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The Julian calendar is the calendar introduced in Rome by Julius Caesar, in which a year has 365 days plus a leap day (an extra day, such a year termed a leap year with 366 days) each four years, for an average of 365.25 days per year (the term Julian year, defined to be 365.25 × 86400 SI seconds, represents this average). The Gregorian calendar, which we generally live by, is a subsequent development that more closely matches the tropical year (aka solar year, the time from an equinox to its subsequent occurrence) by skipping the leap days three times each four hundred years. This gives the Gregorian calendar an average year of 365.2425 days (a Gregorian year), which better matches the tropical-year length which varies slightly, averaging near 365.2422. Both calendars were adjustments to the calendars in use at the time, and in both cases, at the time of adoption, the calendar date was recalibrated to restore the approximate dates of the solstices and equinoxes by adding or skipping some days. (For example, with the initial adoption of the Gregorian calendar, the day following October 4, 1582 was declared to be October 15, 1582.)
The term Julian date is generally used for an indicator of a date by a single number, effectively counting off days, but various fields (science, technology, etc.) have their own notions regarding the date on which to begin the count.