions of the Sun, the Moon and the planets adopted in these modules correspond exactly with Ptolemy’s mean motion tables and the corrections for their eccentric and epicyclic motions are accurately modelled according to the luni-solar and planetary models discussed by Ptolemy in his Almagest.
As Ptolemy’s tables were probably used most often for astrological rather than for astronomical computations, a table displaying the astrological aspects between the luminaries (and the lunar ascending node) with some other astrological quantities is also provided.
At present, latitudes are given only for the Sun (this is always zero) and the Moon. In the near future, latitudes and distances will also be supplied for the other luminaries.
How to Use the Almagest Ephemeris Calculator
When the web page is loaded the ephemeris calculator automatically selects the epoch date for the tables in Ptolemy’s Almagest as the default date. This corresponds with mean noon at the meridian of Alexandria on 1 Thoth 1 Nabonassar (or 26 February 747 BC in the proleptic Julian calendar, around 10h UT). Other dates and times can be inputted directly in the calendar date module (at present only in the “mobile” Egyptian calendar) or can be set with the “Adjust date and time” module that allows for temporal adjustments ranging from 1-second to 100-year intervals.Dates, Calendars and Time in the Almagest
The tables in the Almagest are based on the ancient version of the Egyptian calendar with a constant year length of 365 days and no intercalary days. The epoch of the tables is fixed on the first year of the reign of the Babylonian king Nabu-nasir (Nabonassar, reigned 747-734 BC) as this predated the earliest (Babylonian) observations that were available to Ptolemy. Compared with the Julian calendar, the dates in this calendar shift one day in every four years and it is therefore often referred to as the “mobile” or “wandering” Egyptian calendar.Ptolemy makes no reference in the Almagest to the more modern version of the Egyptian calendar that was introduced in 26 BC (or 30 BC, as some sources suggest) by the Roman emperor Augustus. This calendar, commonly referred to as the Alexandrian or “fixed” Egyptian calendar, remained in step with the Julian calendar (introduced in the Rome Empire in 46 BC by Julius Caesar) by inserting an intercalary day (a 6th epagomenal day) every four years at the end of the year preceding the (Roman) year that contained the Julian leap day. Although no use was made of the Alexandrian calendar in the Almagest, a conversion to this calendar is provided as it was commonly used by many later Greek astronomers and astrologers.
Note that both the Julian as the Alexandrian calendars are here represented in their “ideal” (or proleptic) forms and do not implement the incorrect use of Julius Caesar’s quadrennial leap year rule before 8 AD. For details on the actual use of these calendars in Egypt during the reign of Augustus, cf. Snyder (1943) and Skeat (1993, 2001) – for an alternative view, cf. Hagedorn (1994) and Jones (2000).
The calendar module also provides a tentative conversion to the Greek astronomical calendar of Callippus as described by Geminus (Elementa astronomiae VIII.50-60) and often referred to in the Almagest. The conversion is based on a reconstruction of this arithmetical luni-solar calendar from the data in the Almagest by Fotheringham (1924) and Van der Waerden (1960, 1984), with additional constraints derived from other sources by Jones (2000). It assumes that in every 76-year cycle each 64th day is omitted (as well as the last day at the end of the cycle) and that seven years in each 19-year Metonic sub cycle have an intercalated month (Poseideon II in the years 1, 6 & 14, and Skirophorion II in the years 3, 9, 11 & 17). The Callippic calendar is assumed to commence on 28 June 330 BC (proleptic), at sunset following the New Moon after the summer solstice. Other reconstruction schemes are also possible but this appears to be the simplest one that satisfies nearly all known Callippic dates. The Callippic date converter should be used with caution but it may be useful in identifying new Callippic dates that in turn can help to refine our incomplete knowledge of this calendar.
In Ptolemy’s time the day in the Egyptian calendar started at sunrise and noon was defined as 6 (seasonal or unequal) hours after sunrise. Likewise, sunset occurred 12 seasonal hours after sunrise with midnight occurring 6 nocturnal hours later. The lengths of these diurnal and nocturnal hours depend both on the observer’s latitude as on the season which makes them cumbersome for use in astronomical calculations.
Ptolemy therefore adopted hours of equal length (also known as equinoctial hours) in his tables and shifted the epoch from sunrise to mean local noon (6 hours) at the meridian of Alexandria. Sunrise, as measured in equal hours, will then on average occur around 0 hours but somewhat earlier during the spring-summer seasons and somewhat later during the autumn-winter seasons. Sunset, likewise, will on average occur around 12 equal hours after sunrise but somewhat later during the spring-summer seasons and somewhat earlier during the autumn-winter seasons.
Note that dates between sunset and sunrise are commonly indicated in the Almagest as a double date (the first referring to the current day, the second to the day starting after the next sunrise). For the conversion from Alexandria mean local time to Universal Time (Greenwich Mean Time), a longitude difference of exactly 30º 0' 0" between Alexandria and Greenwich has been adopted.
For conversion to other ancient astronomical prime meridians, contemporary astronomers would have used current estimates for their longitude difference with Alexandria (referred to by Ptolemy as the “Metropolis of all Egypt”). The most comprehensive and authoritative compilation of geographical longitudes and latitudes in antiquity was also compiled by Ptolemy and is commonly known as the Geographica.
The following table lists
some well-known ancient cities that have functioned as astronomical meridians in
the past with their geographical longitudes and inferred time offsets relative
to Alexandria as tabulated in Ptolemy’s Geographica.
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| Babylon |
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| Jerusalem (Hierosolyma) |
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| Alexandria |
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| Constantinople (Byzantium) |
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| Athens (harbour) |
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| Rome |
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Note that Ptolemy’s geographical longitudes are measured eastwards from the “Fortunate Isles”, commonly identified with the Canary Islands. In the near future, a complete online version of Ptolemy’s Geographica will be available at Bill Thayer’s website Ptolemy: The Geography.
Predicting Equinoxes, Solstices, Syzygies, Planetary Stations and Planetary Aspects
The JavaScript modules can also be used for predicting the date and time (to the nearest second if desired, although this accuracy was unattainable with the time-measuring devices then available) of the astronomical seasons, the lunar phases (syzygies), planetary stations (when a planet’s motion in longitude changes from direct to retrograde or from retrograde to direct) and planetary aspects such as conjunctions or oppositions between any pair of planets.For instance, to determine the date of the vernal equinox (when the true solar longitude is 0º) for a given year, input a first estimate (say 25 March, the traditional Roman date) in the calendar module and click on to the solar position module. There a true solar longitude value will be given close to 360º/0º. Adjust the time with the “Adjust date and time” module (only the ±1-day and smaller unit buttons will be necessary) until the true solar longitude is as near to 0º as you want it to be. Return to the calendar module to obtain the predicted calendar date and time (referred to the meridian of Alexandria) for the vernal equinox of that year.
Likewise, the dates of the other astronomical seasons can be derived by homing in to a true solar longitude of 90º (summer solstice), 180º (autumnal equinox) or 270º (winter solstice). For lunar phases, go to the lunar position module (after inputting a first estimate in the calendar module) and home in to a true luni-solar elongation of 0º (New Moon), 90º (First Quarter), 180º (Full Moon) or 270º (Last Quarter).
For planetary stations and planetary aspects, go to the luni-solar and planetary aspect module and follow similar procedures. The latter module can also be used to determine when a specific zodiacal sign (or one of its smaller partitions) is occupied or vacated by the Sun, the Moon or a planet.
A Note on Notation and the Accuracy of the Almagest Ephemeris Calculator
Angular quantities calculated by the various JavaScript modules are expressed in degrees and decimal parts while angular quantities in the text are generally expressed in sexagesimal notation (this is indicated by the use of the semicolon ; separating the integer number of degrees from its sexagesimal parts).Note that in order to obviate the necessity of including bulky double-argument tables for correcting the mean positions of the Moon and the planets to their true positions, Ptolemy introduced some simplifications that made it possible to considerably compress the correction tables in the Almagest. The results obtained from this web page will therefore slightly deviate from the results obtained directly from Ptolemy’s tables but the difference will never amount to more than a few minutes of arc.
The mathematical representations of Ptolemy’s kinematical models used in the JavaScript modules are explained in detail in various publications listed in the bibliography: especially useful in this respect are the works of Neugebauer (1957, 1975), Pedersen (1974), North (1976), Evans (1998) and Jacobsen (1999). Computer animations of Ptolemy’s kinematical models have been prepared by Glenn van Brummelen (1998) and can be downloaded from the website mentioned in the bibliography.
Calendar Date Module
Click here for the precession in longitude and the obliquity of the ecliptic, the position of the Sun, the Moon, Mercury, Venus, Mars, Jupiter, Saturn or a table listing the luni-solar and planetary aspects.