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Here are the objectives for today's lesson.
Before you begin to study the lesson, take a few minutes to read the objectives and the study questions for this lesson.
Look for key words and ideas as you read. Use the study guide and follow it as you watch the program.
Some students find it helpful to make a note in the margin which pertains to a particular objective or a study question.
Be sure to read these objectives in the study guide and refer to them as you study the lesson.
Focusing on the learning objectives will help you to study and understand the important concepts.
Compare the objectives with the study questions for the lesson to be sure that you have the concepts under control.
1. Aristarchus
2. Eratosthenes
3. Euclid
4. Hipparchus
5. Ptolemy
Heliocentrism Rejected
The Size of the Earth
Elements of Geometry
The Astronomy of Hipparchus
The Ptolemaic System
Before we're done with this lesson we will have seen how Aristotle's paradigm influenced the Greek culture in the city of Alexandria and eventually became forged into the Ptolemaic system through the contributions of the Hellenistic scientists with names like Aristarchus, Eratosthenes, Euclid, Hipparchus, and Ptolemy. We'll see how the Ptolemaic system became the astronomy of choice, and how it differed from the cosmology of Aristotle.
With the death of Alexander the Macedonian Empire fell apart. The Greek city states had lost the magic and no longer held the allure of the intellectual. With their decline, a new center of civilization began in the city of Alexandria, built by Alexander in honor of himself.
Who can really explain the details of social conditions that cause the decline of one region while another flourishes. History is full of such happenings, and it is not our purpose to trace their detailed demises and developments.
It is important for this the purposes of this course to see how the ancient ideas adapted to the various values of these different cultures which modified them. Only then can we understand the importance of the great revolution in science, the topic of the seven lessons in section two.
The essence of Athenian culture died with Alexander in 323 B.C. and Aristotle in 322 B.C.
The death of a leader of Alexander's strength, and the political glue and a philosopher of Aristotle's stature, who provides the philosophical strength of an empire can have devastating effects, especially when their lives were so closely linked and ended almost at the same time.
Consider the effect that the death of JFK had on the United States as a small scale parallel. Social unrest, dissent, threat of revolution, ethnic issues all arose. Think of the same effect some 2300 years ago, when an invading army of Macedonians accomplished what three generations of war with Sparta had not been able to do. Ravaged by war, and torn by social dissent, the Greek culture was undermined by the new dominion of Phillip.
There was little left of it by the time Alexander died, and without his leadership, which wasn't that good to begin with, it never made a comeback.
The center of Greek culture shifted to Alexandria, a seaport on the Nile delta, on the opposite side from where the city of Cairo is today.
The Hellenistic civilization, named after a character in
Greek mythology, grew from Alexander's efforts to spread the Greek culture as he conquered the world. After his death dynasties were established that brought back political disunity in the empire, while promoting the Greek unity of trade and learning.
A new culture developed in art, letters, and science, in Alexandria and in other cities in Alexander's former empire.
In Alexandria the standard of living was high, especially for the well to do who were generally well educated. Literature was abundant and the concept of learning, as distinct from knowledge and philosophy, came into being.
The Alexandrians built libraries, compiled anthologies, developed the art of sculpture and studied mathematics and science. When life is good, there is more time for curiosity, just like in the early years of the golden age.
So successful were they in spreading that Greek culture that the triumph of Rome was due in part for their ability to absorb rather than eradicate the Hellenistic influence.
Two of the most famous scientists in Alexandria were Archimedes, whose understanding of buoyancy saved his king a fortune, and Euclid, whose geometry was so precise that it is still taught today almost exactly as Euclid presented it.
The atmosphere in Alexandria was much more practical and much less metaphysical than in Athens.
The mechanical arts such as painting and sculpture became popular as the disapproval of manual work disappeared.
The making of instruments and maps had advanced to a very high level of sophistication by the first century A.D. so that detailed planetary observations could be recorded. A new interest in understanding the world and in doing experiments and taking measurements began to grow.
The Library of Alexandria was the greatest repository of knowledge anywhere in the world, then and until the nineteenth century. It was a huge and elegant building in the classical Greek style with large columns, you know the kind you see in Washington, D.C. and all over Europe.
Among other works, the library contained most if not all of Aristotle's writings. Some of them were original and only copies. There were just too many of them to make enough copies for everyone.
They have copying machines and faxes and other things hard as it might be for us to imagine a world without electronics.
The library was destroyed in a series of raids over 350 years. The first came when it was burned by Septemia Zenobia in 269 A.D.
Septemia Zenobia. I love that name, it sounds like someone you would meet in the Starlight Cafe, or in a chat room on the intergalactic internet. I wonder what a little research on her would reveal. Who was she?
The library was sacked in A.D. 415 by mob of orthodox Christians fighting against heresy and pagan learning. They burned tens of thousands of documents, Aristotle's writings no doubt among them. We have no way of knowing what things were destroyed.
To finish the job the library was completely destroyed in an invasion by Moslems from the east in A.D. 640. Many of the documents were salvaged by the troops and made their way eastward where the new center of civilization was developing.
But that's another story, and one we'll get to in the next lesson.
Aristarchus was one of the earliest of the Hellenistic scientists. His main contribution was his questioning of Aristotle's geocentric system.
Aristarchus considered the possibility of a rotating earth revolving around the sun. He asked what kinds of effects we would observe. It was a wonderfully unbiased account, although he still considered only circular motion to be the only and true motion for the heavens. He did not collect data, he just considered what the universe would appear to do if the sun was the center of heavenly motion for earth and the planets.
The heliocentric universe of Aristarchus was discounted.
More than anything else, Aristarchus started a debate which led to the rejection of the heliocentric theory and a strengthening of the geocentric theory of Aristotle.
There were three principle reasons why the heliocentric theory of Aristarchus was rejected
It was a qualitative system. There were no calculations of planetary paths, and he offered no way to calculate them. It's like saying "on a merry-go-round it appears as if the earth is spinning around you, but you might really be spinning instead"
On a rotating platform like a merry-go-round there are other clues that tell us we are moving and not the world around us. We will explore this idea later.
For now let's just note that the Greeks also looked for some signs of evidence for one theory over the other, but really couldn't find it.
For one thing, there was no parallax. Parallax is what happens when you point at something with one eye closed then switch eyes. Come on, try it. It's OK to point if it's for science. If you're family or friends think it's weird, just tell them it's for science, then they'll understand.
So what happens is that the finger you were pointing with can't line up with both eyes and the same object at the same time.
It's geometry.
Pointing with one eye closed keep your hand still and move your head from side to side. See, the finger doesn't point at the same place when you move because you are seeing the two objects, (the finger and whatever it is lined up with) from a different perspective.
If the earth is moving in a circular orbit, then the same thing should happen with the background of stars when earth is in different parts of its orbit.
This is important because we know today that the stars are so far from us and so far apart that the parallax is too small to be seen without sensitive telescopes and tracking. In fact it was not observed until the nineteenth century despite the fact that the geocentric model had been discarded two hundred years earlier.
One type of parallax is alignment parallax. This is where two objects line up at one time and not at another as circular motion occurs.
To see this, close one eye, hold up your index fingers on both hands, one in front of the other, like this (demonstrate).
Now move your head from side to side and watch how the two fingers appear to move in relation to one another. The greater the distance apart (try it) the greater the apparent movement.
Angular parallax is similar, and a little harder to explain how to demonstrate it. I'll let you figure this out if you want to try it.
Angular parallax does not require that the two objects ever line up. Whether or not they are lined up, the apparent angle between them will appear to change even if they are the same distance from us.
Besides that the heliocentric paradigm would not fit in with Aristotle's System of the World, which required homocentric and geocentric spheres to explain the motion of the planets. The whole cosmology of the system depended on that planetary motion being as Aristotle described it.
"A chain is no stronger than it's weakest link." What does that mean?
What is the chain and what is the link? This might be a good topic to write about.
The whole idea of heliocentrism would violate the obvious distinction between the earth and the heavenly realm, although it is doubtful that many knew where that connection came from.
Among other things, Aristarchus was criticized for "putting in motion the hearth of the universe" and violating a sacred principle, as if he had barbecued a sacred cow.
His ideas were rejected largely on the basis of these three criticisms.
Probably the most influential of the residents of Alexandria was Euclid, the father of geometry.
Euclid collected the entire Greek knowledge of geometry and formalized it, stating axioms, developing methods of proof, relating methods of construction of geometric figures of all kinds, analyzing the properties of different shapes, calculating their perimeters and area, to name a few.
His book, entitled simply Elements of Geometry has has more influence on more people than any other book except maybe the bible. It has been used in schools for more than 230o years and still is the best overall reference for so called plane geometry, or two dimensional geometry.
The use of the logic of analysis in problem solving is just as important to our modern science as is the geometry. Solving mathematical problems can often be made easier if there is a problems solving model to follow. Several of these will rear their heads during our studies, and we will deal with them when they pop up.
Constructing geometric figures is more than just drawing lines. There are very precise methods for bisecting a line segment, drawing equilateral figures like triangles and squares (equilateral means the sides are all the same length, like the polygons of Pythagoras), constructing right angles. There are also rules of proportion for triangles which contain angles of the same measure, and so on.
We will not undertake a detailed study of these rules, but some of them will return to haunt us later. Like the problems solving, we'll deal with these when we get to them.
These are good words to know. We will see them again with Newton, who relied heavily on Euclid's methods in analyzing the motion of the planets while formulating his description of gravitation.
An axiom is a statement of assumed fact. A hypothesis that is not subject to proof, but which seems to be true.
An example. On a flat surface, parallel lines never cross or converge.
This states a property of the flat surface at the same time it defines the concept of parallel lines.
A theorem is a statement which is subject to proof using formal methods of logic and previously proved statements.
A proof involves some method of showing that a statement such as a theorem is true for all situations. Such a statement might be that similar triangles have proportional sides.
A corollary is a statement which follows from a proven theorem. An example is: the ratio of the sides of a triangle depends on the angles in the triangle
We'll leave Euclid behind for now. He will resurface again and we will see how his method works.
Hipparchus easily ranks among the greatest astronomers of all time. His contributions and inventions took astronomy to a level never before obtained, and not matched until Tycho Brahe in the late 16th century.
His list of accomplishments is impressive. Using instruments such as the astrolabe which he designed and constructed he made careful observations and published the first detailed star catalog complete with grid locations. Hipparchus also invented the system of latitude and longitude that we use today for location on the earth and in the heavens.
He rejected the heliocentrism of Aristarchus with well written arguments. His model of the planetary motions was accurate, but not very. His system was geocentric, but not homocentric as he replaced Aristotle's system with epicycles and eccentrics.
The many modifications he made reached their culmination with the publication of the Almagest which became the standard astronomical reference until the time of Copernicus.
The more accurate the measurement the more difficult it is to find a model that will reproduce them to a sufficient level of accuracy. The astrolabe is like a surveyor's transit, but with a sighting tube instead of a telescope. The telescope would not be invented for another 1750 years. These were the high tech instruments of the time, and there was nothing like them anywhere else in the world.
Using these instruments, Hipparchus made many careful and accurate observation of the motions of the stars and planets. They were the most detailed observations ever made up to that time and as a result they showed that previous calculations did not really predict the motions of the planets very well at all.
Using the times and durations of solar eclipses, Hipparchus found a method for calculating the distance to the sun and the moon. This is a remarkable accomplishment. How many of us could figure out how to do that. For one thing, it requires an understanding of the cause of eclipses, which escapes many people here at the end of the twentieth century.
Although the distance to the sun was incorrect, the method was sound. Remember that there were no clocks (not until the fifteenth century) and there was no decimal number system in use until the Arabs modified the Hindu decimal symbols sometime between 400 and 800 A.D.
There are about three thousand stars visible to the naked eye, not counting the milky way. Locating each of them and recording its coordinates was an awesome job. Ptolemy's catalog contained around 500 of them.
The idea of a grid system for describing location on a sphere is one that we take for granted today with our maps of earth and the heavens.
But to think of it for the first time? I don't know, would you have done that. Once again we see the power of the invention of genius in propelling us into a different kind of understanding of our world. Once a grid system was in place, it because much easier to record the locations of the stars, and to keep track of where on earth you were, which, of course is important to navigation. In a seafaring economy, those who don't get lost, get the goods shipped and stay in business.
Hipparchus was among the chief detractors of the heliocentrism of Aristarchus, largely because of the lack of parallax, but also for the other reasons mentioned earlier. Go back and review this if you've forgotten. Another important is the failure to observe the crescent of Venus. Because Venus is closer to the earth than the sun, in a geocentric system it must always be between the earth and the sun and so should not exhibit a full range of phases.
Since the phases of Venus were not observed until Galileo pointed the telescope at the planet in 1609 we'll hold off a bit on getting into the details of it.
Instead of the homocentric spheres of Euxodos and Aristotle, Hipparchus found the calculations easier if he thought of the motions as circles which did not have to have the same centers.
Abandoning the homocentric model of Euxodos and Aristotle, he still had to account for retrograde motions. This is where the concept of circles on circles originated, in the form of the epicycle.
To visualize an epicycle, imagine a small wheel, like a shopping cart wheel, with a white dot painted on it. When the wheel rolls on a flat surface, the white dot moves in a circle.
What happens when the wheel rolls around the outside of a larger circle, such as around the outside of a large can; what shape does the white dot make then?
Can you see that it makes a series of loops. You will notice that in certain cases, depending on the sizes of the wheel and the drum, and the speed of rolling, the white dot sometimes reverses its direction and moves backwards, just like a planet during retrograde.
The modifications to Aristotle's System of the World made by Hipparchus and others after him reached reached culmination with Ptolemy. By the time it got there it had little in common with the original.
About the only thing Ptolemy's system had in common with Aristotle's was that it is geocentric, and, oh yeah, circular.
Claudius Ptolemy was a mathematician, astronomer, and geographer. He systematized, recorded, and added the the data and doctrines known to Alexandrian scientists.
His Almagest, widely translated and influential in Europe until the sixteenth century, presented Ptolemaic system of astronomy, based largely on the concepts and data of Hipparchus.
There are few personal records of Ptolemy's life, so we know next to nothing about him, except through his publications. The dates of his birth and death are approximations.
Ptolemy published several major works, largely textbooks on a variety of astronomical and geographic topics. They were quite mathematical, and his skill as a mathematician was quite good, although he did make a lot of errors. Oh well, what do you expect, he still didn't have numbers. All calulcations were done from truth tables, like finding the distance between two cities on a road map distance table. It took a long time and was not very precise as the concept of decimals did not exist yet in Western thought.
His Geography text was the most comprehensive of its time and had detailed maps of the entire Mediterranean region.
Ptolemy's greatest work was the Almagest. It was a combination textbook, encyclopedia, and astronomical almanac.
It was a remarkable piece of work, despite the errors. The Greek title was "Great Syntaxis" or "Great Compilation" but the European title comes from the Arabic Al Majisti, the same root word as majestic and majesty, essentially "The Greatest." That gives you a clue that the folks thereabouts thought quite highly of it.
It was essentially a collection and compilation of data, calculations, methods of observations and calculation, and tables of planetary locations. Basically, a compendium of six hundred years of Greek astronomy as well as new results of his own work on planetary motion.
It also contained an updated star catalog, with several hundred new stars discovered and located by himself and others since Hipparchus's time nearly two hundred and fifty years before.
The book defined the basis of mathematical astronomy and remained the best and simplest until Copernicus described his heliocentric methods in the sixteenth century
The system of Ptolemy, not surprisingly, has come to be called, the Ptolemaic System
The Ptolemaic system became the standard because it was the best at the time, and Western civilization went into rapid decline as the unity of the Roman Empire weakened before anyone could come up with a better system. Ptolemy marks the last of the great Alexandrian tradition of science and learning.
Since Ptolemy's system had such a strong and lasting influence on the Western world, we should take a few minutes to look at its structure. What we are going to do is analyze what Ptolemy had synthesized.
One of the important differences between Ptolemy's system and that of Aristotle is the linkage of motions. Remember that in Aristotle's universe nothing happened without cause and independent from the whole.
By Ptolemy's time the cosmology had been lost, completely. Never mind what caused the planets to turn, it was hard enough figuring out how to describe how they did turn and calculating where they were going to be.
We might say that Ptolemy kept the mathematics without concern for the cause. That might have been fine for the growth of science, but there was the Church, who who would become greatly concerned with cause, but would remain ignorant of much of Aristotle's work until the twelfth century.
There I go, getting ahead of things again.
What Ptolemy did was very Platonic. He saved the appearances elegantly in a masterpiece of mathematical artistry. He used mathematics to turn the perfect circular motions of the heavens into the observed paths of the planets, retrograde motion and all.
His calculations for planetary tables and navigation were the most accurate of his time, and at least as accurate as the observations anyone could make at that time.
His system accounted for the retrograde motion of the planets
and it accounted for their changing speeds against the background of fixed stars.
It even accounted for the changing brightness and sizes of the planets during their retrograde phases.
One of the tools Ptolemy developed was really a gift from Hipparchus. Ptolemy refined the use of what he called 'devices'.
What does the word device bring to mind? How is a device different from a tool. Look up the two words in the dictionary.
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Here are the retrograde loops as formed by a single epicycle on the deferent. Notice how the motion creates a single and symmetrical set of loops. |
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Here is the motion represented by two epicycles. Note how the first epicycle serves
as a deferent for the second. This produces a much more complicated motion and allows
for a smoother path. Imagine what the path might look like with as many as 8 epicycles. |
Here is an animation of a single epicycle where you can see how it explains the retrograde motion.
(This animated gif was obtained from Astronomy 161 web site at The University of Tennesee, Knoxville.)
Sometime it was necessary to use a device for a time, then not use it at all for awhile. Sometimes Ptolemy would allow the eccentric to change locations.
It was quite difficult to do the calculations and even harder to know when to do what kind. Only the most accomplished mathematicians could even do it at all.
Ptolemy used one device to calculate the speed of a planet at a given time and another to calculate the planet's distance and size.
If the formula gave the correct results, then it was the correct formula whether or not it made sense.
The underlying physics and metaphysics of how and why it worked was simply not important. It worked and that is what mattered.
In addition to the calculations Ptolemy went to some lengths to justify a stationary earth.
Just so we don't forget, many of us learned that up until the time of Christopher Columbus people thought the world was flat. That's not entirely true. As we have seen Eratosthenes recognized the roundness and even measured it's circumference by measuring the lengths of shadows at noon in different locations a know distance apart.
Ptolemy's arguments for a stationary earth were quite convincing and relied heavily on Aristotle's theories of motion, even as he was ignoring, or was unaware of the cause of planetary motion.
Ptolemy's system also allowed him to calculate eclipse times, something that earlier theories did not do well.
Ptolemy also issued one of the first statements of parsimony in his justification for not including a discussion on the cause of motion.
The simplest system was the one which worked best with the smallest number of complications. In fact, his model was extremely reductionistic, having removed from it all that was not necessary to make it work.
It is difficult to criticize Ptolemy or those who adopted his system. It was a good system.
It wasn't until long after Ptolemy's death around A.D. 140 that the metaphysics began to creep back into the theory. Later astronomers tried to integrate a coherent physical system, for example posing and debating questions as to the nature of the material of the spheres and the space between the spheres.
Medieval philosophers had little luck in providing answers to those questions.
In this lesson we have learned how Aristotle's System of the World became corrupted as the Hellenistic Greek culture continued in Alexandria after the death of Alexander and Aristotle, late in the third century B. C.
We followed the river of thought through contributions by Aristarchus, Euclid, Hipparchus. We saw the major tributary of Ptolemy's grand synthesis adding the best of Greek astronomy to to the stream of history, where it would soon replace the lost cosmology of Aristotle which the fresh view of \Christianity.
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