What is this?
The French Republican calendar was a calendar created and implemented during the French Revolution.
It is also frequently referred to as the French Revolutionary Calendar, but this is a misnomer: year 1 of the calendar started on 22 September 1792, the day after the abolition of the monarchy and the founding of the French First Republic.
How does it work?
A year consists of 12 months of 30 days each, divided into three décades of 10 days each, followed by 5 complementary days (6 in leap years).
The year starts on the day of the autumnal equinox at the Paris Observatory (longitude 2°20′14.03″ E). A leap year follow directly from this definition: a year is a leap year when the next autumnal equinox happens 366 days later instead of the normal 365. By this definition, the year will never drift with respect to the seasons.
The 12 months are: Vendémiaire, Brumaire, Frimaire, Nivôse, Pluviôse, Ventôse, Germinal, Floréal, Prairial, Messidor, Thermidor, Fructidor. Every three months represent a season, and the endings of the names reflect this fact.
The complementary days are: la Fête de la Vertu, la Fête du Génie, la Fête du Travail, la Fête de l'Opinion, la Fête des Récompenses, and la Fête de la Révolution (leap years only).
What's so special about this version?
Most versions of the calendar floating around doesn't use the original definition above.
Most versions uses the so-called Romme method for leap years, using the same leap year rules as the Gregorian calendar, i.e. every year divisible by four, except century years not divisible by 400. This method might make sense, except years 3, 7, and 11 were leap years under the original rules and were observed as such in real life, but the Romme method instead makes years 4, 8, 12 leap years instead.
This version uses the original rules. The JPL's DE440 and DE441 ephemerides were used to calculate the exact timings of the autumnal equinoxes between the Gregorian years 13201 BCE and 17191 CE (corresponding to the French Republican years -14991 to 15399). The times were then converted to UT1+00:09:21, the exact local time at the Paris Observatory. UT1 was chosen to keep track of the Earth's rotation without having to worry about the issues posed by leap seconds in UTC. Note that due to the uncertainty over ΔT — the difference between UT1 and Terrestrial Time (TT) used in the ephemerides — it is theoretically possible for there to be inaccuracies when the equinox occurs very close to midnight.
For more details about how I calculated this calendar, please see my blog post on the topic. This is the fourth part of a series on time-keeping, and you are highly encouraged to read the first three parts for a more complete understanding.
What are those names above the Gregorian date?
Those are the names of the days in the rural version of the calendar. This was intended to replace the Catholic Church's calendar of saints, as the French Revolution wanted to reduce the influence of the church. Every day of the year has a unique name associated with the rural economy and these names are supposed to correspond with the season.
Every quintidi is named after an animal, every décadi is named after an agricultural tool, and the remaining days are named after various plants or produce. The only exception is the winter month of Nivôse, which has the remaining days named after minerals.
What are those numbers below the Gregorian date?
The five (or more) numbers separated by dots is the corresponding Mesoamerican Long Count calendar date. This is commonly known as the “Mayan calendar.” This calendar is not available for dates before August 11, 3114 BCE (25 Thermidor -4905).
What is decimal time?
Decimal time is a time system used during the French Revolution that divided the day into 10 hours, each with 100 minutes, which contained 100 seconds each.
The result is 100,000 seconds in one day, compared to the 86,400 seconds with the normal 24-hour system. This makes it very easy to denote time as a decimal fraction of a day. For example, decimal time 5:67:72 (around 13:37:31) on January 1, 2000 can be represented as
Also note that each decimal hour is 2.4 normal hours, each decimal minute is 1.44 normal minutes, and each decimal second is 0.864 normal seconds.