Servois' 1813 Perpetual Calendar, with an English Translation - Gergonne's Footnote

Gergonne, editor of the Annales, attached the following footnote to page 88 of Servois' [1813] article:

I do not know if it has yet been remarked that the Persian intercalation, I mean that of 8 days over 33 years, slightly more accurate than the Gregorian intercalation, could be brought back in a completely remarkable way because of its precision and uniformity. To do this would require adding one day every four years, to delete it every century, to restore it every four centuries, to delete it every ten thousand years, to restore it every forty thousand years, and so on. Indeed, this would give the length of the mean year as, \[365^j + \frac{1}{4} - \frac{1}{100} + \frac{1}{400} - \frac{1}{10000} + \frac{1}{40000} - \ldots\] or \[365^j + \left(\frac{1}{4} + \frac{1}{400} + \frac{1}{4000} + \ldots \right) - \left(\frac{1}{100} + \frac{1}{10000} + \ldots \right)\] or \[365^j + \frac{25}{99} - \frac{1}{99} = 365^j + \frac{24}{99} = 365^j + \frac{8}{33}.\]

Here, Gergonne used the superscript \(j\) to abbreviate the word jours (days).

The process of intercalation refers to the insertion of a leap day in a given year to keep the calendar synchronized with the astronomical year [Richards 1998]. For instance, the Julian calendar had an intercalation of one leap day added every four years, which gave a mean of 365.25 days per year. The Gregorian intercalation consists of a leap day added only in years that are divisible by 4, except that years that are divisible by 100 do not get a leap day unless they are divisible by 400. This intercalation rule gives the Gregorian calendar a mean of 365.24 days per year. However, the Gregorian intercalation is off by 26 seconds per year, so that after about 3000 years, the calendar would be about one day ahead of the astronomical year [Richards 1998].

In his footnote, Gergonne referred to the Persian intercalation, which is eight intercalations in thirty-three years. In other words, every thirty-three years eight leap days would need to be added. This gives an error of 19.45 seconds as opposed to the Gregorian error of 26 seconds [Richards 1998]. This intercalation is attributed to the mathematician Omar Khayyam (1048–1131), who was appointed by Sultan Malik Shah of Khorasan (in Persia) to reform the calendar in about 1079 CE [O’Connor and Robertson 1999]. According to O'Connor and Robertson [1999], Khayyam measured the length of the year to be 365.24219858156 days, which was very precise given that modern astronomical precision allows us to measure the solar year at 365.242190 days.