12 MATHEMATICS IN MESOPOTAMIA procedural texts and ephemerides,tables of positions of the heavenly bodies at various times.The procedural texts show how to compute the ephemerides. The arithmetic behind the lunar and solar observations shows that the Babylonians calculated first and second differences of successive data,ob- served the constancy of the first or second differences,and extrapolated or interpolated data.Their procedure was equivalent to using the fact that the data can be fit by polynomial functions and enabled them to predict the daily positions of the planets.They knew the periods of the planets with some accuracy,and also used eclipses as a basis for calculation.There was,how- ever,no geometrical scheme of planetary or lunar motion in Babylonian astronomy. The Babylonians of the Seleucid period did have extensive tables on the motions of the sun and moon which gave variable velocities and positions. Also special conjunctions and eclipses of sun and moon were either in the data or readily obtained from them.Astronomers could predict the new moon and eclipses to within a few minutes.Their data indicate that they knew the length of the solar or tropical year(the year of the seasons)to be 12 +22/60 +8/602 months (from new moon to new moon)and the length of the sidereal year(the time for the sun to regain the same position relative to the stars)to within 4 1/2 minutes. The constellations that lent their names to the twelve signs of the zodiac were known earlier but the zodiac itself first appears in a text in 419 B.C. Each sector of the zodiac was a 30 arc.Positions of planets in the sky were fixed by reference to the stars and also by position in the zodiac. Astronomy served many purposes.For one thing,it was needed to keep a calendar,which is determined by the positions of the sun,moon,and stars The year,the month,and the day are astronomical quantities,which had to be obtained accurately to know planting times and religious holidays.In Babylonia,partly because of the connection of the calendar with religious holidays and cere monies and partly because the heavenly bodies were be- lieved to be gods,the priests kept the calendar. The calendar was lunar.The month began when the crescent first appeared after the moon was fully dark (our new moon).The day began in the evening of the first appearance of the crescent and lasted from sunset to sunset.The lunar calendar is difficult to maintain because,although it is convenient to have the month contain an integral number of days,lunar months,reckoned as the time between successive conjunctions of sun and moon (that is,from new moon to new moon),vary from 29 to 30 days. Hence a problem arises in deciding which months are to have 29 and which 30.A more important problem is making the lunar calendar agree with the seasons.The answer is quite complicated because it depends upon the paths and velocities of the moon and sun.The lunar calendar contained extra months intercalated so that 7 such intercalations in each 19 years just about