Gregorian Calendar, Leap Years, Solstices and Equinoxes

Every question about dates, days of the week and seasons rests on two sets of facts: how the calendar in use was built, and how the Earth's motion fixes the solstices and the equinoxes. Both are completely stable, and both are tested in SSC, Railway and State papers almost every year, sometimes as general knowledge and sometimes as a calendar reasoning sum. This note sets out the calendar first, then the astronomy, then the arithmetic that joins them.

From the Julian to the Gregorian calendar

Julius Caesar reformed the Roman calendar in 46 BC on the advice of the astronomer Sosigenes, fixing a year of 365 days with one extra day every fourth year. That gave an average year of 365.25 days. The true tropical year, the time the Sun takes to return to the same equinox, is about 365.2422 days, roughly eleven minutes shorter, so the Julian calendar slipped by about one day every 128 years. By the sixteenth century the spring equinox had drifted ten days away from its assigned place, which mattered because the date of Easter was computed from it.

Pope Gregory XIII therefore issued the bull Inter gravissimas in February 1582 and brought in the calendar that bears his name. Ten days were struck out of that October: Thursday, 4 October 1582 was followed immediately by Friday, 15 October 1582, with no break in the sequence of weekdays. The leap year rule was tightened at the same time, and the reform gave a mean year of 365.2425 days, which is within half a minute of the tropical year.

Catholic countries changed over at once; Protestant and Orthodox countries took far longer. Britain and its empire, which then included India, adopted the new calendar only in 1752, by which time eleven days had to be dropped, so 2 September 1752 was followed by 14 September 1752. India today uses the Gregorian calendar for official business, alongside the national Saka calendar adopted on 22 March 1957 on the recommendation of the Calendar Reform Committee chaired by Meghnad Saha. The Saka era is counted from 78 AD, and its first month, Chaitra, begins on 22 March in an ordinary year and on 21 March in a leap year.

The leap year rule

The effect is that a 400 year cycle contains 97 leap years instead of 100, that is 146,097 days in all, and the calendar repeats itself exactly after 400 years. A leap year carries 366 days, the extra day being 29 February, and the quarters of a leap year are not of equal length. Candidates should also know the odd days idea used in calendar sums: 365 divided by 7 leaves one, so an ordinary year has one odd day and a leap year has two. A century has five odd days, two centuries three, three centuries one and four centuries none.

Solstices and equinoxes

The Earth's axis is tilted about 23.5 degrees from the perpendicular to its orbital plane, and it keeps pointing the same way as the Earth goes round the Sun. That tilt, and not the changing distance from the Sun, is what produces the seasons and the four turning points of the year.

The word equinox comes from the Latin for equal night, and on those two days the Sun rises due east and sets due west everywhere on the Earth. The word solstice comes from the Latin for the Sun standing still, because the Sun's noon position stops climbing or falling and turns back. Everything is reversed in the Southern Hemisphere, where 21 June is the shortest day and 22 December the longest. The two solstices and two equinoxes are not the same as the Earth's nearest and farthest points from the Sun: the Earth is at perihelion, its closest, around 3 January and at aphelion, its farthest, around 4 July, which is the opposite of what most candidates expect.

Numbers to keep ready

• A tropical year is about 365 days, 5 hours, 48 minutes and 46 seconds.
• The mean Gregorian year is 365.2425 days and the mean Julian year was 365.25 days.
• The Earth takes about 365.25 days to revolve round the Sun and 23 hours 56 minutes 4 seconds to rotate once with respect to the stars.
• The axial tilt is 23.5 degrees, which also fixes the latitudes of the two tropics and the two polar circles at 66.5 degrees.
• Indian Standard Time is taken from the meridian of 82 degrees 30 minutes east and is five and a half hours ahead of Coordinated Universal Time.
• A leap second is occasionally added to Coordinated Universal Time to keep clocks in step with the Earth's slightly irregular rotation; it has nothing to do with leap years.

How examiners mix the two halves

A general knowledge paper will ask who introduced the Gregorian calendar, how many days were dropped in 1582, which century years are leap years, or on which date the Sun is overhead at the Tropic of Capricorn. A reasoning paper will ask which year repeats a given year's calendar or what day of the week a distant date falls on, and both are solved with the leap year rule and the odd days table. Learning the two halves together means one revision answers questions in two sections of the paper.