Mathematical Methods in the Study of Historical Chronology Florin Diacu ome ancient and medieval historical dates historian Ephorus of Cyme (ca. 405–330 BC), who we learn in school may stick in our minds: provided many dates in his work on universal 1193 BC (the fall of Troy), 753 BC (the history. His more famous predecessor Herodotus founding of Rome), AD 1 (the birth of (ca. 484–425 BC) had been far less interested in Jesus), or AD 1066 (the battle of Hastings), placing events in time. Sextus Julius Africanus Sto mention just a few. Most of us take these dates (ca. AD 160–240), often called the first true for granted, but are they correct? Anatoli Fomenko, chronologist, is considered the father of biblical a fellow of the Russian Academy of Sciences, history, an independent field from which the finds them fictitious and promotes an alternative science of historical dating would later borrow its chronology that often differs from tradition by principles. In his work Chronologia, he put together more than a millennium [1]. dates from Hebrew, Egyptian, Greek, Persian, and The concerns about the correctness of ancient other sources by connecting various pieces of and medieval dates are not new. Isaac Newton also information, as can be noted in the example below opposed them. In his last book, which he finished [15]: preparing for publication just before his death, he If one computes backwards from the end argued that ancient Greek history should be about of captivity, there are 1,237 years. So, by 300 years shorter than claimed [9]. Since the global analysis, the same period is found to be the chronology is based on the Greek chronology, the first year of the Exodus of Israel under Moses foundation of its edifice appears to be shaky. from Egypt, as from the 55th Olympiad to It is thus natural to ask how historical events Ogygus, who founded Eleusis. And from were dated, why they are controversial, and if here we get a more notable beginning for mathematics might play any role in raising or Attic chronography. solving these problems. To lay the background The work of Julius Africanus was continued by towards throwing some light on these questions, Eusebius, Bishop of Caesarea (ca. AD 260–341), let us first make an incursion into the origins of who came up with tables that compared parallel history’s timetables. The material that follows is major events. For instance, he informs us that based on [1], the second edition of a book originally Jesus was born in the year of the 194th Greek published by Knopf Canada. Olympiad, which took place 2,010 years after the birth of Abraham, in the forty-second year A Brief History of Historical Dating of Augustus’s reign and the twenty-eighth year The first writer whom we know to have been after the Roman subjugation of Egypt, or the year interested in historical chronology is the Greek when Antony and Cleopatra died. All of Eusebius’s Florin Diacu is professor of mathematics at the Pacific In- entries start with “the year of Abraham,” i.e., 2010 stitute for the Mathematical Sciences and University of BC. Unfortunately the calculations are based on the Victoria. His email address is [email protected]. unrealistic life spans mentioned in the Bible, such DOI: http://dx.doi.org/10.1090/noti962 as Adam, 930 years; Noah, 950 years; Abraham, April 2013 Notices of the AMS 441 175 years; or Moses 120 years, numbers upon Julian epoch to noon on Monday, January 1, 4713 which later chronologists built history. BC, which thus became a convenient reference Among the medieval researchers who followed point for all his computations [13]. A method he this tradition were Martin Luther and Gerardus then devised, based on the twelfth-century work Mercator. But in spite of their contributions, of Roger of Hereford and used extensively by chronology remained a gathering of disparate Scaliger’s follower Dionysius Petavius, was that dates used mainly for religious purposes, such of combined cycles. This method employs the as making sense of the biblical stories or decid- numbers 19, 28, and 15 in terms of congruences: ing Easter Day. The current consensus among 19 stays for the lunar cycle, i.e., the number of researchers, as expressed by Denys Hay, a profes- integer years the moon takes to complete an integer sor at the University of Edinburgh, is that “[i]n number of orbits (235) around Earth; 28 stays for classical antiquity there was virtually no system of the solar cycle, i.e., the minimum number of years chronology available to historians” [7]. after which the Gregorian calendar repeats itself; The breakthrough came in 1583, when the French and 15 stays for the Roman indiction, a taxation scholar Joseph Justus Scaliger published a seven- cycle established in AD 313 and used as late as volume work titled De Emendatione Temporum, the sixteenth century in some places. This method which provided dates for the main historical events assigns to each date in history its Julian count of humankind. This colossal enterprise treats in (the number of years since the Julian epoch). Every detail the astronomical bases of more than fifty Julian count up to 7980 has a unique triplet of calendars. To achieve this feat, Scaliger had taught numbers resulting from the remainder obtained himself thirteen languages, Hebrew and Arabic when dividing the Julian count by 19, 28, and 15. So among them, becoming one of the most respected if a certain event can be associated with the lunar philologists of his time. He embarked on his cycle, the solar cycle, and the Roman indiction, its chronology work aiming to understand the flow year can be determined. of ideas and who had influenced whom in ancient But Scaliger’s most important contributions literature. He ended up sorting out the problems are with understanding calendars, most of which of history, an achievement for which he received were long forgotten during his time. Such studies in 1593 a research professorship at the University are multidisciplinary. To reach the point when of Leiden, where he worked until his death in mathematics can be of any use, he had to first 1609. His second and last treatise was Thesaurus unravel the calendar’s language and the deeper Temporum, a collection and arrangement of all meaning of the nomenclature. Scaliger started available ancient chronological sources. almost from scratch. An example of the difficulties Despite their deficiencies, his historical time- he encountered are made evident in the lines he tables survived and were later used to build all wrote in 1568 prompted by some third century AD Eastern chronologies, including those of India, statements he disagreed with [5]: China, and Japan. Anthony Grafton, Professor of European History at Princeton University, remarked I do not see how the month of April can recently that “[t]he few modern historians who derive its name from aperio [to open, to mentioned Scaliger described him as a brilliant discover]. First of all, since the year initially innovator who created a discipline in the teeth of had only ten months, they must have always ferocious opposition” [6]. No doubt, this was the wandered and had no fixed position in the new science on which history would rest. year…. [In fact] aprilis comes from aper, which is boar. Early Use of Mathematics Once the calendrical language was clear, many Apart from simple arithmetic operations aimed at other difficulties occurred. The rules of the old understanding various calendars, the early use of Roman (pre-Julian) calendars, for instance, often mathematics in chronology was mostly reduced changed according to the interests of the political to the computation of eclipses. Scaliger based leaders. These alterations are not only hard to trace his method on the astronomical and calendrical but are also detrimental in chronology studies information he gathered from ancient sources, because they can lead to misinterpretations that aiming to fix some historical landmarks, which he may give rise to large errors in time. Apart from could then connect to other events. Since total solar such subjective issues, another problem Scaliger eclipses, for instance, could be computed with had to deal with was the type of calendar he tried reasonable accuracy, he interpreted the ancient to decipher: lunar, solar, lunisolar, or of some descriptions of such celestial phenomena to place arbitrary type, most of which required not only them in time. solid knowledge of astronomy, but some algo- Scaliger also intuitively applied congruences rithmic sophistication as well. The computational and the Chinese Remainder Theorem to fix the difficulties are easy to overcome today, so it’s no 442 Notices of the AMS Volume 60, Number 4 wonder that calendrical calculations now belong to the first volume of a book that considered the the realm of computer science [14], but Scaliger had issue by looking at historical solar eclipses [11]. to approach them without proper mathematical Five years later, he came up with a second volume, training, thus having at his disposal only some which approached the problem from the point of rudimentary tools. view of lunar observations. His conclusion was that the behavior of D00 could be explained only Modern Developments by factoring in some unknown forces [12]. Scaliger’s work was attacked from the very begin- Fomenko found the idea of unknown forces ning, but apart from being subject to fine tuning, it outlandish, so he used his own chronology to redate survived unscathed. Historians based their theories Robert Newton’s astronomical records, which led on it, and today every student of history takes it for to the conclusion that D00 was almost constant granted.