use the angular size of the Moon and observations of how far it travels over a specific period (in terms of angles), and from these, determine the approximate orbital period of the Moon around the Earth.
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The angular size of the Moon, as seen from Earth, is roughly half a degree.
This can be measured with instruments like a sextant (since roughly the 16th century) and it was calculated well before that by covering the Moon with standard-sized objects (e.g., coins) then measuring the distance form the coin to the eye.
The average size could be calculated using the diameter of the Moon and the average distance:
3475 km / 384,400 km = 0.00904 radians = 0.518 deg. = 31'
By observation, you can tell that the Moon moves (relative to the fixed stars) by its own apparent diameter in one hour. To find out how many hours it takes for an orbit, you have to divide a full orbit (360 degrees) by the rate per hour (0.518 deg. per hour) to get:
360 / 0.518 = 695.0378 hours = 695 h 02 m 16 s = 28 days 23 h 02 m 16 s
Of course, this is just an approximation (the real average time it takes for the Moon to move by its own diameter changes with where the Moon is in its orbit: when it is closer to Earth, it goes faster AND it has a larger apparent diameter.
The real average sidereal month is 27 d 07 h 43 m 12 s
You could get this accuracy by measuring the average time for "one diameter" over very long periods. You would discover that the average time it takes for the Moon to move by its average apparent diameter - relative to fixed stars - is actually 56 m 28 s, not one hour.
This is how ancient astronomers did it. No instrument needed, but you would have needed lots of observations over very long periods (years) to get this kind of accuracy.
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If you want to measure the synodic month (cycle of Moon phases) you have to do it relative to the Sun. One way is to measure the average time between transits of the Moon (transit = Moon passing over your meridian).
This can be measured with instruments like a sextant (since roughly the 16th century) and it was calculated well before that by covering the Moon with standard-sized objects (e.g., coins) then measuring the distance form the coin to the eye.
The average size could be calculated using the diameter of the Moon and the average distance:
3475 km / 384,400 km = 0.00904 radians = 0.518 deg. = 31'
By observation, you can tell that the Moon moves (relative to the fixed stars) by its own apparent diameter in one hour. To find out how many hours it takes for an orbit, you have to divide a full orbit (360 degrees) by the rate per hour (0.518 deg. per hour) to get:
360 / 0.518 = 695.0378 hours = 695 h 02 m 16 s = 28 days 23 h 02 m 16 s
Of course, this is just an approximation (the real average time it takes for the Moon to move by its own diameter changes with where the Moon is in its orbit: when it is closer to Earth, it goes faster AND it has a larger apparent diameter.
The real average sidereal month is 27 d 07 h 43 m 12 s
You could get this accuracy by measuring the average time for "one diameter" over very long periods. You would discover that the average time it takes for the Moon to move by its average apparent diameter - relative to fixed stars - is actually 56 m 28 s, not one hour.
This is how ancient astronomers did it. No instrument needed, but you would have needed lots of observations over very long periods (years) to get this kind of accuracy.
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If you want to measure the synodic month (cycle of Moon phases) you have to do it relative to the Sun. One way is to measure the average time between transits of the Moon (transit = Moon passing over your meridian).
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