Foundations of Science/First Steps - Astronomy/Ptolemy

Introduction
In First Steps - Astronomy we briefly described the Aristotelian model of the universe in which the planets are arranged about a central Earth (a geocentric model) according to their orbital periods and the outer sphere, the Primu Mobile or Prime Mover is responsible for the 24 hour revolution common to all the heavenly bodies. An elaboration of this model was described by Claudius Ptolemy of Alexandria (2nd century AD) whose work survives today as the Almagest. The Aristotelian model as described in the previous section was only concerned with the general motions of the planets. The Ptolemaic model, on the other hand, was a real attempt to describe the detailed motions of the planets as accurately as possible using mathematics. When we look up at the night sky and observe the planets, it’s difficult to imagine how Kepler came to know that they travel in ellipses. As a first step towards this understanding, let’s get an idea of how Ptolemy dealt with planetary motion. To do this, we’ll look at the motions of the Sun and of Jupiter.

The Motion of the Sun
As everyone is aware, the overall height or position of the Sun in the sky varies over the course of the year, being relatively high in the sky during summer with long days and short nights, and low in the sky during winter with correspondingly short days and long nights. This variation becomes more apparent the further you are from the equator. If you are on or very near the equator, then all of your days and nights are the same 12 hours in duration and the Sun simply moves north of the equator from April to September and is south of the equator the rest of the year. The furthest points north and south are the summer and winter solstices. The crossing of the Sun across the equator marks the spring and autumnal equinoxes, when the day and night are 12 hours in length for everyone no matter where they are in the world.

This annual motion of the Sun is vitally important, for it is the cause of the coming and going of the seasons and as such was carefully watched. The spring equinox was used by many ancient peoples to mark the passage of a year. One method of observing the equinoxes used by the ancient Greeks was to use an equatorial ring, a circular strip or belt of metal mounted so that the plane of the circle was parallel to the plane containing the earth’s equator. When the Sun crossed the equator, the shadow of the southern edge of the circular belt would be exactly centred on the northern half of the belt (in the northern hemisphere). Using these observations they could determine the timing of the spring and autumnal equinoxes to within a half day or better.

While the annual motion of the Sun was carefully observed, the Moon provided a better tool to mark the passage of days within the year. The year was divided into 12 months alternating in 29 or 30 days in length, since the lunar cycle is about 29½ days. The new year began with the new moon of the lunar cycle containing the spring equinox. Unfortunately, a twelve month period would be 11 days short of a solar year. Thus to keep the lunar year synchronised with the solar year, an extra month would be added to the end of the year in February every 2 or 3 years.

The Romans made substantial changes to this arrangement. With the advent of societies that had good record keeping and communications, the lunar cycle was less important as a time-keeper and by ignoring the moon, the calendar could be made more uniform. The best estimate for the length of the year made by Callipus (Fourth century BC) was 365.25 days. Thus the 11 days left after 12 lunar months were added to the months, making them alternately 30 and 31 rather than 29 and 30 days. Since they were only 11 extra days for the 12 months, one would expect that February, the last month, would not get an extra day. But worse than that, it actually lost a day, and it would seem to have gone to January or August. This then would give a 12 month year of 365 days, albeit the months were no longer aligned with the lunar cycle. As mentioned, the year is on average 365.25 days and to make up the extra quarter day, one full day would be added to the end of the year every four years, and February could again enjoy its former fullness. Theses changes to the calendar were made during the time of Julius Caesar and thus it was named the Julian calendar. Through the widespread rule of the Romans, the Julian calendar was instituted across the Mediterranean and a large part of Europe.

As early as the 2nd century BC, Hipparchus had determined that the year appeared to be shorter than 365¼ days by about 1/300 of a day. Ptolemy used observations taken by Hipparchus compared with his own observations about 285 years later, that about 19/20 of a day shorter thus, the year is about 1/300 of a day shorter than 365¼ days. We now know the year is about 1/128 of a day shorter than 365¼ days so that it would appear that either the length of the year, when measured in days, seems to be gradually decreasing, or that Ptolemy had deliberately misrepresented his data to achieve results in line with those of Hipparchus.

This small adjustment to the calendar was not formulated until the sixteenth century. By this time the calendar had advanced about 12 or 13 days, so that the spring equinox was being observed on or around March 12. So in 1582, following the work of Christopher Clavius who had been charged with calendar reform, Pope Gregory XIII decreed that October 4 would be followed by October 15, thus getting rid of 10 extra days and that henceforth every leap year divisible by 100 would not have the extra day added, unless it was divisible by 400, in which case it would be a regular leap year, thereby dispensing of 3 days every 400 years. Thus the year in the Gregorian calendar is 3/400 or approximately 1/133 days/year less than the Julian year, still not quite right, but, it seems, close enough.

Excerpts from the Almagest