1. Centennial Lecture in Mathematics
S. Patricia McKenna, CSJ,
Centennial Lecture in Mathematics
e Past.
CELEBRAS…..TING the Present. ENVISIONING the Future.

2. In Praise of Pure Reason
How the Greeks Deduced the Sizes of the Earth, the Moon and the Sun as well as the Distances Between Them
D.N. Seppala-Holtzman
St. Joseph’s College
faculty.sjcny.edu/~holtzman

3. Step into my Time Machine
Set the dial to 600 BC
Destination Greece and the Eastern Mediterranean Basin
Here we meet Thales (625 – 547 BC), generally regarded as the person to first introduce the notion of mathematical proof.

4. The Maturation of Mathematics
Over the next 8 centuries, the Greeks nurtured and cultivated this powerful form of human reasoning.
It was a gift to the world of extraordinary value.

5. The Greeks and Astronomy
Impressive as this is, it is not the main topic of today’s presentation.
Instead, we turn to the extraordinary contributions that the Greeks made to the field of astronomy.

6. Pondering the Seemingly Unknowable
The Greeks, as did most ancient cultures, pondered the mysteries of the sky.
What is the nature of the Earth on which we stand?
What can be said of the Sun?
What mysteries does our Moon hold?

7. Mathematics Proves a Most Powerful Tool
Quite astoundingly, the Greeks, despite having no telescopes or other modern tools, were able to determine, with amazing accuracy, the sizes of and distances between the Earth, the Sun and the Moon.
Mathematics was, essentially, their only tool.

8. Similar Triangles I
One of the main mathematical tools that was used was the notion and properties of similar triangles.
The definition is quite simple: Two triangles are said to be similar if all of the three angles of one triangle are equal (the technical term is congruent) to the corresponding angles of the other triangle.

9. Similar Triangles II E B C A F D

10. Similar Triangles III
Essentially, two triangles are similar if one is a blown up or shrunk down copy of the other.
They have the same shape and proportions.
Their only difference is size.
Corresponding sides of similar triangles are proportional.

11. The Gnomon I
A moment ago, it was claimed that the Greeks had essentially no tools other than mathematics to aid them in their study of astronomy.
This is not entirely true.
They had a device that served as a crude clock, calendar, compass and sextant.
It was a called a gnomon.

12. Gnomon II
The Gnomon was a stick.

13. Gnomon III

14. The Gnomon Evolved into a Sundial

15. Gnomon IV Place a straight stick, vertically in the ground.
Observe and measure the shadows that it casts at different times.
At sunrise, the stick casts a long shadow in the western direction.
As the day progresses, the shadow becomes shorter and curves around towards the north (we assume, throughout, that we are above the Tropic of Cancer.)

16. Gnomon V
Later in the day, the shadow would begin to lengthen again and point in the north-eastern direction.
At sunset, the shadow would be long and pointing in the eastern direction.
None of this is news. Why are we making a big deal of this?
We are making a big deal because a great deal of information can be deduced from these observations, if you are clever.

17. Gnomon VI
The moment, on any given day, when the shadow is the shortest is noon: the gnomon is a clock.
Repeating this process daily gives the days of the solstices. When the noontime shadow is the longest, we get the winter solstice and when it is the shortest, we get the summer solstice. The days midway between the solstices are the equinoxes. The gnomon is a calendar.
The direction of the shadow at noon on the two solstices is due north. The gnomon is a compass.

18. Gnomon VII
The angle of elevation of the sun with respect to the gnomon on either of the solstices can be used to determine the latitude. The gnomon is a sextant.
In other words, a great deal of information can be deduced from this simple stick if you are sufficiently clever.
The Greeks were clever.

19. The Solstices I Why are we making such a big deal about the solstices?
Consider the following diagram.

20. The Solstices II

21. The Solstices III
At the solstices, the plane that contains the Earth’s axis and is perpendicular to the ecliptic plane (the plane of the Earth’s orbit) also passes through the center of the Sun.
Thus, the shadow that the gnomon casts aligns with a meridian on the solstices.

22. Observations and Assumptions I
Not all that was known or discovered at this time was deduced.
Some things were observed or assumed.
To begin with, the Greeks realized that the Earth was a sphere.
Their evidence included the fact that ships disappeared over the horizon hull first and mast last.
In addition, lunar eclipses showed that the shadow that the Earth casts on the Moon is round.

23. Observations and Assumptions II
The Moon is clearly, observably, a sphere.
Moonlight is light reflected from the Sun, not generated by the Moon. This is obvious from the phases of the Moon.
The Sun is very large and very far away. The solar eclipse tells us this.

24. Enter Aristarchus (310 – 230 BC)
Eratosthenes
Archimedes

25. The Moon I
By a very happy coincidence, the Moon and the Sun have the same apparent size.
During a solar eclipse, the Moon almost precisely covers the Sun.
They both subtend an arc of roughly ½ degree.

26. The Moon II
Simple geometry tells us that, as a consequence of the angle subtended by the Sun being ½ degree, that the shadow of every object cast by the Sun is 108 times the diameter of the object.
This includes the Earth, itself.

28. The Moon III
Armed with this knowledge, Aristarchus deduced the size of the Moon as well as its distance from Earth in terms of the Earth’s diameter.

29. The Moon IV Aristarchus waited for a lunar eclipse.
Then he measured how long it took for the Moon to go from just entering Earth’s shadow to being fully engulfed by it.
Then he measured how long it took from being fully eclipsed to when it began to emerge from the Earth’s shadow.

31. The Moon V
He measured that the Moon spent 2 ½ times longer to cross the shadow of the Earth than to enter it.
Thus, the shadow at the path of transit must be 2 ½ times wider than the moon.
This led him to the following diagram.

33. The Moon VI
This diagram has three similar triangles, all of them isosceles with apex angle ½ of a degree.
This allowed Aristarchus to compute the diameter of the Moon and its distance to the Earth in terms of the Earth’s diameter.

34. The Distance to the Sun I
Using these results, Aristarchus, deduced the distance from the Earth to the Sun as follows.
He began with the observation that the light from the Moon was really reflected sunlight. Thus, when the Moon was in half phase, a right triangle was formed with the centers of the Sun, the Moon and the Earth as vertices.

35. The Distance to the Sun II

36. The Distance to the Sun III
Aristarchus measured the base angle of this right triangle and got a value of 87 degrees.
Simple geometry was now all that was required to deduce the distance from the Earth to the Sun.
Alas, although his idea was brilliant, his measurement was slightly flawed. The true value is degrees.

37. The Size of the Sun I
It is now very simple to deduce the size of the Sun from the previously stated fact that the apparent sizes of the Moon and the Sun are nearly identical.
The only tool needed at this point is that of similar triangles.

38. The Size of the Sun II

39. The Size of the Sun III
From the similar triangles shown in the previous slide, we deduce that the distance from the Earth to the Sun is to the diameter of the Sun as the distance from the Earth to the Moon is to the diameter of the Moon.

40. Filling in the Missing Piece
Aristarchus was able to determine the sizes of the Sun and Moon and their distances to the Earth in terms of the Earth’s diameter.
If only we knew this key piece of information!

41. Enter Eratosthenes (276 – 195 BC)
Aristarchus
Eratosthenes
Archimedes

42. Eratosthenes
Eratosthenes was a brilliant mathematician and Chief Librarian at the famous Alexandria Library.
His famous “sieve” gave a method for finding the prime numbers “hidden” amongst the integers.
He came up with a method to deduce the size of the Earth.

43. The Size of the Earth I
Eratosthenes learned that there was a well in the town of Syene, situated several hundred miles due south of Alexandria, where, at noon on the summer solstice, the Sun illuminated the water at the bottom of the well, i.e. the well was situated on the Tropic of Cancer.
He exploited this information to deduce the size of the Earth.

44. Water Wells

45. The Size of the Earth II
Eratosthenes placed a gnomon vertically in the ground in Alexandria and measured the angle between the Sun’s rays and the stick at noon on the summer solstice.
He measured an angle of 7.2 degrees.
He took as an assumption that the Sun was so large and the distance between Alexandria and Syene so small in comparison, that the rays of the Sun were essentially parallel.

46. The Size of the Earth III
Some very simple geometry led Eratosthenes to the conclusion that the angle between Syene and Alexandria, measured at the center of the Earth, must also be 7.2 degrees.
7.2 must be to 360 as the distance from Alexandria to Syene is to the circumference of the Earth.
As 7.2/360 = 1/50, Eratosthenes deduced that the distance from Alexandria to Syene must be one 50th of the circumference of the Earth.
The value he obtained, 24,500 miles, is correct to within 2.3%.

47. The Size of the Earth IV

48. Size of the Earth V

49. The Size of the Earth VI
Knowing the circumference of the Earth gave the diameter of the Earth as an immediate corollary, since C = π D.

50. Putting it All Together
Plugging this value of the diameter of the Earth into the earlier results of Aristarchus, one can deduce the sizes of the Moon and Sun as well as their distances to Earth.

51. The Weakest Link
Although the ideas were brilliant and the logic was pristine, the weakest links were the measurements.

52. Measuring Time with a Water Clock

53. Measuring Distance with a Camel Caravan

54. How Fast Does a Camel Walk?

55. Brilliant Results Nonetheless
Thus we see that, armed with an analytic mind-set and the tools of deductive reasoning, the ancient Greeks were able to conclude, with astounding accuracy, the sizes of and distances between the Earth, the Sun and the Moon.

56. Conclusion
One can only respond with awe.