Mathematics and Astronomy in Ancient Egypt and Greece

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Mathematics and Astronomy in Ancient Egypt and
Greece
Steven EdwardsSouthern Polytechnic State
University
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Writing in Egypt 3200 BC
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Rhind Papyrus 1650 BC
87 problems involving fractions, area, volume
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Except for 2/3, Egyptians used 1/n for
fractions. The Rhind papyrus has a table of
calculations of 2/5, 2/7, 2/9, . 2/99.
How to divide n loaves between 10 men where n 1,
2, 6, 7, 8, or 9.
For 7 loaves, each man receives 2/3 1/30.
Find the value of heap if heap and seventh of
heap is 19.
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Rhind Papyrus Algebra
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Area of a triangle with side 10 and base 4 is 20.
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Thales of Miletus (640 BC) The Ionic
School A circle is bisected by its
diameter. Base angles of an isosceles triangle
are equal. Opposite angles of intersecting lines
are equal. The angle in a semicircle is a right
angle. SAS congruence for triangles.
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Mathematics in Greece was spurred by efforts to
solve three famous problems, using Euclidean
geometry or other methods.
1. Square the circle, i.e. construct a square
whose area is equal to a given circle (or vice
versa). 2. Duplicate the cube, i.e. construct a
cube whose volume is double a given cube. 3.
Trisect a given angle.
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Pythagoras 580 BC
Pythagoras traveled to Egypt. Pythagoras
discovered the construction of the cosmic
figures, according to Proclus.
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Ancient Egyptian Patterns
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Euclid of Alexandrias figure for the proof of
thePythagorean Theorem (300 BC)
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Area of a circle in the Rhind Papyrus
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Euclid X, Proposition 1, The Method of
Exhaustion Two unequal magnitudes being set
out, if from the greater there be subtracted a
magnitude greater than its half, and from that
which is left a magnitude greater than its half,
and if this process be repeated continually,
there will be left some magnitude which will be
less than the lesser magnitude set out. (Due to
Eudoxus)
Euclid XII, Proposition 2 Circles are to one
another as the squares on the diameters.
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Archimedes 287 BC
Compare the area of a circle to the area of a
right triangle with one leg equal to the radius,
the other leg equal to the circumference of the
circle. If they are not equal, then one is
larger. Suppose that the circle has larger area.
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Each triangle has area less than 1/2 the radius
times the base.
The inscribed figure has area less than 1/2 the
radius times the circumference, which in turn is
less than the previously mentioned triangle. But
with enough sides, the inscribed figure has area
arbitrarily close to the circle, so greater than
the triangle.
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Conclusion the area of a circle is 1/2 r
C. This is possibly the first proof that pi shows
up in both as the ratio of circumference to
diameter and in the formula for the area.
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The Law of Sines from Ptolemys Almagest (100
AD) Hipparchus (135 BC) provided much of the
source material for Ptolemy
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From 3000 BC, Egyptians had a 365 day year.
After 1460 years, this year resynchronized with
the seasons
In 239 BC, Ptolemy III tried to introduce the
leap year. Augustus imposed it 200 years later.
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Relative to the sun, any given star rises later
and later each day.
Sirius from the Hubble
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From Luxor,now in Paris
Luxor
Rome
Central Park
Obelisks
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Temple of Ramses II
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The Inner Sanctum
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Illuminated by the sun twice a year
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DecanChart2100 BC
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How is time measured?
Before 1967, one second was defined as
1/31,556,925.9747 of the tropical year 1900.
Now 9,192, 631, 770 periods of radiation
corresponding to the transition between the two
hyperfine levels of the ground state of the
cesium 133 atom
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Thales of Miletus 624 BCBrought the 365 day
year from Egypt
Anaximander of Miletus 610 BCThe earth is the
center of the universe
Pythagoras of Samos 572 BCThe earth is a
sphere.
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Eudoxus 408 BC Motions of each heavenly body
moved by several spheres. One sphere for the
fixed stars, three for the sun, three for the
moon, and four for each planet, for a total of
27 spheres.
Heraclides of Pontus 388 BCThe Earth spins on
his axisVenus and Mercury orbit the sun
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Aristarchus of Samos 310-230 BC On the Sizes and
Distances of the Sun and Moon
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When the moon is half full, the great circle
dividing light from dark is in the same plane as
our eyes
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The distance from Earth to Sun is 19 timesthe
distance from Earth to Moon.
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During a solar eclipse, the moon blots out the
sun.
Earth
Earth
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The ratio of the distance to the moon to the
diameter of the moon is the same as the ratio of
the distance to the sun to the diameter of the
sun.
M/m S/s
S/M s/m
If the sun is 19 times as far away from us as the
moon,then the suns diameter is 19 times the
moons.
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Positions of Sun, Moon, and Earth at total solar
eclipse
S/s M/m
Diameter Distance to
Earth Sun 1,392,000 km 148,000,000 km Moon
3500 km 400,000 km
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Earth
Moon
Using the angle from our eye to the moon as 2
degrees, Aristarchus gets the distance from us
to the moon is between 25 and 33 times the moons
diameter.
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During a lunar eclipse, the diameter of the moon
is half the width of the shadow of the earth at
the moon.
This relationship allowed Aristarchus to estimate
the size of the earth in relation to the
distances to the sun and moon.
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By Aristarchus calculations The suns diameter
is about 7 times as big as the earths. The
earths diameter is about 3 times as big as the
moons. The Suns diameter is about 19 times as
big as the moons
The current estimate is that the suns diameter
is 110 times the earths.
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The accepted value for the angle at the sun is 10
seconds. If Aristarchus had used this angle, then
his calculations would putthe sun 344 times
farther away than the moon, rather than 19
times.The accepted value is 370 times.
This would also give him a much larger sun.
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In the geocentric universe, all objects revolve
around the earth, which does not move.
Aristarchus was the first to hypothesize a
heliocentric universe, with the earth and planets
moving around the sun.
In 1543, Copernicus published his version of the
heliocentric theory.
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Eratosthenes of Cyrene 275 BC Measured the Earth
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Apollonius of Perga 262 BCInvented Epicycles
Hipparchus of Rhodes 190 BCCalculated trig
tablesDiscovered precession of the
equinoxRefined the epicycle theory
Ptolemy 85 ADWrote The Almagest, using epicycles
to describe astronomical motion in a geocentric
universe.The standard theory for 1500 years
More epicycles
Epicycles