1. Method: by Ancient Greeks
Distant of Moon
Method: by Ancient Greeks
Present: So
Leo
Leon
Leong

2. Introduction I : UN-AIDED OBSERVERS
Imagine a time before satellites, planes,telephones, telescopes…
What would you conclude about the world using just your own senses?
Earth is at rest (i.e., motionless)
Earth is flat
Sun, Moon, planets, stars move in the sky (from East to West)
Occasional bizarre things happen (comets, meteors)

3. Historical Astronomy part1: Ancient (Greek) astronomical measurements of the size of the Earth, Moon, distance to Sun
Aristarchus (c. 280BCE)
Use shadow of Earth on Moon
Earth round,
Earth 3x size of Moon (actually 3.7x)
Use 1st qtr Moon angle to estimate Sun is ~20x farther than the Moon (actually 400x)
Eratosthenes (c. 200 BCE)
Used shadow angle of pole to measure radius of Earth (6,900 km), actual 6,400km (8% error!)

4. Cosmology of Eudoxus and Aristotle
Fundamental “principles”:
Earth is motionless
Sun, Moon, planets and stars go around the
Earth: geocentric model
Eudoxus ( B.C.) & Aristotle ( B.C.)
Proposed that all heavenly bodies are
embedded in giant, transparent spheres that
revolve around the Earth.
Eudoxus needed a complex set of 27
interlocking spheres to explain observed
celestial motions
E.G., need to have 24-hr period =day and 365-day
period=year for the Sun

5. Geocentric-model

6. Aristarchus of Samos (310-230 B.C.)
Using eclipse data and geometry:
 Measured relative sizes of Earth, 􀀁Moon
 Curvature of Earth’s shadow on Moon during lunar eclipse ⇒
REarth=4×Rmoon
 Measured distance to Moon
 (duration of eclipse)÷(1 month)=
(2REarth)÷(circumf. of Moon’s orbit)
 Attempted to measure distance to
Sun
 Need to measure (using time interval ratios) the angle of
Sun when Moon is exactly at 1st or 3rd quarter
 Then use trigonometry and known Earth-Moon distance to
get Sun’s distance
 Meaured angle was too small, but still
concluded Sun was very distant
from Earth (20×Moon’s distance) and
larger than the Earth (5×Earth’s diameter)
 Note: true distance & size 20×larger
gthought/

7. Heliocentric model of Aristarchus
Observations implied Sun is much larger than
Earth
Therefore proposed the first heliocentric
model
Sun is the center of the Universe
Everything goes around the Sun
Never accepted by others of his time
 inconsistent with apparent perception of stationary Earth
 No apparent shift in stellar positions could be observed over
course of seasons
 Prevailing culture was uncomfortable with the idea that Earth
was not central to the Cosmos

8. More Assumptions The Moon receives its light from the Sun;
The earth is in center of the sphere that carries the Moon.
At the time of a Half Moon, our eyes are in the plane of the great circle that divides the dark from the bright portion of the Moon.
At the time of a Half Moon, the Moon's angle from the Sun is less than a quadrant (90°) by 1/30 of a quadrant [that is, the angle is 90° - 3° = 87°].
The breadth of the Earth's shadow when the Moon passes through the shadow during a lunar eclipse is two Moons.
Both the Moon and the Sun subtend 1/15 of the sign of the zodiac [that is, with 12 signs of the zodiac spaced around the ecliptic, (360° / 12) x 1/15 = 2°].
The Earth is a sphere.
The Sun is very far away.
The Moon orbits the Earth in such a way that eclipses can occur.

9. General Method
Moon moves (relative to stars) at 360 degrees/month ~ 0.5 degrees/hour.

10. General Method
2. From duration of lunar eclipse, can infer angular size of Earth as seen from Moon.

11. General Method
Compare to angular size of Moon as seen from Earth.

12. General Method Aristarchus concluded:
Earth diameter = 3 x Moon diameter (close to true value).

13. General Method
Combine with Eratosthenes measurement of Earth diameter to get Moon diameter in stadia (or km).
i.e km

14. General Method
6. Knowing angular size of Moon (0.5 degrees) and physical size of Moon, find the distance to Moon by:

16. Q&A

17. Reference

18. The End