### Chinese Remainder Theorem: an Application to Chronology

I am reading an unusual book on an extraordinary weird subject. A Canadian mathematician Florin Diacu's *The Lost Millennium* collides two points of view on the existing chronology: one would shorten it by about a thousand years. Along the way, Diacu meticulously pursues the origin and evolution of chronology as a science. This is a fascinating book that deserves a careful reading and a detailed review. I hope to give it its due shortly. Meanwhile, here's a little piece of mathematics that was used by one of the fathers of the modern chronology Denis Pétau, a Jesuit theologian and philologist born in Orléans, France, in 1583. In 1627 Pétau - better known by his Latinized name of Dionysius Petavius - published a fundamental work on chronology, *Doctrina Temporum* (*On the Doctrine of Chronology*). Petavius

... laid more weight on astronomical phenomena than his predecessor (Joseph Justus Scaliger) had, coming up with ingenious ways of relating dates to the motions of celestial bodies. Petavius used the

combined-cycles methodextensively - lunar (19), solar (28), and indiction (15) a system Scaliger had developed from the twelfth-century work of Roger of Hereford.The rationale for those cycles can be discussed succinctly.

19 is the smallest number of full years the Moon takes to complete a full number of orbits (namely, 235) around Earth.

28 is the smallest number of years after which the calendar repeats itself, with the dates matching the days of the week.

15 represents the *Roman indiction*, a taxation cycle introduced by the Emperor Constantine, starting with AD January 1, 313. This period became a standard throughout the Eastern Roman Empire.

Every year, starting with 4713 BC which counts as 1, is assigned its Julian count. For example, the year 753 BC has the Julian count 3961 (= 4713 - 753 + 1). Every Julian count has a unique triplet of numbers which are remainders of the division of the count by 19, 28, and 15. Any such triplet uniquely determines a count up to 7980. (Do you see why?)

This is an instance of what is nowadays known as the Chinese Remainder Theorem. The applicability of the method depends of course on the presence of documents that relate a historical event with a year in a lunar cycle, another in the solar cycle, and yet another in the indiction cycle. I understand that there is a sufficent number of events with a complete set of triplet associations to make the combined-cycles method useful. Obviously, for the sake of an exercise, one may not need to refer to a well documented historical event.

Carnival of Mathematics 85......a little piece of mathematics in the history of chronology given in Florian Diacu's......

April 5th, 2012 at 3:44 am