In Praise of Pure Reason

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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. Josephs College
• faculty.sjcny.edu/holtzman

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Step into my Time Machine

• Set the dial to 625 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 deductive reasoning and proof as
  opposed to experiment and intuition.

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The Birth of Real Mathematics

• Prior to Thales, substantial use was made of
  calculation in many cultures around the world.
• This is not the same as mathematics.
• Thales gave us deductive systems of theorems and
  rigorous proofs.

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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.

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Greek MathematicsSome Milestones

• Thales geometry and deductive reasoning
• Pythagoras (ca. 540 BC) number theory geometry
• Eudoxus (ca. 400 BC) method of exhaustion
• Plato (ca. 380 BC) mathematics and mental
  training (The Academy)
• Euclid (ca. 300 BC) The Elements
• Archimedes (ca. 225 BC) considered the greatest
  mathematician of antiquity

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Archimedes

• Archimedes deserves a slide of his own.
• He contributed to and enormously advanced all
  branches of mathematics of his day.
• Amongst other things, he, along with Eudoxus (ca.
  400 BC) laid out the foundations of the calculus
  (some 1800 years before Newton) and used it to
  obtain some extraordinary results.

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The Greeks and Astronomy

• The contributions of the individuals listed,
  along with quite a number of others,
  fundamentally changed the nature of pure
  mathematics.
• Impressive as it is, it is not the main topic of
  todays presentation.
• Instead, we turn to the extraordinary
  contributions that the Greeks made to the field
  of astronomy.

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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?

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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.

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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.

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Similar Triangles II

• 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.
• Theorem Corresponding sides of similar triangles
  are proportional.

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Similar Triangles III
E
B
C
A
F
D
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Similar Triangles IV

• In the previous slide, triangle ABC is similar to
  triangle DEF.
• Their sides are proportional.
• Thus, if side DE is k times the length of the
  corresponding side AB, then DF will also be k
  times the length of AC and EF will be k times the
  length of BC.
• All of these ks are the same.

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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 stick.

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Gnomon II

• Place a straight stick, vertically in the ground.
  Surround the stick with sand.
• Observe and record in the sand the shadows that
  it casts at different days and times.
• At sunrise, the stick (called a gnomon) 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.)

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Gnomon III

• 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.

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Gnomon IV

• The moment, on any given day, when the shadow is
  the shortest is noon the gnomon is a clock.
• The direction of the shadow at noon is geographic
  north the gnomon is a compass.
• 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.

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Gnomon V

• Observing the angle of elevation of the sun with
  respect to the gnomon on either of the equinoxes
  gives 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.

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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.

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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.
• Indeed, it is so large, that its rays are
  essentially parallel.

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Eratosthenes

• Eratosthenes (ca. 300 BC) 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.

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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.
• Eratosthenes knew that this never happened in
  Alexandria because it lay above the Tropic of
  Cancer.
• He exploited this information to deduce the size
  of the Earth.

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The Size of the Earth II

• Eratosthenes placed a gnomon vertically in the
  ground in Alexandria and measured the angle
  between the suns rays and the stick at noon on
  the summer solstice.
• 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.
• He measured an angle of 7.2 degrees.

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The Size of the Earth III

• Some very simple trigonometry 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.
• As 7.2 must be to 360 as the distance from
  Alexandria to Syene is to the circumference of
  the Earth, he concluded that this value was
  24,500 miles.
• 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.
• As the commonly accepted value today is 25,060,
  he was right to within 2.3.

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The Size of the Earth IV
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The Size of the Earth V

• Knowing the circumference of the Earth gave the
  radius of the Earth as an immediate corollary.
• Since C 2 p R, we get a value for the radius,
  R, of roughly 4,000 miles.

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The Size of the Moon I

• Now that he knew the size of the Earth,
  Eratosthenes was able to deduce the size of the
  moon.
• Taking the rays of the sun to be roughly
  parallel, the width of the shadow of the Earth
  should be approximately equal to the Earths
  diameter.
• Timing the transit of the moon through this
  shadow during a lunar eclipse gave the result he
  sought.

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The Size of the Moon II
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The Size of the Moon III

• In the previous slide, it is seen that it takes
  about 50 minutes for the moon to go from just
  touching the shadow to being fully engulfed in
  it.
• It takes 200 minutes for the moon to begin to
  emerge from the shadow.
• Conclusion the diameter of the moon is ¼ of the
  diameter of the Earth, or roughly 2,000 miles.
  (The correct value is a tiny bit more 2,088
  miles.)

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The Distance to the Moon I

• Now all Eratosthenes had to do to deduce the
  distance from the Earth to the moon was to
  exploit the properties of similar triangles.
• If one extends an arm full length, one observes
  that the moon has the apparent size as that of a
  fingernail.
• As an arms length is roughly 100 times the
  height of a fingernail, we deduce that the
  distance to the moon is roughly 100 times its
  diameter or 200,000 miles.

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The Distance to the Moon II
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The Distance to the Sun I

• Using the results already established, a
  contemporary of Eratosthenes, 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.

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The Distance to the Sun II
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The Distance to the Sun III

• Aristarchus measured the base angle of this right
  triangle and got a value of 87 degrees.
• Simple trigonometry 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 89.85 degrees giving a distance of roughly
  93,750,000 miles.

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The Size of the Sun I

• It is now very simple to deduce the size of the
  sun from the well-observed fact that the apparent
  sizes of the moon and the sun are nearly
  identical.
• During a solar eclipse, the moon fits almost
  perfectly over the sun.
• The only tool needed at this point is that of
  similar triangles.

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The Size of the Sun II
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The Size of the Sun III

• From the similar triangles formed during a solar
  eclipse (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.
  
• This calculation gives a result of roughly
  86,875,000 miles for the diameter of the sun.

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Conclusion

• 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.
• One can only respond with awe.

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Further Reading

• Big Bang by Simon Singh