Mathematics and Astronomy in Ancient Egypt and Greece Steven Edwards Southern Polytechnic State University

Writing in Egypt 3200 BC

Rhind Papyrus 1650 BC 87 problems involving fractions, area, volume

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.

Rhind Papyrus Algebra

Area of a triangle with side 10 and base 4 is 20.

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.

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.

Pythagoras 580 BC Pythagoras traveled to Egypt. Pythagoras discovered the construction of the cosmic figures, according to Proclus.

Ancient Egyptian Patterns

Euclid of Alexandria’s figure for the proof of the Pythagorean Theorem (300 BC)

Area of a circle in the Rhind Papyrus

Euclid XII, Proposition 2: Circles are to one another as the squares on the diameters. 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)

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.

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.

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.

The Law of Sines from Ptolemy’s Almagest (100 AD) Hipparchus (135 BC) provided much of the source material for Ptolemy

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.

Relative to the sun, any given star rises later and later each day. Sirius from the Hubble

From Luxor, now in Paris LuxorRome Central Park Obelisks

Temple of Ramses II

The Inner Sanctum

Illuminated by the sun twice a year

Decan Chart 2100 BC

How is time measured? Before 1967, one second was defined as 1/31,556, of the tropical year 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

Thales of Miletus 624 BC Brought the 365 day year from Egypt Anaximander of Miletus 610 BC The earth is the center of the universe Pythagoras of Samos 572 BC The earth is a sphere.

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 BC The Earth spins on his axis Venus and Mercury orbit the sun

Aristarchus of Samos BC On the Sizes and Distances of the Sun and Moon

When the moon is half full, the great circle dividing light from dark is in the same plane as our eyes

The distance from Earth to Sun is 19 times the distance from Earth to Moon.

During a solar eclipse, the moon blots out the sun. Earth

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 If the sun is 19 times as far away from us as the moon, then the sun’s diameter is 19 times the moon’s. S/M = s/m

Positions of Sun, Moon, and Earth at total solar eclipse S/s = M/m Diameter Distance to Earth Sun1,392,000 km 148,000,000 km Moon 3500 km 400,000 km

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 moon’s diameter. Earth Moon

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.

By Aristarchus’ calculations The sun’s diameter is about 7 times as big as the earth’s. The earth’s diameter is about 3 times as big as the moon’s. The Sun’s diameter is about 19 times as big as the moon’s The current estimate is that the sun’s diameter is 110 times the earth’s.

The accepted value for the angle at the sun is 10 seconds. If Aristarchus had used this angle, then his calculations would put the 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.

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.

Eratosthenes of Cyrene 275 BC Measured the Earth

Epicycles More epicycles Apollonius of Perga 262 BC Invented Epicycles Hipparchus of Rhodes 190 BC Calculated trig tables Discovered precession of the equinox Refined the epicycle theory Ptolemy 85 AD Wrote The Almagest, using epicycles to describe astronomical motion in a geocentric universe. The standard theory for 1500 years