Calendar

The calendar is a way to reckon time, specifically the passage of days in relation to months and years. The choice of calendar determines when the anniversaries of events will take place. Hence, it determines when and how long their celebrations—the feasts and fasts—will be, and even whether those take place at all.

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The Church calendar

Main article: Church calendar.

Gregorian reforms

Main article: Gregorian Calendar.

Origins

Over centuries the Julian calendar drifts relative to the heavenly bodies, resulting in the seasons no longer changing on the same dates. This means that the vernal equinox defined in March for calculating the date of Pascha is no longer a physical equinox, and also results in the dates for planting, etc. in agriculture moving over time. The implications are discussed in more detail in the appendix below.

The papacy decided to compensate for the drift by instituting an immediate shift in the calendar, bringing dates back in line with the historical date of the equinox, and adjusting the leap year calculation to more accurately follow the astronomical equinox.

The New Calendar

Main article: Revised Julian Calendar.

Astronomical appendix

The Copernican revolution

The Keplerian model

The mathematician Kepler, working with measurements made by Brahe, developed the first heliocentric model capable of actually predicting the positions of planets—but not the moon—with any appreciable accuracy. He achieved this by replacing the idealized circular motions of Copernicus with ellipses, where the planets speed up when closer to the sun and slow down when further away. The variation in speed is such that the line between the planet and the sun traces out an equal area over an equal length of time.

Classical physics

Perturbative models

The method of perturbations is commonly used to predict the long-term behavior of astronomical systems. This involves first performing a clever coordinate transform. A baseline motion is thus established—probably a Keplerian orbit—and other effects are studied as deviations from this baseline. For example if one wishes to study how the motion of Saturn is influenced by Jupiter, the gravitational pull from Jupiter may be modeled as a perturbation of the Keplerian motion of Saturn. The result produced is roughly correct on average over an orbit, so the trend over many years is made apparent. Although we can say in some sense that the results are averaged out and not exactly correct for a specific date, the effects so studied are small and it would not normally matter.

Perturbation is critical for understanding the motion of the moon because it is subject to unusually large and difficult to calculate perturbations. Hence it impacts the date of Easter. Providentially we do not need to perform these calculations; an accurate understanding of the moon's motion from first principles would have required the ancient fathers to have understood Newtonian physics and have had access to advanced computers. Instead the net result is that complex interactions correspond to relatively simpler perturbations, which result in even simpler precessional motions, which can be predicted centuries in advance without complex tools—as has been done for millennia.

Nevertheless the ancients should be credited for developing methods so advanced as to even predict eclipses. This is often glossed over in modern education and was more difficult than it is made to sound.

Modeling limitations

Chaos theory

Chaos theory is a field of mathematics which studies unpredictability. In chaotic systems there is a compounding of errors over time, which makes it impossible to predict the future even if those systems are behaving totally in accordance with predictable rules.

As an analogy, suppose that you are driving a car in a straight line at exactly 100 km/h on an infinite flat plane. After an hour you are 100 km away, and after 100 hours, 10 000. If there is an uncertainty of 1% in your speed, then after an hour you would be 99 to 101 km away, an error of 1%. After a million hours you would be 99-101 million kilometers away, which is still an error of 1%. If instead you drove on the equator of a spherical earth with a circumference of 40 000 km, an error of 1% would still result in you being after an hour 99 to 101 km away, or 1%. After 400 hours you would be back where you started, except that you could be ahead or behind by 400 km. After 800 h you would have completed two revolutions and could be ahead or behind by 800 km. After 20 000 hours you would have completed 50 revolutions, and now you could be ahead or behind by up to 20 000 km. However, 20 000 km is the distance to the opposite end of the earth. This means that your distance that you could be ahead or behind have met, and you could be anywhere in the system (equator of the earth in this case). This concept is called Lyapunov time, and it limits the precision with which astronomical calculations can be performed into the future because even arbitrarily small measurement errors result in eventually zero knowledge of where objects will end up.

Lyapunov time for the positions of the planets in their orbits (in the Newtonian model) is relatively long—on the order of a billion years. For other motions it can be much shorter. Regardless, systems behave unpredictably when considered over several Lyapunov times. Studying it by Monte Carlo simulation, the most likely scenario for a metastable system such as the solar system would be that, after several Lyapunov times, the system would violently rearrange into a new metastable configuration, typically by way of flinging planets into outer space and rearranging the orbits of those that remain. The Church believes in Providence rather than chance, so such a scenario does not frighten her. It does however do two things: it places an absolute upper limit on how long any calendar can predict the seasons and it undermines Darwinian natural history. (Newtonian physics works mathematically the same when calculated forwards or backwards in time, and would in any case apply if you started with the solar system several billion years ago and calculated forwards until today. We would not expect to survive under materialistic assumptions.)

As an aside, the Lyapunov time for weather prediction is very roughly a week. If you ever wondered why you feel like you cannot get useful predictions, that is why.