# Time and its milestones

Harmony/Cape: 1997. Pp. 192 $17.95, £12.99 Marking the passage of time: a Breton portolan and calendar showing the cycle of the Moon, 1546 (above) and a modern copy of an Aztec calendar (right). Credit: BRIDGEMAN ART LIBRARY/MUSEE CONDE, CHANTILLY With only two years to go to the next millennium, Stephen Jay Gould has been moved to ponder the what, when and why of this supposed watershed in human affairs. Gould points out that apocalyptic prophecies of the end of the world have been with us since the beginning of Christianity, and the idea of the Second Coming to be followed by the blissful years of the new millennium is part of the Christian tradition. Gould discusses a few of these ideas, which have appealed more to the poor and oppressed who welcome a change than to the Princes of the Church, happily ensconced in their palaces. The Book of Revelation in the Bible suggests that history unfolds in 1,000-year cycles, or millennia. Several seventeenth-century divines addressed the date of the Creation, when the cycles started. A favourite approach was to tot up the ages of the biblical patriarchs and try to assign historical dates to biblical events. Bishop Ussher made one of the best- known attempts. He calculated that the Creation began 4,000 years before the birth of Christ, at noon on 23 October 4004 BC. The odd four years are attributed to an error made by Dionysius Exiguus, who invented the Christian era in the sixth century. Thus Gould calculates that the Second Coming occurred a few weeks ago, on 23 October 1997. This date seemed to pass uneventfully, but the desire to celebrate the start of the two-thousandth year of the era remains, with a subtle change in the meaning of ‘millennium’. Credit: BRIDGEMAN ART LIBRARY/PRIVATE COLLECTION But when, exactly, should we open the champagne? Turning to calendrics, Gould points out that since about 1690 there has been an argument between those who supposed that the dawn of a new century should be celebrated on 1 January 1701 or on 1 January 1700, and so on. The proponents of high culture have, with few exceptions, espoused the former date, which marks the elapse of a whole number of centuries since the start of the Christian era, but the latter date, a nice round number, has been favoured by the populace. This is a sterile debate, and Gould wisely does not take sides. Gould writes with humanity and with his usual flair and style. He provides a dozen or so illustrations of the Day of Judgement, and an index but no bibliography. He ends with a moving account of some idiot savants who, with impaired mental abilities, can nevertheless instantly calculate the day of the week of any date. For those convinced that our mental abilities can be encapsulated in a single parameter, this story provides a counter-example of someone who can perform a mental feat that eludes most but who is unable to master other mental skills. To find out who, you must read the book. The French Revolution sparked off a widespread interest in the calendar and its reform. Many articles were published on the subject, including a method by Lewis Carroll (Nature 35, 517; 1887) for calculating the day of the week. The most recent comprehensive treatise on the calendars of the world, Friedrich Ginzel's Handbuch der Mathematischen und Technischen Chronologie, dates from 1914. ## Calendrical Calculations Cambridge University Press: 1997. Pp. 307. £40,$64.95(hbk)

Since then we have seen the development of the digital computer, which now allows the instant transformation of a date in one calendar to a date in another. Nachum Dershowitz and Edward M. Reingold have provided us with a comprehensive set of computer programs to do just that, and, as a bonus, to calculate the dates of various festivals including Easter. They have chosen to present their algorithms in LISP, and one can download them from their World Wide Web page for one's personal use. But perhaps LISP is not the most widely understood programming language, and some will regret that the algorithms are not presented in a more accessible form.

Dershowitz and Reingold consider a representative selection of arithmetic calendars in which a date, most often in the form of a triplet of numbers representing the day, month and year, is determined by well-defined arithmetical rules; the calendars considered include the Julian, Gregorian, Coptic, Ethiopian, Islamic, modern Persian, Baha'i, Jewish, Mayan and Old Hindu. They go on to a further set, the French Revolutionary, Chinese and Modern Hindu calendars, in which the start of a year or month is fixed by astronomical observation or at least by an accurate astronomical theory, and provide routines for translating each date into a day number and back again into a date. As the epoch of their day count, they chose the proleptic start of the Gregorian calendar, 1 January 1 AD. In this they differ from others who have used the Julian day number, as used by astronomers.

Dershowitz and Reingold discuss briefly the history of each calendar, and provide selected references for further reading, and there are apposite illustrations. The logic of the calendars and the LISP algorithms are discussed adequately, if sometimes tersely. Some 100 pages of appendices are devoted to a full LISP listing and a table of selected days rendered in each of the calendars. They mention that there is at least one error, but point to an errata to be found on their Web page. Mayan calendricists should note that they have used the correlation of the epoch of the long count with Julian Day 584283, as opposed to 584285 which is favoured by some.

For those prepared to come to terms with the code, this book must surely become the standard work on calendar conversions. No historian, chronologist or recreational mathematician should be without it.

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