The Handy Math Answer Book

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the heliacal rising (when a star is finally seen after being blocked by the Sun’s light) of
the star Sirius, the brightest star in the sky located in the constellation of Canis Major.
This occurred (and still occurs) in July, with the Nile flooding shortly after that, which
made it a perfect starting point for the Egyptian calendar. The Egyptians divided the
calendar into 365 days, but it was not the only calendar they used. There was also one
used for planting and growing crops that was dependent on the lunar month.

What is a lunar-based calendar?
A lunar calendar is a calendar based on the orbit of our Moon. The new moon (when
you can’t see the Moon because it is aligned with the Sun) is usually the starting point
of a lunar calendar. From there, the various phases seen from Earth include crescent,
first quarter, and gibbous (these phases after a new moon are also labeled waxing, such
as waxing crescent). When the entire face is seen, it is called a full moon; from there,
the phases are seen “in reverse,” and are labeled waning, such as waning crescent.
Overall, the entire moon cycle takes about 29.530589 days. This cycle was used by
many early cultures as a natural calendar.

What was the problemwith the lunar-based calendars?
Nothing is perfect, especially a lunar month. The biggest drawback with using a lunar
calendar is the fractional number of days, which makes a lunar calendar quickly go
out of synch with the actual phases of the Moon. The first month would be off by
about a half a day; the next month, a day; the next month, a day and a half; and so on.
One way to help solve the problem was to alternate 30 and 29 day months, but this,
too, eventually made the calendars go out of synch.

To compensate, certain cultures added (intercalations) or subtracted (extracala-
tions) days from their calendar. For example, for more than a thousand years, the
Muslims’ lunar calendar has had an intercalation of 11 extra days over a period of 30
years, with each year being 12 lunar months. This calendar is only out of synch about
one day every 2,500 years: To see this, mathematically speaking, the average length of
a month over a 30-year period is figured out with the following equation: (29.5 360)
11/360 29.530556 days, in which 11 is the number of intercalated days, 360 is the
number of months in a 30-year cycle (12 months 30 years), and 29.5 is the average
number of days in the calendar month, or (29 30) / 2.

What is a solar-based calendar?
A solar-based calendar is one based on the apparent movement of the Sun across the
sky as we orbit around our star. More than 2,500 years ago, various mathematicians
and astronomers were basing a solar year on the equinoxes (when the Sun’s direct
60 rays are on the equator—or the beginning of fall and spring) and solstices (when the

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