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Why does February get a 29th day in some years but not in 1900? Because the year is not a whole number of days, and getting its length right took centuries of careful measurement. One of the most careful measurers was al-Battani, working in Syria around 900 CE. This post covers who he was, what he measured, and works through his numbers against modern values, using the same unit conversions you need in physics.
Who was al-Battani?
The MacTutor biography gives his birth as about 850 at Harran, in present-day Turkey, and his death in 929 at Qasr al-Jiss in Iraq. In Latin he became Albategnius. His father was probably an instrument maker, and he was skilled at making instruments himself. He observed at Antioch and, mostly, at ar-Raqqah on the Euphrates. A 10th-century catalogue of books, the Fihrist, mentions observations from 880 and 900.
His major work, the Kitab al-Zij, has 57 chapters of astronomy, mathematics and tables. MacTutor credits it with a catalogue of 489 stars and these key values:
Length of the year: 365 days 5 hours 46 minutes 24 seconds.
Precession of the equinoxes: 54.5″ per year.
Inclination of the ecliptic, the tilt of Earth's axis: 23°35′.
What is the year he measured?
The year that matters for seasons is the tropical year, which NASA's calendar notes define as "the mean interval between vernal equinoxes". Their formula gives 365.2421897 days at the year 2000. The NASA Earth fact sheet lists it as 365.242 days, and the sidereal year, measured against the stars, as 365.256 days.
Worked example: how close was he?
Convert al-Battani's extra time to seconds. 5 h 46 min 24 s = 5 × 3600 + 46 × 60 + 24 = 18,000 + 2760 + 24 = 20,784 s.
Convert to days. 20,784 / 86,400 = 0.240556, so his year is 365.240556 days.
Convert the modern value back. 0.2421897 × 86,400 = 20,925 s = 5 h 48 min 45 s.
Difference. 20,925 − 20,784 = 141 s, about 2 min 21 s.
What it would do to a calendar. An error of 141 s a year adds up to one day (86,400 s) in about 86,400 / 141 = 613 years.
Compare the Julian calendar's year of exactly 365.25 days: too long by 0.0078 days, about 11 min 15 s a year, which drifts a full day in about 128 years. The Gregorian calendar, as the same NASA page explains, makes century years leap years only if they divide by 400, giving an average of 365.2425 days. That is why 1900 was not a leap year but 2000 was.
Worked example: precession and tilt
Precession. At 54.5″ a year, the equinox moves 1° (3600″) in 3600 / 54.5 = 66 years, and a full circle in 360 × 66 ≈ 23,800 years.
Tilt. NASA's Earth fact sheet gives an obliquity of 23.44°, which is 23°26′. Al-Battani's 23°35′ is about 9′ larger. Part of that gap is real: the tilt changes slowly. MacTutor quotes the historian W. Hartner, who notes that al-Battani corrected Ptolemy's conclusion that the obliquity never changes.
Why were his measurements so good?
MacTutor reports two reasons. He used trigonometry rather than Ptolemy's geometric methods, with relations such as b sin A = a sin(90° − A) for right triangles, which you would now write as tan A = a/b. And one study suggests his solar measurements beat those of Copernicus because ar-Raqqah lies further south: at the winter solstice the Sun stood higher in his sky, so atmospheric refraction distorted his readings less.
He also showed that the Sun's greatest distance from Earth varies, so annular eclipses of the Sun are possible as well as total ones.
Which mistakes does this help you avoid?
Confusing tropical and sidereal years. 365.242 and 365.256 days differ by about 20 minutes. Seasons follow the tropical year.
Dividing time by 100. 46 minutes is 46/60 of an hour, not 0.46.
Forgetting the century rule. A year divisible by 100 is a leap year only if it is also divisible by 400.
How did his work spread?
The Kitab al-Zij was translated into Latin by Plato of Tivoli in 1116 and printed in 1537 and 1645, MacTutor records, and it influenced Tycho Brahe, Kepler, Galileo and Copernicus. Our posts on Ptolemy's Almagest and Copernicus cover the before and after.
What can a student learn from al-Battani?
He respected Ptolemy's framework but checked Ptolemy's numbers against his own sky, and corrected them where they failed. Keeping a method while testing its results is a good way to treat any formula sheet.
Unit-conversion slips cost marks in physics too. The Fermi Problem Library has a step-by-step solver that explains the reasoning, so you can see exactly where a conversion went wrong.
Try it now: convert the sidereal year of 365.256 days into days, hours, minutes and seconds.
