Mersenne (Titanic) PrimeModulusVikeGridTerminal/Command LineCondorSieve of Eratosthenes.
Eratosthenes and China.ppt - University of Maryland ... · Eratosthenes Columbus went even smaller...
Transcript of Eratosthenes and China.ppt - University of Maryland ... · Eratosthenes Columbus went even smaller...
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• For First year book and James Randi must meet for discussion with Dr. Peel. See e‐mail from him for details
James Randi – dinner tonight at North Campus Diner, meet after class today
First year book – Coffee at the Stamp 3 pm tomorrow
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• No SAB next week
Conflicts with National Geographic excursion
If signed up for excursion and can’t make it let us know by 3 PM tomorrow (have people on wait list)
• Still a few slots open on Greenbank trip
• Leonids is last excursion!
• Watch e‐mail for voting on charity
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• If missed deadline for studentinterviews\talks\chats
We still want to see you
Will only get partial credit
Which is better than no credit!
• Potential Service Day make‐up evening of Friday 2 November
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• Next Project Deadline
• Rough draft of paper • Due Thursday 1 November
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• How big is the Universe?
• Is this an “impractical” question? Intellectual curiosity? Our place in the grand scheme of things? Does the answer to this question affect your attitude towards life? Spiritual/religious question?
• Is this a “practical” question? Economic Political
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• Bound the problem Could be infinite in size Could be finite in size
• If it is infinite: Don’t have to worry about a boundaryWe can’t be at the center (boo)
• If it is finite:We could be at the center (yea) Have to worry about the boundary (maybe) Could have no boundary, but then can we be at the center?
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• The smallest the Universe can be is? The size of the EarthMoon, Sun, Planets, Stars just things up in the atmosphere
• Even if don’t buy that need the size of the Earth and the distance to the Sun to get the distance to the closest stars
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EarthEarth
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Earth Sun Earth
• Do you always have a no shadow point?
• Infinite sized Earth• Finite sized Earth
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Flat Earth infinite size
Sun
Flat Earth finite size
Sun
• Can you determine the size of the Earth? If not, can you bound the size?
• Can you find the distance to the Sun?• Does it matter how far away the Sun is?
• Do you need a no‐shadow point?
• Can you find the size of the Sun?
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• Do you always have a no shadow point?
• Can you determine the size of the Earth?
If not, can you bound the size?
Does it matter how far away the Sun is?
Do you need a no‐shadow point?
• Can you find the distance to the Sun?
• Can you find the size of the Sun?
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• Greek, born in 276 BCE in Cyrene, now Shahhat, Libya
• Studied in Athens and Alexandria in Egypt
• Became the director of the Library in Alexandria
• Died in 197 BCE in Alexandria He became blind in his old age Said to have committed suicide by starvation
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• If you take intro astronomy will probablybe taught that he was best known for his accurate calculation of the Earth’s circumference
Within 10–15% of modern day value
Noticed that on the first day of summer, the noon sun was reflected in a well dug at Syene (modern Aswan)
On the same day, and at the same time, the Sun was south of overhead by 1/50th of a full circle (7.2º) at Alexandria
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• Huh?
• Hint: Sun is so far from the Earth thatthe rays from the Sun striking the Earth can be thought of as parallellines
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A
B
Two Parallel Lines – 1
What can you say about angles A and B?
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Two Parallel Lines – 2
A
B
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Two Parallel Lines – 3
Well atSyene
Zenith
SUNA
B
Alexandria
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Ta Da
7.2 º = Distance from A to S360 º Circumference of the Earth
Circumference of the Earth = 360 × AS7.2
Notice do not use !
Think in ratios!
Well atSyene
Zenith
SUNA
B
Alexandria
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Shadow
Gno
mon
C
D
Final Test: The Question
Which angle do you Want?
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Final Test: The Answer
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Is the Earth Flat or Round?
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Is the Earth Flat or Round?
NO! Because the Sun’s rays are parallel, if the world was flat when the Sun was directly overhead at Syene, it would also be directly overhead at Alexandria
Well at Syene
SUN
Alexandria
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Near Sun & Flat Earth:1. Distance to Sun2. Size of Sun3. Size of Sun to minimum
size of the Earth
Far Sun & Round Earth:1. Size of the Earth
No (hidden) assumptions?
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• That the angles were the same? Went to school and studied Euclid Same as you
• That the Sun’s rays could be assumed to be parallel?Went to school and studied Aristarchus of Samos Not same as you Introductory astronomy books skip this part, i.e. you are not supposed to be smart enough to question this 29
• Lived from 310 to 230 BCE Thirty four years old when Eratosthenes born
Generation between Euclid and Archimedes
• Exponent of a Sun–centered universe
• Only surviving work:“On the Sizes and Distances of the Sunand Moon”
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• Lunar eclipses caused by Moon passing through Earth’s Shadow Only occur at full moon
Don’t happen every month, orbital plane of the Moon tilted 5º with respect to Sun’s
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Moon EarthSun
28.5
1Sun
Earth
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Comes from Size of Sun in the sky is 2 degrees
Bounds the Earth–Moon distance! Moon not further than 28.5 Earth diameters away Moon not closer than 1 lunar diameter
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By observations during a lunar eclipse he determined that the size of the Earth’s shadow at the Moon’s orbit was 2 lunar diameters
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1
A
B
CD
E
F
2 Earth
Moon
And then a miracle occurs…Size of the Sun and the Moonin the sky are about the same
For similar triangles, ratios of sides is persevered on all sides:
3 × CE = 28.5
CE 9.5
1
A
B C
D E
F2c
C
E
1d1e
2d2e1c
28.5
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2
1
AECE
EFDE
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Lunar Eclipse – 5
B
CD
E
F
Earth
Moon
28.5
9.5
Note all distances in units of Earth diameters
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Reality
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Size of Sun in the sky is really ½ degree
Know he later changed it from 2 degrees to ½ degree
At Moon’s orbit shadow cone is 2 ½ lunar diameters
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1Sun
Earth
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Earth Sun Distance – 1
Earth
9.5
Sun
Moon
A
• Sound theory
• But it is a hard observation to do In real life angle A is 89.83º Too close to 90º for naked eye observation
• To get an idea of real scale of problem, imagine a scale drawing with 1 million miles = 1 inch Earth–Moon distance = ¼ inch Earth–Sun distance = 93 inches = 7 feet 9 inches 41
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• What he got instead was:
Earth–Sun distance = 19 × Earth–Moon distance
Earth–Sun distance = 180 Earth diameters
• Note all distances in units of:
Earth Diameters (absolute value not known)
• Once Eratosthenes figures this out:
Know the Earth–Moon distance in absolute units
Know the Earth–Sun distance in absolute units
Know the sizes of the Moon and Sun in absolute units 42
• Synergistic
One absolute measurement gives you a lot more
Still true today
Relative measurement between things is easy
Which is hotter
Which is bigger
Etc.
Relative measurement to an absolute standard easy
Determination of absolute standard hard 43
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Does this affect your world view?
Source Moon – Earth Distance
Earth – Sun Distance
Aristarchus 9.5 180
Hipparchus 33 2/3 1,245
Posidonius 26 1/5 6,545
Ptolemy 29 1/2 605
Reality 30 1/5 11,726
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Note all distances in units of Earth diameters
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Source Earth – Sun Distance
(Earth Diameter)
Earth – Sun Distance (Miles)
Minimum Volume of the
Universe
The Earth 1 7,920 2.08 1012
Aristarchus 180 1,430,000 1.22 1019
Hipparchus 1,245 9,860,000 4.02 1021
Posidonius 6,545 51,900,000 5.84 1023
Ptolemy 605 4,790,000 4.61 1020
Reality 11,726 92,900,000 3.36 1024
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Source Earth – Sun Distance
Velocity(mi/hr)
Aristarchus 180 1.87 105
Hipparchus 1,245 1.29 106
Posidonius 6,545 6.79 106
Ptolemy 605 6.27 105
Reality 11,726 1.22 107
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The Gnomon: It’s Multicultural
• How can you explain that?
The same size gnomon
On the same day
Casts different shadow lengths at two different places
• Well… how about The Earth is flat and the Sun is close
It’s a consistent model with the data
Can get the Earth – Sun Distance and size of the Sun 51
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• True for all problem sets
They do count towards your grade (see syllabus)
SHOW ALL WORK! Right answer with no derivation = ½ possible points
Group work OK, even encouraged, BUT
Max 5 people per group
Only one set (with full name, first and last) per person
Must know what each person does
Employ one person as a number checker 52
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ShadowNo Shadow
SlantDistance
VerticalDistance
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So now you know the size of the world, the distance to the moon, etc.
So what?
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• What
• Why
• How
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• The educated class at the time of Columbusknew that the world was round
• The problem really was that if you used Eratosthenes’ determination of thecircumference before Columbus could reachChina he would run out of supplies
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• So the argument about funding Columbus was
An argument about the distance to travelSize of the Earth & the size of Asia
An argument about risk vs. payoff
• Size of the Earth
Ptolemy had the Earth being smaller than Eratosthenes
Columbus went even smaller than that58
• Size of Asia and Marco Polo Know that Columbus had a copy of Marco Polo Don’t know if he had it prior to his first voyage
• Marco Polo It’s a fun read Mix of fact and fiction Written in collaboration Does not have a distance in it for the Europe–Asia land mass, has instead how many days it took Had an overestimate for the distance of China to Japan By the time of Columbus, over a 100 years old
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Fact or Fiction?
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• So Columbus used A very small size of the Earth (wrong) A large size for the Asia land mass (wrong) A large size for the distance from China to Japan (wrong)
Even then some evidence that he fudged those numbers to make the voyage seem possible (wrong)
Four wrong’s make a right?
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• An argument about risk vs. payoff So he dies, what do you lose? But what if he is right, what do you gain? When do you play the lottery? And while not right, it worked!
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