ASTRONOMY

Introduction.
Astronomy (from Greek astronomos, star-law, "law of the stars") is the oldest of the physical sciences, originating among the Ancient Babylonians, Egyptians, and Greeks. Over four millennia ago, the Babylonians and the Egyptians began to accumulate observations of stellar and planetary positions. The astronomy of the Babylonians was religiously motivated and, like other peoples of the ancient world, they worshipped the sun, moon, planets, and the stars as gods. The job of the priest of this astral religion in Babylon, called Chaldeans by the Greeks, was to know the future positions of the sun, moon, planets, and the stars so that could know the future and the will of the gods. Remnants of this astronomy-religion of the Babylonians exists today as astrology. But the astronomy of Babylonians was also practically motivated. As early as 3000 B.C., they had developed solar calendars using the sun and moon to keep track of time and to mark the seasons. Marking the seasons was particularly important to these agrarian cultures for which planting and harvest times had to be determined. At the beginning the measurements were crude. But over the centuries their measurements became more accurate. By 800 B.C., they were able to find the basic cycles of celestial motion and predict planetary positions. The ancient Greeks used these observations, along with their own, to formulate explanations (theories) of the movements of the heavens. Even though little of these Babylonian observations have come directly to us, their system of angular measurement has. Our present system of angle measurement is based on their system, although the names of units have changed. The Babylonians developed a system of numbers, now called the sexagesimal system, based upon the number 60, that counted by sixes. They divided a circle into 360 units which we now call the degree, each degree into sixty units which we call minutes of arc (to distinguish it from minutes of time), and each minute of arc into sixty seconds of arc (to distinguish it from seconds of time). To avoid confusing minutes of arc with minutes of time and seconds of arc with seconds of time, the following notation will be used:
1 degree = 1º = 1/360th of a full circle,
1 minute of arc = 1' = 1/60th of a degree,
1 second of arc = 1" = 1/60th of a minute of arc,
so that,
360 degrees = 6 * 60 degrees = 1 full circle,
60' = 1 degree,
60" = 1'.
When it is said the two stars are 4 degrees apart, this means that, if imaginary lines are drawn from each star to our eye, the angle between the two lines at our eye is 4 degrees. Equivalently, if an imaginary circle is drawn in the sky through the two stars with our eye as the center of imaginary circle, the arc distance between the two stars is 4/360 of the circumference of this circle, or 4 degrees. How big is the degree? The observed angular diameter of the sun and moon is about 1/2 degree. To get some idea of the size of this degree measurement, go out at night, extend your arm out toward the sky, holding your thumb up, and observe the thumb against the background of the stars, as you first close the left eye and then the right eye. The position of your thumb will appear to shift against the star background by about 5 degrees. (This apparent shift of position of an object against the background stars when viewed from two different positions is called "parallax") The angle between the two outer stars in the cup of the Big Dipper (the "pointer stars", which point to the North Pole Star) is almost exactly 5 degrees. What is the smallest angle that the human eye can distinguished? The smallest angle that the average human eye can resolve is 10 minutes of arc. The second star from the end of the handle of the Big Dipper is a double star, whose two stars are separated by 11 minutes of arc. In ancient times the Roman army used this star to test the eyesight of new recruits, asking them if they could distinguish the two stars of this star. Because of this limitation of the human eye, before the invention and use of the telescope in astronomy, the tables of astronomical data were not accurate to better than 10 minutes of arc.
Descriptive Astronomy.
1. The constellations.
  What can be observed of the heavens without optical equipment at night far away from the lights of the city? At first there appears to be no particular order in the uncountable number of stars. But if the stars are studied for a while, they begin to fall into pattern of groups of stars. These groups of stars, with ill-defined boundaries, are called constellations. Ancient observers imagined that they saw mythological or realistic figures in these groups. The oldest known of these constellation figures originated about 3000 B.C. in the Tigris-Euphrates valley of Mesopotamia. The constellations names used today come from the Greeks and Romans. In the northern hemisphere one of the most familiar constellation is the Big Dipper which was called by the Romans Ursa Major or the Big Bear. The shape of these constellations do not seem to change from night to night. And they appear not to change over long periods of time. The stars appear to have fixed positions relative to each other and, hence, are known as the fixed-stars, compared to others who appear to wander, the planets.
2. The Celestial Sphere.
  Even though the stars in their constellations do not appear to change with respect each other, or the constellations with respect to each other, the whole of the heavens seems to move during the night-time from the east to the west. In fact the heavens appear to rotate about one particular star, the North Star or Polaris. The stars appear to be attached to the inside of a sphere centered on the earth, which appears to rotate about an axis through the North Star. This imaginary sphere is called the celestial sphere. This concept was invented by the Greeks, who believed that the celestial sphere was a real sphere with the stars fixed in it. The North Star moves hardly at all. It is always at the same angle above the horizon night after night, throughout the year, at the same latitude on the earth. At another latitude on the earth the angle of the North Star above the horizon is different; in fact, it will be equal to the latitude of the place. In the southern hemisphere the North Star is not visible; at the equator it will be on the horizon. The heavens still rotate from east to west from dusk to dawn about a point in the southern sky at an angle above the southern horizon equal to the southern latitude of the place of observation. This point of rotation is called the south celestial pole and corresponds to the north celestial pole as the other end of axis of rotation. There is no star at this point, no South Pole Star; unlike the north celestial pole which has the North Star or Polaris very near its location. Back in the northern hemisphere, some of the stars and their constellations never drop or set below the horizon but trace out complete circles about the Pole Star; these stars are called the circumpolar stars. It takes 23 hours and 56 minutes for these stars to completely circle the North Star; but they are visible only part of that time, from sunset to sunrise. If a plane is passed through the center of a sphere, it cuts the surface of the sphere in what is called a great circle. The great circle whose plane passes through the center of celestial sphere perpendicular to the axis of rotation is called the celestial equator of the celestial sphere. It would be directly above the equator of the earth, which is half way between the north and south poles of the earth. The angular distance of a star above the celestial equator is called the declination of the star and corresponds to the latitude of a place on earth directly below the star. For example, the star with declination of 90 degrees would be the North Star. The point on the celestial sphere directly overhead on a line through the center of celestial sphere is called the zenith and the point on the celestial sphere directly below one's feet on the line through the center of celestial sphere is called the nadir; one's nadir is thus 180 degrees from one's zenith, on the opposite side of the celestial sphere. The angular distance of a star from one's zenith is called the zenith angle and is the complement of the declination of the star; that is, the sum of the zenith angle and declination angle of a star is 90 degrees and therefore the zenith angle is equal to 90 degrees minus the declination angle.
3. The Sun.
  The sun, in addition to its diurnal (daily) motion, moves roughly 1 degree each day eastward on the celestial sphere. Observations made just after sunset or just before sunrise show that the sun gradually slips toward the east with respect to the stars day by day about one degree per day. Over a year the sun describes a second great circle among the stars. This yearly path of the sun among the stars is called the circle of ecliptic. The ecliptic is a great circle on the celestial sphere that crosses the celestial equator twice, at an angle of 23 1/2 degrees, at points called the equinoxes. The position of the sun on this path with respect to the celestial equator marks the annual seasons. As spring advances on summer, the sun moves to the northerly constellations of the Pleides and Orion where on about June 24 it is 23 1/2 degrees north of the celestial equator at a point called the summer solstice. This is the longest day of the year in the northern hemisphere and the shortest day of year in the southern hemisphere. As the sun moves on along the ecliptic, it approaches the point where the ecliptic crosses the celestial equator from north to south on about Sept. 23, called the autumnal equinox. At this point the day and night are of equal lengths. As fall advances on winter, the sun moves to the southerly constellations where on about Dec. 22 it is 23 1/2 degrees south of the celestial equator at a point called the winter solstice. This is the shortest day of the year in the northern hemisphere and the longest day of year in the southern hemisphere. As the sun moves on along the ecliptic, it approaches the point where the ecliptic crosses the celestial equator from south to north on about March 21, called the vernal equinox. At this point the day and night are again of equal lengths. The sun does not always move at the same rate along the ecliptic; it moves along the ecliptic a little more quickly in the winter than in the summer. Winter is, consequently, three days shorter than summer. And this is why winter has one short month, February. Along the ecliptic there are twelve constellations forming a band of about 8 degrees on either side of the ecliptic called the zodiac (from the Greek, zoidiakos, "circle of living figures"). These zodiacal constellations are: Aries, "the Ram"; Taurus, "the Bull"; Gemini, "the Twins"; Cancer, "the Crab"; Leo, "the Lion"; Virgo, "the Virgin"; Libra, "the Scales"; Scorpio, "the Scorpion"; Sagittarius, "the Archer"; Capricorn, "the Goat"; Aquarius, "the Water-Bearer"; and Pisces, "the Fish". The sun is "in" each of these about 1/12 of each year, approximately one month each. Outside the zodiac all motions are simple and regular where there are only stars. The sun, moon, and the planets move along the zodiac.
4. The Moon.
  In addition to its daily motion, the moon's position from evening to evening slips eastward with respect to any particular "fixed" star at an angular speed of 1/2 degree per hour. Thus the moon moves gradually west to east during the month across the entire zodiac in about 27.3 days. This time interval between repetitions of any position of the moon with respect to the stars is called the sidereal month, and has an average length of 27.3 days. But this is not the time interval between consecutive phases of moon, between new moons or between full moons. The time interval between equivalent phases of the moon is called the solar month and is 29.5 days long. Thus there are two kinds of months. During the solar month the moon's surface is illuminated by different amounts called phases that follow a definite sequence. When the moon rises at sunset, its face is completely illuminated; this phase is called a full moon. About 14.5 days later, the moon face is completely dark and the moon is not visible; this phase is called the new moon. A few days later the moon reappears at sunset partially illuminated and is called a crescent moon. If the moon is high in the sky at either sunset or sunrise, the moon is said to be in a quarter phase; first quarter if the moon is waxing toward full moon and third quarter if it is waning toward a new moon. The moon moves on the celestial sphere on a great circle that crosses the ecliptic at an angle of about 6 degrees. Thus the moon and the sun do not trace out the exactly the same path through the constellations of the zodiac. Occasionally, where their paths intersect, if the moon crosses in front of the sun, the sun will be eclipsed (a solar eclipse) but, if the sun crosses in front of the moon, the moon will be eclipsed (a lunar eclipse). Since these eclipses occur only where the moon's path intersect the sun's path, the name of sun's path is called the ecliptic.
5. The Planets.
  There are five heavenly bodies visible to the naked eye as bright stars, whose motion is highly irregular. They did not twinkle like the other stars and their brightness varied over long period. Like the sun and moon, they have diurnal (daily) movement across the sky, and from evening to evening they slipped eastward with respect to the fixed stars, eventually completing whole cycles of heavens. Their movement is also strictly confined to that region of the sky traversed by the sun; that is, their positions are never far from the circle of ecliptic in the band of constellations called the zodiac. But, unlike the sun and moon, over a period of time they sometimes slowed down in their eastward movement, come to a full stop and reverse their direction of motion toward the west; and after period of time they again slow down in this westward movement, come again to a full stop and reverse their direction of motion, resuming their "normal" movement toward the east. This peculiar movement of these heavenly bodies is called retrograde motion, in contrast to direct motion of the sun and moon. Because of this peculiar movement, the Greeks called these heavenly bodies "wanderers" or planets, which is the Greek word which means "wanderer". The Romans, like other ancient peoples, named them after their principal gods: Mercury, Venus, Mars, Jupiter, and Saturn, which is our names for these planets. The explanation of the wandering of these planets became the central problem of astronomy.
  These planets can be divided into two groups: one group consisting of Mercury and Venus are never very far from the sun, and a second group consisting of the rest of the planets, Mars, Jupiter, and Saturn, can be found any distance along the zodiac from the sun. The first group of planets, Mercury and Venus, are always close to the sun, rising above the horizon shortly before sunrise, and setting below the horizon after sunset. Venus, for example, moves with sun but at various distances from the sun. It moves back and forth across the sun, sometimes ahead, sometimes behind the sun. But it never gets more than 46 degrees from the sun. When Venus is 45 degrees west of the sun, this angular distance is called its greatest western elongation. In this configuration west of the sun Venus is called the "morning star", rising in the morning about three hours before the sun. Venus, like the sun, moves eastward among the stars of zodiac but slightly faster than the sun, so that it approaches closer to the sun, rising each day nearer and nearer to sunrise. Eventually Venus catches up to the sun and gets lost the sun's glare. Later Venus reappears east of sun as the "evening star". Ancient astronomers thought that the morning star, which they called "Lucifer", and the evening star, which they called "Hesperus", were two different planets. It was a great step forward when the ancient astronomers understood that they one and same planet. As the evening star, Venus continues to move eastward faster than the sun until it reaches its greatest eastern elongation; there its angular distance from the sun is 46 degrees east of the sun and sets about three hours after the sun. Venus then slows its eastward movement, so the sun catches up to the planet. Soon Venus stops its eastward movement and starts to move westward among the stars. This reversal of direction is the start of the retrograde movement of Venus. While Venus is moving westward, the sun is still moving eastward along the ecliptic so that they approach each other very rapidly. In a few weeks Venus again becomes the morning star again, stops its western, retrograde movement, and starts moving eastward again. Then the cycle starts over again. Note that Venus is in the middle of its retrograde motion as it passes the sun. The period of the complete cycle is about 584 days and is called the synodic period of Venus. During this synodic period Venus passes the sun twice; these passes are called the conjunction of Venus with the sun, meaning "together with" the sun. The first conjunction after the greatest western elongation is called superior conjunction and the second conjunction after the greatest eastern elongation is called the inferior conjunction. The period between consecutive superior conjunctions or inferior conjunctions is equal to the synodic periods and is about 584 days for Venus. Mercury goes through a similar cycle, except that the western and eastern elongation is not as great as Venus' being about 23 degrees. Thus the synodic period of Mercury is also not as long as Venus' being only about 116 days.
  The second group of planets, Mars, Jupiter, and Saturn, unlike Mercury and Venus, do not follow the sun around the sky. Since they do not stay close to the sun, they may be visible all through the night, rather than for a few hours before sunrise or after sunset. Even though their motion is not uniform, their average time for each to travel through all twelve constellations of the zodiac can be determined: Mars takes 1.88 years, Jupiter takes 11.86 years, and Saturn takes 29.46 years. Like Mercury and Venus, Mars, Jupiter, and Saturn also stop their eastward motion from time to time, retrogress westward, then resume their eastward motion through the zodiac. They retrogress, or move westward, only a small part of time so the net result is an overall eastward motion. Mars, for example, retrogress only about two months every two years. Unlike Mercury and Venus that retrogress near the sun, Mars, Jupiter and Saturn are in the middle of retrogression when they are farthest from the sun, 180 degrees around the zodiac from the sun. This position of the planet is called opposition and at this place in the zodiac they are highest in the sky at midnight. The synodic period of these planets is the interval of time between successive oppositions, or, to put it in other words, between two successive times when the planet is directly opposite the sun on the celestial sphere. The synodic periods for Mars is 779.88 days, for Jupiter is 398.8 days, and for Saturn is 378.1 days. At opposition, when they are retrogressing and are farthest from the sun, all three of them are the brightest.
Theoretical Astronomy.
Theoretical Astronomy, sometimes called cosmology (from Greek, cosmos-logos, "order of the world"), attempts to explain the observations of the movement of the stars, the sun, the moon and the planets, made by Descriptive Astronomy. Before presenting the modern Copernican theory of the heavens and their movements, the history of Greek astronomy will be presented and in particular the Ptolemaic theory, which dominated the medieval period until 17th century.
1. Greek Theoretical Astronomy.
  Because of their religious view of astronomical events, that the sun, moon, and the planets are gods, the Babylonian astronomer-priest did not attempt to find an explanation for the observed movements of the sun, moon, and the planets. The Greeks developed, however, not only explanations of the movements of the heavenly bodies, but a new and entirely different form of theoretical astronomy, one based on geometrical models that went beyond just finding cycles of recurring events. In spite of the beginnings by the Pythagoreans in developing a mathematics of numbers, they gave a geometrical interpretation to numbers which was in line with the Greek understanding of mathematics as geometry. In fact the Greek word which is translated "mathematics" is geometrein, which literally means "to measure the earth or land". Our word "mathematics" is from the Greek mathematika, which literally means "learning". This Greek understanding of mathematics as geometry originated with the first Greek philosopher, Thales of Miletus. He was born about 650 B.C. in the Ionian city of Miletus, possibly of Phoenician parentage. He traveled and studied in Babylonia and Egypt. While in Egypt he learned Egyptian surveying methods. Upon returning to Miletus he founded Greek geometry by demonstrating and proving several theorems that latter were incorporated in Euclid's geometry. Although he, like the Babylonian, thought that the earth was flat, he used their astronomical method to predict successfully a solar eclipse for May 28, 585 B.C. This established his fame throughout Ionia. He was accused of being, as a philosopher, "impractical", because he lived in near-poverty. To disprove the accusation, he predicted a great olive harvest in the coming year using Babylonian star tables and, cornering all the olive-presses in Chios and Miletus, he made a lot of money renting them at the olive harvest. As Aristotle later said, he showed the world that a philosophers can easily be rich, if they like, but their ambitions are of another kind.
  1. The Pythagorean School.
    1. Pythagoras (572-497 B.C. or 580-500 B.C.).
      Pythagoras was born in the Ionian city of Samos but because of his dislike of the policies of the Samian tyrant Polycrates he migrated to Crotona, on the "instep" of the Italian peninsula. There at the age of 50 he founded his philosophical school as a religious and ascetic community. It got involved in the politics of the city and certain Cylon stirred up a revolt against them, forcing Pythagoras to flee to Metapontum. The Pythagorean society was deeply influenced by the cult of Orpheus, called Orphism. The followers and students of this school pledged themselves to the master of the school and to each other to keep a series of strict regulations concerning their diet, dress, and deportment. They believed that salvation was obtained by purification of the body and by the knowledge obtained from the study of mathematics. The central doctrine of their philosophy was that all things are numbers and are explained by numbers. They regarded numbers concretely, not abstractly, and represented them spatially, expressing them by dots or by pebbles or by marks in the sand. The number one is a dot. Since a dot occupies space, it is not the point of later Greek geometry, which has no size, but only location. All numbers are generated by adding one to the preceding number. Thus the number one is the source and generator of all the other numbers. The number two is two dots and geometrically a line. The Pythagoreans regarded the line, not as later Greek geometry, as having width and thus occupying space. The number three is three dots and geometrically a triangle (one dot above the other two dots in a line). The number four is four dots and geometrically a square (two of the dots in a line above the other two dots). And so for the other numbers. Thus numbers were viewed geometrically, as lines, triangles, squares, etc. The numbers 3, 6 and 10 are triangular and the numbers 4, 9 and 16 are square. The Pythagoreans regarded the triangular number 10 as particularly significant. It was the sum of the first four number: 1 + 2 + 3 + 4 = 10, and thus combines the dot or point, the line, the triangle and the square. They called this number tetrakus and believed that it was the nature (phusis) from which all things originated or spring; and that it is master-key for understanding all things. By assigning certain qualities to each of these first four numbers, the Pythagoreans thought that the tetrakus explained all things. The basic and first tetrakus is the plane figures: the point, line, triangle and square; the second tetrakus is the four elements: fire, air, water and earth; the third tetrakus is the four solids: the tetrahedron or pyramid, the octahedron, the icosahedron and the cube; the fourth tetrakus is "of things that grow": the seed, growth in length, growth in width and growth in height; the fifth tetrakus is of society: the individual, the family, the village and the state; the sixth tetrakus is the four cognitive faculties: reason, knowledge, opinion and the senses; the seventh tetrakus is the four seasons: spring, summer, autumn and winter; the eighth tetrakus is the four ages of man: infancy, youth, adult and old age. Thus all things could be explained by numbers. Thus the number ten seemed to be the ideal number and embraced the whole of nature. The Pythagoreans were the first to classify numbers as even and odd; all numbers, except the number one, is even, if it can be divided by two, otherwise it is odd; odd numbers cannot be divided by two. Square numbers are generated by adding successive odd numbers. That is, one plus three is four, one plus three plus five is nine, and so on. The oblong or rectangular numbers are generated by adding successive even numbers. That is, two plus four is six, two plus four plus six is twelve, and so on. The Pythagoreans also were the first classify numbers as prime and non-prime; prime numbers are those who factors are one and the number itself (the factors of a number are the set of numbers when multiplied together give the original number). That is, three is prime because three and one are the only numbers when multiplied together give three. On the other hand, the number four is not prime, since the numbers four and one are not the only numbers when multiplied together give four; two times two also gives four (the factors do not have to be different). The numbers one, two, three, five and seven are prime; the numbers four, six and eight are not prime. The Pythagoreans also discovered what they called perfect numbers; perfect numbers are those whose factors, including one, when added together, the result is the number itself. That is, the factors of six are one, two and three, the sum of which is six, the number itself; thus six is a perfect number. The Pythagoreans also discovered the following number theorem that related prime and perfect numbers. Construct a sequence of numbers starting from one by doubling at each step the previous number; that is, one, two, four, eight and so on. Now begin to add the series. Whenever the sum is a prime number (as one plus two is three which is prime), this prime number multiplied by the last number added will be a perfect number (that is, three, the prime number, times two, the last number added, is six, a perfect number). In addition to these contributions to number theory, Pythagoras discovered a very important geometrical theorem about right triangles, which is called the Pythagorean Theorem: the square of the hypotenuse of a right triangle is equal to sum of squares of the other two sides or legs of the right triangle. When the Greeks drew a right triangle they put the long side opposite the right or 90 degree angle down so that the long side was the side under the right triangle, hence the name hypotenuse, which in Greek literally means "the side under". In this position the other two sides of right triangle looked like the two legs of a man spread apart, hence the name of these two sides as the legs of the right triangle. This theorem was considered the crowning discovery of their mathematics, because of the importance of the triangle to their philosophy of numbers. This theorem so delighted the master that he sacrificed a hundred oxen to the gods. That is, until he discovered that their were certain right triangles whose hypotenuses did not come out to be a whole number or a ratio of two whole numbers. For example, if the two legs of are equal, say one unit long, there is no whole number when multiplied times itself would equal the sum of the square of those legs of that right triangle; that is, the square of the hypotenuse is equal to one square plus one square or two. Since there is no whole number multiplied times itself that would equal two, the length of hypotenuse cannot be found. Or, to use Pythagorean geometrical understanding of numbers, no square of dots could be constructed on the hypotenuse of right triangle that will be equal to sum of the two squares of dots constructed on the equal legs of the right triangle. This result was a scandal to the Pythagoreans because their conception of numbers, that numbers are whole numbers or ratios of whole numbers, did not allow for non-ratio or non-rational numbers, or, as they were later called, the irrational numbers. (They did not have zero or negative numbers.) They consider this violation of their philosophy of numbers a scandal (they called these irrational numbers arrhetos, unspeakables) and they made them a secret that was not to be revealed to outsiders; if any member would reveal it, that one would be excommunicated from the society. It is said that Hippasos, a member that let the secret leak out, was put to death.
      The Pythagoreans also discovered number relations in the musical scales (harmonia); they found that ratios of the octave, the fourths, and the fifths, 2:1, 4:3, 3:2, contained the harmonic proportion, 6:4:3. They also discovered the arithmetic and harmonic mean. Pythagoras also believed that the earth, sun, moon and planets were spheres and moved in circular orbits within a spherical universe. Since circles and spheres are "perfect" geometrical figures, they are suited to celestial bodies and their motions. He pictured the heavens as series of concentric spheres in which each of the seven heavenly bodies are carried by a sphere separate from the sphere of the stars, so that the motion of heavenly bodies results from the independent rotation of the different spheres. The friction between the spheres make harmonic sounds, the music of the spheres, which only the gifted can hear. He believed that the radii of spheres are proportional to the successive strings of a stringed instrument.
    2. Philolaus (c.450 B.C.).
      Another member of the Pythagoreans school was Philolaus who lived in the next century. He was a native of Tarentum; lived in Thebes for a number of years and at the end in Boeotia. He was a contemporary of Socrates. He was the first to introduce the concept of the earth in motion. For the first time the rotations of the heavens is explained by a moving earth rather than by a revolving heavens. On the basis that the number 10 is the ideal number, Philolaus asserted that the number of bodies moving through the heavens is ten. But since only seven are visible, the sun, moon, and the five planets, and considering the moving earth as the eight; he postulated existence of two more heavenly bodies: the "central fire" as the ninth and a counter-earth as the tenth, both of which are always invisible from the earth. He postulated that the earth is not the center of the rotation of the heavens, because the earth is too gross to be the center of the heavens, and that the earth, like the other heavenly bodies, moves about a "central fire", which burns at the center of the universe. The earth moves about this central fire in 24 hours, always keeping one face of the earth toward it. This central fire is never seen from the earth, the inhabited side of earth being always turned away from it. He postulated that the counter-earth is on the opposite side of central fire from the earth and equidistant from it. The earth and the counter-earth share the same orbit about the central fire, but on opposites sides from each other, revolving about the orbit once a day, from west to east. Philolaus regarded the celestial sphere as motionless and its apparent rotation as a result of the revolution and rotation of the earth. He proposed that the moon, the sun and the five planets moved about the central fire in their respective spheres outside the common orbit of the two earths. This system is neither heliocentric nor geocentric.
    3. Hicetas and Ecphantus of Syracuse.
      These two Pythagoreans proposed the view that the earth, which is at the center of all things, rotates daily on an axis of its own and the sphere of the stars is fixed and immobile. The sun and moon were required to revolve in circular orbits about the earth, the sun yearly and the moon monthly. According to this view, the stars being perfect do not move at all and the planets, sun and moon, being less perfect move but much less than the earth which is the least perfect of the heavenly bodies; it moves more rapidly.
  2. The Athenian School.
    1. Plato (427-347 B.C.).
      Plato is one of greatest of the Greek philosophers. Although some parts of the Pythagorean astronomical system were considered heretical by some of the Athenian philosophers, Plato adopted and elaborated the Pythagorean concept of perfect circles and spheres. According to Plutarch, Plato also absolved astronomy from heresy by returning to the idea that the earth does not move, that the earth is the center of the universe and that the stars move daily about the earth. He deplored the time wasted in making astronomical observations, considering the formulation of a cosmic view of what is seen in the heavens as more important. His ideal of heavenly motion as circular and uniform (constant speed) was more difficult to apply to the real motion of the heavenly bodies. In particular the irregular motion of the planets would not completely conform to the ideal of circular and uniform motion. Plato considered this problem of the planets to be the central problem of astronomy and he succinctly stated that the goal of astronomy was to "save the appearances", that is, to devise a model that would explain the observed motion of the planets. This has remained the goal of astronomy for many centuries. Another related problem was the variations in the brightness of the planets; not just between different planets but as the planets moved through the heavens.
    2. Eudoxus of Knidus (408-355 B.C.).
      Eudoxus was an outstanding mathematician who proposed a 4 year solar cycle, 3 years of 365 days and one of 366 days. This was later adopted by Julius Caesar for his calendar. Eudoxus was one of the Plato's students who pursued Plato's goal of astronomy and proposed an influential system of cosmology. He followed his teacher's advice and rejected the moving-earth hypothesis of Pythagoreans, and returned to the earth centered conception of the universe. He proposed that the motion of the heavenly bodies be represented by combination of rotating spheres. According to Eudoxus, the stars were carried on the celestial sphere, which was the accepted theory. In addition, he proposed that a separate series of spheres be used for each of the planets, the sun, and the moon. Each sphere in each series would be pivoted at two points (or poles) on opposite sides of the next inner sphere; the outermost sphere would carry the planet (or sun or moon) itself, and the innermost one was attached to the celestial sphere that carried the stars which is centered on the earth. By adjusting the rates of rotation and the inclination of the axis of the spheres, he was able to reproduce approximately the complicated apparent motion of the heavenly bodies. To explain the retrograde motion of the planets, the poles of planetary spheres are attached to larger sphere, concentric with the first, which rotates at different speeds around its poles from that of the first. If this was not sufficient for explanation, he attached the poles of the second sphere to a third larger concentric sphere, which rotates at a different speed. These spheres not carrying a planet were called starless. His system required 27 spheres, 1 for the stars, 3 each for the sun and the moon, 4 each for the five planets. Later, Callippus further refined the system by adding 7 more spheres, for a total of 34. This was a geocentric system but it raised several questions: What was the cause of the rotation of the sphere? What is the material of the spheres and what was its thickness and the diameter of the spheres? Were the spheres just a device for calculation or did they physically exist? The system did not answer these questions.
    3. Aristotle (384-322 B.C.).
      Aristotle was the most famous student of Plato and wrote on almost every subject, including astronomy, summarizing all of the previous knowledge of the Greeks on the subject. He devised a complicated geometric model based on the concept of uniform and circular motion. To explain the motion of the planets, especially the retrograde motion, his model had a total 56 concentric spheres centered on the earth. As complicated as this model was it did not account for the motion of heavenly bodies very well, as even as Aristotle admitted. But it incorporated some physical ideas that made possible to answer the questions raised by Eudoxus' model. Aristotle viewed the cosmos as being divided into two distinct realms: the imperfect realm of change near the earth and the perfect and eternal realm of heavens. The earthly realm was composed of the four elements: earth, water, air and fire. Each element had it own place and had a natural motion to that place. The place of the element earth was at the center of the cosmos and its natural motion was downward. The place of the element water was above the place of the element earth and below the place of the element air; the natural motion of the element water was down to its place above the element earth and the natural motion of the element air was upward to a place above the elements earth and water. The place of the element fire was just below sphere of the moon and its natural motion is up to that place. The natural motion of physical bodies depended upon which element predominated in its mixture of the elements. The heavens were composed of the fifth element, the quintessence, which was immutable and transparent; its natural motion is circular and uniform. The celestial spheres were composed of this fifth element. The most distant sphere of the stars rotated fastest in circular and uniform motion, which the unmoved-mover or God impart to it from his place in a sphere outside the sphere of the stars. The spheres of the planets, the sun and the moon are also in circular and uniform motion, which is imparted to them by contact with the outer sphere of the stars. Thus no forces were needed to move the sun, the moon, and the five planets around the earth. In the terrestrial realm, no force is needed to cause natural motion which is vertical either up or down depending upon which element predominated in the physical object. Force was necessary only for unnatural or "violent" motion, motion not vertically up or down. As soon as the force is removed the physical body stops moving if it is in its natural place. Thus Aristotle attempted to explain the nature of the two realms and the motion natural to each. This explanation dominated physics and astronomy until 17th century and Isaac Newton.
      Aristotle was right about some astronomical ideas. He argued that the sun is farther from the earth than the moon, because
      (1) the moon's crescent phase shows that it passes between the earth and the sun and
      (2) the sun moves more slowly in the heavens than the moon.
      This second argument is not as rigorous as the first. Aristotle also attempted to prove the spherical shape of earth by three arguments:
      (1) the curve shape of earth's shadow cast upon the moon during a lunar eclipse,
      (2) as a ship sails away from port the hull disappears below the horizon before the masts,
      (3) the angular distance of the pole star above the horizon increases as one travels north toward the star and decreases as one travels south away from the star.
      But he was wrong when he attempted to prove that the earth did not move, as some of the Pythagoreans assumed. He argued that if the earth moved around the sun there would be an annual shift in the position of stars, known as stellar parallax. Parallax is defined as the angular, or apparent, shift in the position of an object due to a change in the position of the observer. Stellar parallax is the annual shift of the stars that results from the earth's orbital motion. Since no one had observed this stellar parallax, Aristotle concluded that the earth does not move around the sun or any other heavenly body. This conclusion reinforced his geocentric view of the cosmos. Of course he was wrong. Stellar parallax does occur but it is too small to be seen with the naked eye. It was never observed until the invention of the telescope in the 17th century and it was looked for. After years of searching for it, it was discovered in 1838.
    4. Heraclides of Pontus (388-310 B.C.).
      Heraclides of Pontus was a member of the Old Academy, which consisted of the disciples and associates of Plato himself who more or less held to Plato's philosophy; the "Pythagorean" elements in the thought of Plato received particular attention. Heraclides differed with the Plato's firmly held view that the earth stood still, and taught that the earth rotates on its axis once every 24 hours, while the stars stood still. Heraclides also held that Mercury and Venus, which are never far from the sun, revolve in orbits about the sun, and seems to suggest that the earth also revolves about the sun. Like Plato he recognized the importance of the problem of the retrograde motion of the planets in astronomy.
  3. The Alexandrian School.
    After the time of Alexander the Great (356-323 B.C.), who founded the city of Alexandria in Egypt, the study of astronomy in the Greek world shifted to that city and centered on its great library. Here the Greek astronomers Aristarchus and Eratosthenes worked.
    1. Aristarchus of Samos (c.310-230 B.C.).
      Aristarchus, who was born on the island of Samos off the coast of present-day Turkey, extended the Greek quantitative methods in astronomy. In his only surviving work, "On the Sizes and Distances of the Sun and the Moon," he devised an igneous method to find the relative distances from the earth to the sun and moon. When the Library at Alexandria was burned in 641 by the Moslems, all of his major works were destroyed. What we know of his ideas are from quotes and comments by other Greek authors. From these we can reconstruct his cosmological views. Because he thought that the sun is much bigger than the earth, he proposed that the sun, not the earth, is the center of the heavenly system. He apparently held that the moon orbited about the earth and the earth about the sun; and that the earth was planet like the other five planets. But we do not have any evidence that he worked out the planetary motion in detail. His system was a true heliocentric system. For this an outraged critic, who was a contemporary disciple of Plato, declared that he should be indicted for impiety. His system was attacked for two reasons: it contradicted the view of Plato and Aristotle in stating that the earth moved, and that it required stellar parallax, which was not observed. Aristarchus answered his critics by declaring that the stars were so far away and extremely distance from the earth that their parallax can not be observed. Because of the influence of Aristotle's physics his heliocentric system was rejected and ignored until Copernicus revived and expanded it.
    2. Eratosthenes of Alexandria (284-192 B.C.).
      Eratosthenes was a researcher and librarian of the Great Alexandrian library. He is reported to have made a catalog of the 675 brightest stars and to have measured the 23 1/2 degree inclination of the earth's polar axis to the pole of the ecliptic. Eratosthenes is most famous for determining the size of the earth. Being told that at the summer solstices the sun shone directly down a well near Syene (modern Aswan) in Egypt, he noted that on the same date at Alexandria the sun was off the vertical 1/50th of circle. He realized that this difference had to be due to the curvature of the earth. He concluded that the circumference of the earth was 50 times the distance from Alexandria to the site of the well at Aswan. He measured the distance and multiplying it by 50 he got an estimate of circumference of the earth that was within 20% of the actual size. The diameter of the earth then can be found by dividing the circumference by PI = 3.14159. Unfortunately his amazingly accurate measurement was not commonly accepted but the value of later observer, which was about 1/3 of Eratosthenes' value, was more generally accepted. This mistaken value is responsible for the error of Columbus 1700 years later, who thought he had reached the Orient when he reached the West Indies of the Americas; he had only traveled 1/3 of the way to the Orient.
    3. Hipparchus (c.160-127 B.C.).
      Hipparchus was probably the greatest astronomer of ancient times and is certainly one of greatest astronomical observers of all times. From his observatory on the island of Rhodes, he observed the positions of the astronomical bodies as accurately as is possible without the use of the telescope; he prepared a catalog of some 850 stars. He also divided the stars according to their apparent brightness into six categories or magnitudes, and specified the magnitude of each star. With his exhaustive observations along with the material inherited from Babylonia, he was able to predict with reasonable accuracy the future position of the sun and moon for any date. The most important discovery attributed to Hipparchus is what is called the precession of the equinoxes. From his data he discovered that the position in the sky of the north celestial pole had changed over the previous centuries with respect to the fixed stars. Also the autumnal and vernal equinoxes were also shifting with respect to the stars; that is, the times of occurrence of the annual seasons were slowly shifting with respects to the constellations. He deduced correctly that the circle of the ecliptic was not absolutely fixed; the points where the circle of ecliptic crosses the celestial equator (called the equinoxes) were moving a circle and would complete a cycle in about 26,000 years. This small effect remained unexplained until the 17th century.
    4. Claudius Ptolemy of Alexandria (c.73-151 A.D.).
      Two and half centuries after Hipparchus, Claudius Ptolemy was born at Ptolemais on the Nile, hence his name. Little is known of his life. He did his astronomical work at Alexandria from about A.D. 127 to 151. According to one tradition, he worked for 40 years at Alexandria and lived to the age of 78. His great work, which he titled Mathematike Syntaxis, was given the Arabic-Greek title of Al-megiste, "The Greatest", by the Arabic astronomers who admired the work immensely. The Latin speaking scholars of Medieval Europe translated the title as the Almagest; and this is the title by which it is known today. It is a very large book and it is not entirely his own work. Ptolemy frankly gave credit to his predecessors, particularly to Hipparchus. At the start of the Almagest, Ptolemy pays respect to Aristotle and then he defines the problem: to use geometry to describe and explain the astronomical observations, that is, to save the appearances, as Plato had dictated. Using Aristotle's physics of motion, Ptolemy was able to give his system a physical basis; no forces to explain the circular and uniform heavenly motions. The system of the heaven it presents is a common-sense system. The heavens are assumed to move just as they appear to move. The earth appears to be stationary, so the Ptolemaic system assumes that it is motionless. The stars, the sun, the moon, and the planets with their retrograde motion are assumed to move exactly as they appear to move.
      His explanation of the retrograde motion of the planets was a series of circles upon circles; it was system of wheels rather than spheres. In other words, the three-dimensional spheres of Eudoxus were not needed; Ptolemy replaced them with two-dimensional circles, which could be easily drawn on a flat sheet of paper. Hipparchus had applied this basic idea to the motion of the sun. Eudoxus had place the sun on the equator of a rotating sphere and this sphere was attached to the celestial sphere so that the orbit of sun, the ecliptic, makes an angle of 23 1/2 degrees with the celestial equator. Hipparchus noticed that the sun never wanders off the circle of the ecliptic; it always remain in the plane of circle of ecliptic and a sphere was not needed to explain its motion. Hipparchus had replaced the sphere of the sun with a circle, a wheel, centered on the earth and tilted at 23 1/2 degrees to the celestial equator. And if the wheel of sun completed one rotation every 365 1/4 days, the sun's annual rotation around the circle of the ecliptic could be correctly explained. But one circle was not enough to explain the different length of the seasons. Eudoxus' follower, Callipas, had added another sphere to make winter shorter than summer. Hipparchus used a similar device. He proposed that a wheel or circle be attached in place of the sun on the rim of the big annually turning wheel, whose center is on the earth. The sun is attached to this smaller wheel. The big wheel he called the deferent and the smaller wheel was called the epicycle. Thus if the sun moves around its small epicycle twice of a year while the deferent carrying the epicycle rotates once a year, the different lengths of season could be explained. For six months the sun's motion about the epicycle and the epicycle's motion about the deferent in the same direction, so their motions add. During the other six months the motion of epicycle subtracts from the motion of the deferent. Then the motions add, the sun appears to move faster through the heavens; when the motions subtract, the sun appears to move slower. Thus the difference in length of the seasons are explained. By choosing the right size of the epicycle with reference to the size of deferent, the exact difference in the length of the season could be computed. Using this device of epicycles, Hipparchus was able explain the motion of the moon. Ptolemy's great contribution was to realize that epicycles could be used to account for the retrograde motion of the planets and thus to solve Plato's problem of accounting for the retrograde motion of the planets by uniform and circular motion. But he also was able to account for the nonuniform eastward movement of the planets through the zodiac.
      Ptolemy used three geometric devices to account for and describe these variations in the motion of the planets:
      (1) the eccentric, (2) epicycles and (3) equants.
      Like Hipparchus, Ptolemy accounted for the movements of the planets in the heavens by the circular motion of the planet about the earth at an uniform or constant speed. But to account for the nonuniform eastward movement he introduced the eccentric, the offset of the earth from the center of the circle. From the earth the movement of the planet appears to be nonuniform, while according to the theory it is really uniform. Also the motion of the planet is circular about the earth, while the earth is not exactly at the center of the circle. Like Hipparchus, Ptolemy used a smaller circle (epicycle) moving on a larger circle (deferent) to account for the retrograde motion of the planets in the heavens. But to account for variations in the retrograde motion he invented a new geometrical device: the equant. To make an equant Ptolemy started with an eccentric, in which the earth is placed away from the center of the circle of the planet's deferent. Now Ptolemy imagined another point, not at the center of the circle, from which the motion would appear uniform. This imaginary point is the equant point. Ptolemy placed this equant point opposite of the center of the deferent from the earth, which is at the eccentric point. If one stood at the equant point, then he would see the planet move around the sky at an uniform angular speed relative to the stars. From the earth and center of the deferent it is not uniform. This means that with the use of the equant celestial motions no longer had to be uniform about the center of the circles. The equant was a nonphysical, purely geometric device that broke the fundamental principle that planetary motion had to be uniform about the center of the circles. But this is not only place that Ptolemy fudged to get the results he wanted. The motion of the planets Mercury and Venus had to be treated differently from the rest of planets. In order to account for the greatest elongation of Venus and Mercury, Ptolemy demanded that the epicycles of Venus and Mercury always lie on a line between the earth and the sun. So these planets were constrained to stay near the sun. In contrast, since Mars, Jupiter and Saturn may be anywhere on Zodiac relative to the sun, their epicycles could be anywhere on the circumference of their deferents. But to ensure that these planets retrograde at opposition, Ptolemy set the radii of their epicycles parallel to the earth-sun radius. But in spite of these theoretical imperfections, this system of Ptolemy was magnificent accomplishment and remained in use for 1400 years, because it worked - it predicted planetary positions to the accuracy (of a few degree) needed by astronomers who did not have telescopes. Also it agreed with the physics and philosophy of Aristotle and with theology after Thomas Aquinas integrated Aristotle into Christian theology.
      Ptolemy's cosmos was finite and small. Using the distances worked out by Aristotle and Hipparchus in terms of earth radii, he worked out the distance to sphere of the stars at about 20,000 earth-radii. This is about equal to the present known distance from the earth to the sun. It was a small cosmos.