When speaking of the phases of the moon, it is common to include the full moon, first quarter, full moon and third quarter. Technically, these phases only occur for an instant, and the four phases of the moon are more accurately referred to as the waxing crescent, waxing gibbous, waning crescent and waning gibbous. Regardless of the terms used, the following mathematical calculation, from Princeton mathematician John Horton Conway's book "Winning Ways For Your Mathematical Plays," allows you to calculate the position of the moon on any date from any location.
Calculate the remainder (R) by dividing the last two digits of a calendar year (Y) by 19. For example, on April 14, 2010, the Moon is "new", according to the Gregorian calendar.
R (Y/19) = R (10/19) = 0 with a Remainder of 10.
Subtract 19 if the remainder (R) is greater than 9.
R-19 = N, or 10-19= -9; therefore N=-9. (Note the negative sign.)
Multiply the result (N) by 11.
N x 11 = n, or -9 x 11 = -99; therefore n=-99.
Apply modulo 30 until you reach a number between -29 and 29. Modulo means whatever is left over (T) after you subtract (or add) the specified modulo; 31 modulo 20 is 11 (31-20), and 31 modulo 30 is 1. Therefore, -99 modulo 30 is -9.
-99 + 30 = -69; -69 + 30 = -39; -39 + 30 = -9. Do not continue once you have reached the specified number range of -29 to 29.
R (n)/30 = T, or -99/30=-9; therefore the remainder is 9.
Add the day (D) and month (M). For our sample date August 14, 2010, we add 4 for April and 14 for the day.
T+D+M = t, or -9+14+4=9; therefore t=9.
Subtract 8 for 21st century years or subtract 4 for earlier years. The year 2010 is the 21st century, you subtract 8 from your last result (t) to obtain your Moon phase number (P).
t-8=P; 9-8=1; therefore P=1.
Determine the Moon phase from the resulting number (P) using the following phases and event calculations.
Setting the calculation for a new Moon (NM) as 0 and a full Moon (FM) as 15, you arrive at the following calculations for the Moon's phases: "waxing crescent," 3.75; "waxing gibbous," 11.25; "waning gibbous," 18.75; "waning crescent," 26.25.
For the other events use "new Moon," 0; "first quarter Moon," 7.5; "full Moon," 15; "third quarter Moon," 22.5.
So, on April 14, 2010, the Phase (P) of the moon is 1, which is the beginning stage of a "waxing crescent moon" or just after a "new moon" event. (Use Reference 2 to confirm your calculations.)
Modulo commonly refers to positive numbers, (31 modulo 32 is 31); however, for the purpose of Moon phase calculations, modulo add to negative integers while modulo subtract from positive integers. Phase terms include crescent, where the moon is less than half illuminated; gibbous is when the moon is more than half illuminated; waxing means "growing" or expanding of the moon's illumination; and waning is decreasing or shrinking illumination. Moon phases are counterclockwise, starting with new moon and waxing (growing) crescent.