The year is not a whole number of days. Nearly everything strange about the calendar is the long struggle to live with the fraction left over.
A day is one turn of the Earth relative to the Sun. A year is one lap around it. There is no reason these two should fit together, and they don't: a year is about 365.2422 days long. Not 365. Not 365¼. An ungainly, irrational-looking number that no calendar can honour exactly — only approximate, and the history of the calendar is the history of those approximations getting better.
This page rebuilds that arithmetic. Every date below is converted in your browser by the same routine the offline verifier uses; nothing is pre-baked. By the end you'll have watched the spring equinox drift loose and be hauled back, seen ten days deleted from a single October without breaking the week, and met two exact and slightly eerie consequences of the rule we use today.
If you decree a year of exactly 365 days, you lose a quarter-day each year. Within a human lifetime the seasons noticeably slide; within fifteen centuries, midsummer falls in what the calendar calls spring. Julius Caesar's fix (46 BC) was the obvious one: add a day every fourth year. That makes the average year
1461 ÷ 4 = 365.25 days
— four years are 1461 days exactly, three of 365 and one of 366. Beautifully simple, and too long by 0.0078 of a day every year. That sounds like nothing. It is one full day every 128 years, and the Church had noticed: by the 1500s the spring equinox — pinned to 21 March by the Council of Nicaea in AD 325, because Easter is reckoned from it — had slipped to around 11 March. Ten days lost.
Pope Gregory XIII's reform of 1582 kept Caesar's leap day but stole three of them back every four centuries: century years are leap only if divisible by 400. 2000 was a leap year; 1700, 1800 and 1900 were not. That drops the average to
146097 ÷ 400 = 365.2425 days
— 97 leap days per 400 years instead of 100. Now the error is one day in roughly 3,200 years. Watch both rules race the real Sun:
To undo the accumulated slip in one stroke, the reform simply deleted ten dates. By the bull Inter gravissimas, the day after Thursday 4 October 1582 was Friday 15 October 1582. October that year had no 5th through 14th in the countries that obeyed at once (Spain, Portugal, Italy, Poland).
The detail people miss: the week was not touched. Thursday was still followed by Friday. The dates jumped; the cycle of weekdays marched on unbroken — a deliberate, careful choice. Press the button and watch the page enact it.
Because the rule is pure arithmetic, the weekday of any date is computable — past, future, in either calendar. The engine below converts a date to its Julian Day Number (a running count of days used by astronomers since the 1500s) and reads the weekday straight off it. Try the day you were born, or a date centuries out.
Here is the strangest consequence, and it falls straight out of that 400-year periodicity. Since the calendar repeats exactly every 400 years, you can ask an exact question: across one full cycle — 4800 months — how often does the 13th of the month land on each weekday?
You might guess evenly. It can't be: 4800 ÷ 7 isn't a whole number, so some weekday must get more 13ths than the others. The surprise is which one. The chart is counted live in your browser over the real 400-year cycle:
One last thread, because the reform was never only about the Sun. Easter is fixed to the first Sunday after the first full Moon on or after the equinox — so the Church also needed the Moon to keep its place in the calendar. The ancient tool is the Metonic cycle: 19 years and 235 lunar months come out almost equal.
19 tropical years ≈ 6939.60 days
235 synodic months ≈ 6939.69 days
They differ by only about two hours per 19-year cycle — close enough that the same dates carry the same Moon-phases for a generation, not close enough forever, which is why the Gregorian reform also shipped corrected lunar tables (the epact) alongside the leap rule. The calendar you live by is two approximations, Sun and Moon, bolted together.
The fractions that drive all of this — the tropical year of 365.2422 days, the synodic month of 29.5306 days — are measured inputs, taken here from standard astronomical references, not things this page derives. Both drift slowly over thousands of years, so "one day in 3,200 years" is a true order of magnitude, not a fixed appointment.
And which "year"? The mean tropical year is 365.2422 d, but the reform was really chasing the vernal-equinox year (≈ 365.2424 d), to hold Easter to its Nicene date — and against that target the Gregorian 365.2425 is better still, off by a part in ten thousand. The choice of constant changes the headline number, so the page names it rather than hiding behind one figure.
What is exact, and machine-checked, are the pieces of pure arithmetic: the 1461- and 146097-day counts, the dropped days and their unbroken weeks, the 400-year period, and the Friday tally. Those are below.
Fliegel, H. F. & Van Flandern, T. C. (1968). A machine algorithm for processing calendar dates. Communications of the ACM 11(10):657 — the date↔Julian-Day-Number conversion used here.
Inter gravissimas (papal bull, 24 February 1582) — the Gregorian reform; Thursday 4 Oct → Friday 15 Oct 1582.
Calendar (New Style) Act 1750 (24 Geo. 2 c. 23) — Britain & colonies, Wednesday 2 Sep → Thursday 14 Sep 1752.
Meeus, J. (1998). Astronomical Algorithms, 2nd ed., and the US Naval Observatory — mean tropical year ≈ 365.24219 d, synodic month ≈ 29.530589 d.
Brown, B. H. (1933). Note, American Mathematical Monthly 40:607 — the 13th is most often a Friday.
Poole, R. (1995). "'Give us our eleven days!': calendar reform in eighteenth-century England." Past & Present 149:95–139 — the riots as myth.