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The solar cycle is a period of time under a calendar system during which a leap year day takes place on each of the seven days of the week. This means every possible combination of days and dates happens at least once during a solar cycle. Under the current Gregorian calendar system used in most of the world, the solar cycle is 400 years.
Until the 16th century, the solar cycle was much shorter. Before this time, most countries used the Julian calendar, named after Julius Caesar. This had the simple rule of 365 days a year, with an extra day on February 29th every fourth year, known as a leap year. The way the Julian calendar works out means that if February 29th falls on a Monday in a particular year, it will be 28 years until the next time it does so. This period is the solar cycle.
In 1582, several European countries switched to a system known as the Gregorian calendar. Other countries switched to the system over the next 350 years and it is now the standard system in most of the world. The Gregorian system differs slightly to take account of the fact that a "year" in the Julian calendar is actually slightly longer than the time it really takes the Earth to go round the Sun.
To help correct this, the Gregorian calendar does not count years ending in 00 as a leap year unless the year can be divided by four. This means that 1800, 1900, 2100 and 2200 are not leap years, but 2000 was a leap year. The Gregorian calendar still counts a year as slightly longer than it is in reality, but the difference is just 27 seconds a year compared with 11 minutes difference using the Julian calendar.
Because of these differences, the solar cycle is much longer under the Gregorian calendar, meaning it takes 400 years to use every possible set of dates and days. However, this will never be an issue in the lifetimes of most people alive today. Because there are no years which are exceptions to the usual leap year rules between 1901 and 2099, there are only 28 different sets of days and dates in the entire period. This makes life considerably easier for firms which produce printed calendars!
It is important to realize that the phrase "solar cycle" is sometimes also used to refer to the actual period of time it takes the Earth to go around the Sun. As explained previously, this period is slightly shorter than a calendar year. To avoid confusing the two uses of the term, it may be safer to mention the name of the calendar system when referring to a calendar-related solar cycle.