A. The Persistence of Geocentrism
B. The Human Brain, Language and Technology
C. Calendars, Causality and Mysticism
D. Rules of Arithmetic
E. Multiplication is Shorthand Addition
F. Advances in Mediterranean Mathematics
G. Nonwestern Astronomy
The fact that we are curious about our environment should come as no surprise. As a survival skill, the more we know about the resources and hazards of our world, the more tools we have which will aid in our survival.
In order to learn, we must receive, organize, and classify information about our surroundings. This is impossible, of course, without a brain, and ours does it well. So well, in fact that it is safe to say that the human brain is the most remarkable and complex structure in the universe. Around forty to fifty thousand years ago an important development took place that marks an important step in the growth of what we now call science.
The ability to reason and symbolize, to use language prehensile thumbs which allow us to make and use technology are the results of the new brain. It is those qualities which are necessary for systematic learning about our universe.
2.1.1. Why Science
Previously we have seen that the scientific learning we do is only a formalized version of learning in general.
Is that the only distinction between science and generalized learning? What are some of the other differences?
2.1.2. Origins of science are complex, cultural
Everyone learns from culture as well as from experience, so it is not surprising that the method of learning and the perceptions of the universe vary significantly from culture to culture. We would like to have a science which is free of cultural bias, but this is not likely as long as there are humans doing science.
The variety of cultural values, practices, and beliefs among the world's cultures, past and present, is vast. It is not surprising that the origins and growth of science as a formalized system has been so long and complex. Our modern scientific system includes contributions from every major culture. What is surprising is that for the most part, people from all cultures can agree on the truth of our science, although there may still be disagreement on the scope of reality which it represents.
It is arguable that astronomy is the oldest science. It would be in competition with chemistry, especially the process of combustion, but one might argue that the study of fire is technology and not science, at least until the eighteenth century A.D.
Certainly our most distant ancestors must have been aware of the stars and wondered about them since the beginning of consciousness, whenever that was.
184.108.40.206. everyone learns from culture and from experience
220.127.116.11. astronomy is the oldest science
2.1.3. senses and earliest paradigms tell us Earth is stationary
Was it just a coincidence that early astronomers considered the earth to be flat and stationary, or was there a good reason to think so?
From first impressions, anyone would come to the conclusion that we are not moving while the sun, stars, and everything else in the sky moves past us in seasonal cycles.
18.104.22.168. We are not moving, therefore . . .
22.214.171.124. heavenly bodies revolve around us, therefore . . .
126.96.36.199. Earth is center of the universe.
2.1.4. modern paradigm tells us Earth is spherical and sun is at center
Although our modern paradigm sees us on a rotating planet whizzing around the sun at thousands of miles per hour, this paradigm is only a few hundred years old. It replaced the geocentric system which was the standard model for thousands of years.
188.8.131.52. modern paradigm is only a few hundred years old
184.108.40.206. replaced older system of belief which stood for thousands of years
2.1.5. why replace a system at all
At this stage in our studies the relevant question is not the heliocentric, nor even the geocentric model. The question we want to explore for the next four lessons is the way in which the intuitive geocentric model became formalized and even canonized. The social and cultural processes by which intuition, incorrect or otherwise, becomes a paradigm serves as a model for human thinking, scientific or otherwise.
So the question of why we replaced the geocentric paradigm with a heliocentric one really becomes, "how did the geocentric paradigm become so hard to replace?"
220.127.116.11. it is consistent with senses
18.104.22.168. it worked well for 2500 years
22.214.171.124. it is ingrained in culture and religion
126.96.36.199. it is embodied in authority of religion
It is our brains that distinguish us most of all from other animals. It is our brain that allows us to perceive the universe in the way we do, and to classify it, write about it, draw pictures of it, and talk about it.
The development of the human brain took a major leap forty thousand or so years ago with the development of language and the ability for abstract reasoning. The use of meaningful symbols to communicate abstract ideas is the basis for our common reality, which is the physical universe.
In this section we will consider the relationship between the brain and our understanding of the physical universe.
2.2.1. the source of science and knowledge
Without the rational abilities of the human brain there would be no science, just as there would be no art. We could argue that knowledge also requires rational ability. Do you really know something if you do not know that you know it?
A cat knows how to climb a tree, squirrels can open acorns, birds can fly, and fish can swim. Is this knowledge, or is it something else?
2.2.2. abstract brain
The ability for abstract thought is really behind much of our human traits. We could argue that organizing knowledge for later recall requires a brain which is capable of abstraction of the concept of classifying. Sure, computers can classify and organize information, but only after they are told how to do it by a carbon based brain.
The first abstract brain probably developed around one million years ago. We can never be completely sure of that, but human remains and worksites from that time period indicate a fairly high level of abstract concepts, such as religious rituals and burial of the dead.
188.8.131.52. first probably about one million years ago
184.108.40.206. abstract ideas and concepts
220.127.116.11. rational thought and religious concepts
2.2.3. fire and stone tools appeared hundred of thousands of years ago
Fire and stone tools did not develop until long after the religious rituals were well established. It is hard to imagine our barely human ancestors struggling for a few hundred thousand years with the concept of sharpening a stone tool, or of capturing and controlling fire.
That it is so hard to picture this should serve as a reminder that the sharing of ideas from one individual to another and from parents to children is a very powerful adaptive tool. Try to imagine yourself never having been taught that you could cut things with a sharp edge, and that you could make your own sharp edges by carefully shaping certain kinds of rocks. Easy? I don't think so.
This common, shared culture of ideas requires not only a brain capable of processing abstract information. It also requires the use of some kind of symbols to convey information.
18.104.22.168. manipulation of environment
22.214.171.124. technological advancement
2.2.4. language arose forty thousand year ago
All of the symbols used in human language stand for some abstract idea. In the case of Western languages, the symbols represent sounds which when combined in certain sequences, mean certain things. We call these alphabetic languages. In Eastern languages, the symbols are graphic representations of ideas or objects. These are ideographic languages.
That the two very different way arose for written language indicates that spoken language came first.
In the fossil remains of ancient humans, certain anatomical changes correspond with the appearance of cave art and mass human migrations. More than likely this represents the beginnings the abstract brain and its concomitant abilities in language, art, and science.
With language comes the invention and communication of abstract ideas. People can now exchange ideas about their perceptions of the physical and the spiritual world.
126.96.36.199. Homo Sapiens Sapiens
188.8.131.52. anatomical changes coincide with cave art, mass migrations
184.108.40.206. invention and communication of abstract ideas
2.2.5. legends and other knowledge could be passed by oral traditions
Language and the exchange of information is one of the greatest innovations that has occurred in Earth's history. It allows for shared experiences and group concepts. When someone sees something in the sky, it is possible to ask for the observations and opinions of others. Discussion can revolve around explanations and speculations. This would have several effects.
Our curiosity drives us to explanations. When a group has certain shared experiences and are able to discuss them, a consensus will eventually develop which is different from any individual idea. Certainly one individual can persuade others of the veracity of his or her view. This has been the nature of science and learning in historic times and there is no reason to assume it would have been otherwise.
When a consensus is reached by a group, whether formally or informally, the need for understanding, purpose, and symbolism is satisfied and a world view develops which characterized that social group regardless of its size.
220.127.116.11. allowed for shared experiences and group concepts
18.104.22.168. need for understanding, purpose, symbolism
22.214.171.124. invention of explanation in terms of world view
2.2.6. couldn't help but notice and talk about the stars
It is hard to imagine that people would not notice, wonder, and talk about the stars. What are they, where are they, what's between us and them, why do they move, why to they twinkle . . .? The list of questions would be endless, like those asked by curious children.
Many of the earlier concepts are childish. What would we expect. Making the transformation into adulthood does not automatically mean that we give up our childish constructs and concepts. If only we did . . .
Most of us know the date most of the time. But how do we know? If you did not know the date, how would you find out?
You might use your modem to call the national time standard in Boulder Colorado to synchronize your internal clock which measures the time in seconds from January 4, 1904.
Some people might do that. But, others might use a calendar.
What an unusual timekeeping device a calendar is. It has all of the days on it at once, and we pick the one that we know it is from 365 choices. It's kind of like a watch with no hands, yet we still use it to keep track of days, months,seasons, and even years.
As we saw in lesson 4, the calendar is based on the movements of the sky, and the relationship of the sun and moon, and the time and location of sunrise and sunset.
2.3.1. hand-in-hand with civilization
It is difficult to think about civilization without calendars. With few exceptions, every culture has developed some way of keeping track of the seasons and cycles of the stars. The further from the equator, the more important it is to know of the coming seasons so that food can be stored against the harsh winter.
2.3.2. crops, weather, migrations, trade
When to plant and harvest crops, when to store firewood for the winter cold, when to move uphill or downhill to take advantage of the climate, and how to find marine trade routes and return home safely, are all reasons to know when and where you are in time and space.
We take this for granted today, but we certainly rely on the navigational abilities of others when we travel by ship or airplane. Before the days of satellite navigation, beginning in the 1960s, the techniques used for navigation had not changed in a thousand years.
2.3.3. agriculture and settlements
The development of civilization require domestication of crops and animals. This requires timekeeping for planting, harvesting, irrigation, religious festivals, and animal migrations. The better the knowledge of the stars, the more accurate the calendar, and the more successful the planting, tending, and harvesting of crops.
2.3.4. sea trade requires navigation
Some of the earliest explorers traveled by sea to establish trade. Different products can be produced in different regions, and in many cases a region may be uninhabitable without importing essential goods such as water and salt. Trade increases the amount of habitable land, which allows for population growth, the primary goal of most species, man included.
Being able to travel long distances by sea, find a center of trade and return home with new goods requires the skill of navigation. Again, the better the navigation, the more successful the voyages.
The ability to know the future has been sought by mankind ever since the concept of the future was first envisioned. When and where this occurred we will never know. But we could make an educated guess that it was part of the abstract brain package.
The future that we know will happen would be a little less disconcerting if we knew that the sun will come up tomorrow and that the world will not end. This is not to say that we should be concerned about that particular eventuality, but if we didn't know, how might our lives be different?
Generally the sun and stars are regular and predictable. The yearly seasonal cycles are linked more closely to the sun and its position relative to the stars rather than the moon.
The moon is regular, but it more difficult to predict its precise location and rise/set times. From out modern, heliocentric perspective, this is because the moon's orbit is tilted about five degrees from the earth's equator. But without knowing that, it's position is relatively unpredictable even when the phases are predictable.
The planets move irregularly only during retrograde and are generally predictable except during those times. It is this capriciousness which inspired the mystical connection which led to astrology. The unpredictability of the planets is the closest of any of the heavenly objects to the chaos here on earth. For this reason it was thought that the intermittent chaos of the planetary orbits might give some clue to future events here on earth.
126.96.36.199. sun and stars are regular and predictable
188.8.131.52. moon is regular but more difficult to predict
184.108.40.206. planets are irregular but are they predictable
One of the side effects of keeping track of the seasons, is the tendency to connect events causally.
It is part of human nature to be curious. We just can't help but wonder "why"?
If two events always occur in the same sequence, isn't it natural to assume that one causes the other? This is what we mean by causality, or a causal link between events.
We do this in everyday life, and five or six thousand years ago, there was no distinction between learning about the physical world and other types of learning. It is not surprising that our ancient ancestors mystified the stars and seasonal patterns. After all, today we ascribe these same patterns to the sun. The difference is that we distinguish between one event which causes another and two events which are controlled by the same cause. It is a subtle but importance difference.
2.4.1. rising of Sirius causes flooding of Nile
Sirius, the Dog Star, rises just after sunset about twenty degrees to the south of east beginning in February. Or at least that's what it did in central Africa, where the Egyptians living along the Nile depended on the annual floods, five thousand years ago. In the southern hemisphere, February is like July in the northern hemisphere. The summer sun, high in the sky melts the snow in the peaks and valleys of the Mitumba Mountains to the south and the Ethiopian highlands to the southeast. The meltwaters flow into the Nile, across the plains and into the desert where they flood the broad floodplain cut by the Nile into the soft rocks. These annual floods were the lifeblood of the Egyptian society. Without the floods the floodplain would soon become desert and be as barren as the Sahara.
Within a few weeks of the appearance of Sirius at sunset, the floods came, Every year for thousands of years, sometimes later rather than sooner, sometimes large and others barely noticeable, the floods came.
Here we have an interesting coincidence, at least from our perspective. The brightest star in the sky, brighter than some planets, rises in mid summer, just when the snow is melting, and would be melting whether Sirius was there or not. It's the sun that melts the snow, not a distant star.
Of course, the Egyptians didn't know this. From their perspective, the appearance of Sirius always preceded the floods whose source waters was unknown to them. What would you think?
The Egyptians assumed, and very logically so, that Sirius was the cause of the floods.
As such, Sirius was worshiped as a God, responsible through its causal link to the floods, for the success and the fate of the whole society.
2.4.2. alliance of religious/secular in early societies
This connection between the physical and spiritual worlds is common in early societies. Dubbed Paganism by the early Christians, the common belief was that Gods of various types controlled different aspects of the physical as well as the spiritual worlds. Today we might think of this as primitive, but our own religions and associated festivities are not really so far removed. It is only in the past three hundred and fifty years or so that the machinations of the planets were understood in scientific rather than spiritual terms.
2.4.3. supernatural explanations given for heavenly events
Without sophisticated concepts to guide the thinking of these earliest civilizations, it is not surprising that supernatural explanations sufficed to satisfy their curiosity and fear about the world. As a species, we sort of got our brains without an instruction manual for its use, and without a guided tour of the workings of the universe. Without physical evidence and understanding of basic physical principles, the human brain tends to invent explanations. This is much like the monster under the bed explanation for anything unusual in the darkness of the childhood bedroom. Both science and religion spring from the same "fear of things that go thump in the night" motivation. Although a monster under the bed is still fearful, it is an explanation, and even a monster under the bed is better than not knowing what it is. At least the monster will stay under the bed and won't come after you. Yeah, that's it, the monster can't get me on the bed because the bed has a monsterproof shield . . . .
2.4.4. any explanation is better than none
In the complete absence of information, any explanation is better than none. The human brain craves order and classification. Explanations are likely to be in terms of familiar cultural symbols. In modern terms we could cite the sightings of UFOs. There have been unexplained sighting of apparently supernatural phenomena since the earliest written records were kept. Today we intrepet these events, whatever they may be, in terms of space aliens because we live in the space age. Similar sightings at the turn of the last century were attributed to "giant airships". This was the era of balloons and blimps, long before the sight of airplanes was common in the sky. In biblical times, Ezekiel saw a wheel in the sky and Moses saw a burning bush. The interpretation of these events is very different depending on the cultural paradigms of the observers.
Whatever they are, they have always been interpreted in current cultural context wherever and whenever they have occurred. Someday maybe we will understand them. In the meantime they stand to us the way the movements of the stars stood to those ancient souls who wondered about them, and no doubt feared them since they were associated with darkness and the death of the sun.
Mysticism seems to be a part of the human psyche, for whatever reason. Because there are so many things we do not understand, and perhaps cannot understand, there will always be a certain amount of mystery.
What is mysticism?
It is a mystery to most people how a television or computer operating system works, but that is not what we mean.
We have to distinguish between mystery and mysticism. Look up the words in the dictionary.
It is useful to connect mysticism also with the concept of destiny. The connection is that if events are destined, then there might be some way to find out ahead of time what is destined to happen. So looking for clues in the stars, in tea leaves, or in omens just might work.
The concept of mysticism will reappear throughout the course , so it is a good idea to take a few minutes to think about it. When you encounter the term in future lessons you might want to stop and make sure you understand the context.
Danger: Rapids and Waterfalls. Failure to exercise caution could lead to serious problems later. Do Not Memorize. If you attempt to memorize too many things you will be swept away and may never be rescued. Memorizing names and numbers will be hazardous to your learning experience.
The idea of counting and arithmetic is second nature to some of us, even if we are not good at it. Having been taught the rules of arithmetic at a young age, it is difficult for us to imagine how hard it must have been to figure them out in the first place. The importance of numbers, arithmetic, algebra, and geometry can not be overemphasized. Although this course is not about calculations per se, it is necessary for us to understand the important role that early mathematics played in the development of science.
The basic rules of arithmetic are all that is necessary to understand higher mathematics. The properties of numbers and their combinations has spawned a variety of advanced and abstract concepts and methods. These do not concern us specifically, although we will dig a little deeper in subsequent lessons.
The formalization of these rule was long in coming. The development of algebra in the middle ages generalized the rules and stated them in symbolic terms, using letter to replace numbers.
First of all, basic mathematical operations such as addition and multiplication are commutative. That means they can be done forwards or backwards. Two times four is the same as four times two. Five plus three is the same as three plus five. Think of commuting back and forth to work. First one way, then the other.
4.1.2. Multiplication and Subtraction are forms of adding
Although we do not realize it, multiplication is nothing more than a shortcut for adding. Let's look at a few examples. Suppose we add three plus three. The result is six. Adding three again yields nine. Symbolically:
0 + 3 = 3
add three one time
1 three = 3
1 x 3 = 3
3 + 3 = 6
add three two times
2 threes = 6
2 x 3 = 6
3 + 3 + 3 = 9
add three three times
3 threes = 9
3 x 3 = 9
It is just another aspect of counting and grouping, like counting with the stacks and groups of pennies as we saw in program 4.
For small numbers adding repeatedly is sufficient. But the larger the number the more motivation there is for finding another way to keep track of things. Suppose there were forty boxes each containing a dozen eggs. How many would there be? You might say four hundred and eighty without giving it another thought, or you might stop and figure out that you must multiply forty times two and add that to forty times tem, and then you would do it. I can guarantee that few people will count from one to twelve forty times while keeping track with a series of marks.
Division, which is an inverse operation of division, can also be performed by a series of subtractions. For example, how many times will three "go" into nine. Starting with nine, how many times can you subtract three before you run out? Try this one yourself, then look at the operation we call "nine divided by three" and think of it like this: If nine cookies are to be divided between three people, how many will each get?
This is a basic concept, and one which is important to grasp early in the course. It might seem trivial now, but it will help you to understand subsequent course material.
Once we have established the idea of three groups of three, we see that we can think of this arrangement in many different ways. For example, the numbers three and nine bear a special relationship to one another simply because it takes nine blocks of equal size to make a 3 x 3 square. So we can say that nine is the square number generated from three. We can also say three squared and three is the square root of nine. (Think of the word root as the basis for the figure or number. In this case it is the number of blocks in the basic group, which is a row of three. Since we have repeated the operation of adding groups of three two times we can also say that three squared is nine. This is usually written as three to the second power, or 3 squared, or 32.
The idea of repetitive adding can be carried one step further. Suppose that I want to know how many blocks it would take to make a solid cube of a given size. Assume that all the blocks are the same size, and that each block is a cube of equal dimensions.
Let's use the number three again. We saw in the previous section that we can construct a group of nine blocks by using three rows of three, like this ..
To make a cube we could place two more of these 3 x 3 groups on top of one another to make a cube. Quickly, how many blocks are there in the big cube?
You might take it apart and count them all, or you might visualize the inner ones that you can't see. If you have conceptualized the way it is built, then you can count the number in each nine block layer and multiply by three. Nine times three is twenty seven. Symbolically 9 x 3 = 27.
But that's not the only way to do it. Recalling that nine is three threes (3 x 3 = 9), or three groups of three, you might say that the number of blocks is three three threes, that is three groups of 3 x 3 squares, or 3 x 3 x 3 = 27.
Now we have used our concept of grouping precounted things at two different levels. We could write:
(3 x 3) + ( 3 x 3 ) + (3 x 3)
which is really
(3 + 3 + 3) + (3 + 3 + 3) + (3 + 3 + 3).
[At this point it is fair to remind you of the danger of trying to memorize this material. It won't help. If you understand now how it worked it will be much easier downstream.]
Hopefully you can see why we invent symbols to simplify these concepts. A word in an unknown language may sound like an incomprehensible grunt. In the same way a notation such as 33 means nothing without some understanding of the concept. The examples presented previously are one way, but not the only way in which we can see a relationship between numbers and shapes. Because this relationship exists it allows us to understand the symbols by referring to a mental model. It also means that we can think of the concept three cubed and have some idea what it means.
Why would someone care about square roots in three thousand B.C.?
For several reasons, most importantly in building construction. Then, as today, two of the important problems in building a building are keeping it square and estimating how much material will be needed for the project.
How many tiles will be required to tile a room eight feet by twelve feet if each tile is a one foot square? Use the concepts of the previous section we see that it is ninety six because 12 x 8 = 96. There we see twelve rows of eight tiles or eight rows of twelve tiles, which in either case is ninety six tiles.
Here's a different twist. Suppose you had sixty four square tiles and you wanted to know what size room to build so that they would cover the floor. Sure, you could lay the tile first and then build the room around them, but that's only a temporary solution to a recurring problem.
What we have here is a square root problem. What will be the size of a square (equal sides) which will contain sixty four one foot squares?
You might have said 8 x 8 intuitively, or from your prior experience with numbers. But what if I said the number of tiles is eight hundred and forty one. Can you tile a square patch of floor with that number, and if so how big will it be?
The solution is easy if you have an electronic calculator with a square root key. It is not so easy otherwise.
You can imagine many different uses for the squares and square roots, other than the curiosity that the relationships exist.
The other problem, that of building square buildings, also has a solution in the concept of squares and square roots. We will learn of Pythagoras and his mathematical philosophy in the next lesson. For now we are interested in the relationship of certain triplets of numbers, such as 3, 4, 5. At first glance one notices that they are in sequence, but that's not the magic.
If you are laying out the floor of a new building, you can use these numbers to make perfectly square corners.
To lay out a square, tie five knots in a piece of string or rope. Be sure that the knots are equally spaced.
Start from a point and count three knots. From that point stretch four knots. When five knots span the distance then the angle is square, or 90°, one quarter circle, or perpendicular, however you prefer to think of it, like this:
There are other triples of numbers which have the same quality, for example 5, 12, 13.
Are these numbers random, or is their some relationship between them which can be easily characterized.
Why this works was unknown to the Egyptians, but it did work well enough for them to build the pyramids and other structures with perfectly square bases. It is not clear whether they knew the relationship between the numbers, but probably they did not. They only knew that it worked.
4.4.3. The Relationship
That relationship is really quite simple. It has to do with areas and squares, but in a way which is not immediately obvious:
32 + 42 = 52.
9 + 16 = 25.
It is not just these numbers. In fact any three numbers for which this relationship is true will form a right triangle. You may want to test for yourself that in general three numbers will not have this relationship and will not form right triangles.
4.4.4. General Relationship
The general relationship was not discovered until the fifth century B.C. in Greece. In today's symbols we express it generally as a2 + b2 = c2, where a, b, c represent any numbers at all. We say that any three numbers for which this relationship is true will form a right triangle with sides of length a, b, c.
This relationship is known as the Pythagorean Theorem and it will play an important role in Newton's synthesis of physics in the seventeenth century A.D., nearly 2200 years after it was discovered.
It is true that every culture has an astronomy. We focus on the early Mediterranean because it was from those early cultures which our modern scientific world view grew. These astronomers planted the seeds which through a series of historical accidents, eventually sprouted and grew into the scientific revolution which spread throughout Europe two thousand years later.
It is not our purpose to memorize the details of the cultures and their contributions, but rather to trace the development of the astronomy and mathematics in the thread of heritage as we unravel the nature of science.
Later in this lesson we will consider the astronomy of some other cultures.
The Sumerian culture flourished in Mesopotamia as early as 5000 B.C. They developed considerable power in the Mediterranean region due to their irrigated agriculture. At this time pottery making and metallurgy became fine arts and cuneiform writing became perhaps the first written language. Conquered by their rivals, the Semites, a brief uprising failed in 2120 and the growth of Babylonia ended the Sumerian nation.
5.1.1. urban civilization, bronze metallurgy by 3000 B.C.
Only two thousand years after the beginnings of agriculture in the fertile triangle of Mesopotamia, and extensive urban civilization had congregated into lush cities. The cities thrived on commerce driven by the high tech agricultural practice of irrigation in the fertile valleys of the Tigris and Euphrates rivers. The Sumerians ushered in the bronze age with ores traded with Cyprus, far away in the Mediterranean.
5.1.2. cuneiform writing by 2500 B.C.
By 2500 B.C. the writing known as cuneiform was well developed. Although the spoken language had about 600 phonetic symbols, the writing had 2000 symbols, compared to our modern English alphabet of 26 symbols. How many phonetic symbols (called phonemes in modern parlance) do you suppose there are in the English language?
Sumerian Cunieform Tablet
Obviously only those who were well schooled and well practiced could read and write. Some of the world's oldest known literature dates from this period, most notably a work called The Epic of Gilgamesh.
220.127.116.11. 600 phonetic symbols
18.104.22.168. writing had 2000 symbols
22.214.171.124. only extremely educated could write
5.1.3. basic mathematics
The Sumerian mathematics were basic, but quite sophisticated.
126.96.36.199. number system based on 60
Even today we bear traces of he Sumerian number system, based on 60 (compare with our system based on 10). There are 60 minutes per hour and 60 seconds per minute. And in angular measure there are 60 minutes of arc per degree and sixty seconds of arc per minute.
Why sixty? It's not known for sure, but one good reason is that sixty is a very utilitarian number. For example, it is the smallest number which is divisible by most of the small integers.
All of these numbers divide into sixty with no remainder: 1, 2, 3, 4, 5, 6, 10, 12, 15
It makes it easy to do arithmetic if their are no remainders.
188.8.131.52. multiplication tables
Extensive multiplication tables for all of the common integers were common among the Sumerian priests, who were in charge of accounting and inventory of grain and other products.
184.108.40.206. formulas for areas and volumes
The Sumerians had developed formulas for calculating areas and volumes of basic shapes. This is not such an easy thing to do. If you don't believe us, try coming up with your own formula for the area of a circle.
220.127.116.11. p = 3 (integer rounding)
The number known as pi plays a significant role in the development of mathematics, astronomy, and physics. Today we know it as 3.1415927 . . . to a million digits or so, more that is needed for any calculation. It is the ratio of the circumference to the diameter of a circle, p = C/D. Further significance of this number is beyond our scrutiny here, but it is an interesting topic which deserves further attention. It would make a nice topic for the final research paper for the course.
5.1.4. accurate calendars, some astronomical data, navigation
The Sumerians were not great astronomers, but they had developed a reasonably accurate calendar and they were great navigators and thus established a strong sea trade over wide areas of the Mediterranean.
The Mesopotamian Empire known as the Babylonians centered around the city of Babylon, on the Euphrates river in what is now Iraq. The empire flourished for 2500 years before falling to a Persian invasion in 538 B.C.
5.2.1. Babylonians were close observers
Some of the Babylonian records of astronomical observations have survived the millennia. These records show that Babylonian astronomers were making accurate observations by 1000 B.C, and by 700 A.D. had developed systematic recording of planetary positions and star catalogs. Many of our constellations, especially those of the Zodiac were recognized and named during this period.
18.104.22.168. some records survive
22.214.171.124. accurate measurements by 1000 B.C.
126.96.36.199. systematic recording by 700 B.C.
5.2.2. calendar based on lunar month
The Babylonian calendar was very sophisticated considering the non integer nature of the heavenly rhythms. A year is not exactly divisible by either days or months. A year is 365 days 5 hours 48 minutes and 46 seconds long. With our modern calendar we solve this problem by adding days every four years (in leap years) and subtracting days every 400 years (any century year divisible by 400 is not a leap year, for instance 2000 won't be a leap year.
The lunar month is 29.5 days, so 12 lunar cycles is 354 days 8 hours and 48 minutes. We solve this by having some months with thirty days and some with 31 days, and February.
The Babylonian calendar consisted of a 7 day week, which is the closest whole number to a quarter of a moon cycle, and a year of 360 days. There were twelve months, each 30 days long (the closest whole number to the actual month) Extra days were added when necessary to keep everything in synch with the sun and moon.
188.8.131.52. lunar month = 29.5 days
184.108.40.206. 18 year lunar cycle
220.127.116.11. 7 day week
18.104.22.168. 360 days (12 months of 30 days)
22.214.171.124. extra days added to keep in time
5.2.3. sunrise, sunset predictions
The observations allowed them to develop tables of sunrise and sunset locations for various locales which could be used as navigational aids.
5.2.4. sun, moon, 5 planets, zodiac
The Babylonians knew of seven heavenly objects and had named twelve constellations of the zodiac, one for each month. In this system the position of the sun and moon in relation to the various constellations took on mystical significance as astrology was born
Please distinguish between astrology and astronomy. Astronomy is the study of the location and movement of the stars and planets. Astrology is the attempt to attach mystical significance to the locations and movements with the belief that human affairs are linked to the heavens. It is an important distinction.
Astrology really began with the Babylonians because of their culture based on myth, superstition and a strong belief in the supernatural.
In the Babylonian cosmology the stars were gods who emerged daily. The heavens were seen as sandwiched between two layers of water, one of which held the earth and its seas. These gods controlled earthly affairs by some system that they thought could be figured out. Thus the motions of the heavens were tied somehow to human destiny. In the eyes of Babylonian priests who were also the astronomers and mathematicians, destiny would reveal itself when and if the motions were understood. Here we see the motivation behind the development of astrology
126.96.36.199. stars were gods who emerged daily
188.8.131.52. heaven sandwich on water bread
184.108.40.206. Gods controlled earthly affairs
220.127.116.11. motions of heavens tied to destiny
18.104.22.168. beginnings of astrology
5.2.6. more advanced mathematics
The study of mathematics was significantly advanced by the Babylonians. By 2000 B.C. they had developed the concept of fractions and division. They made tables of squares, square roots, cubes and cube roots, developed procedures for drawing geometric figures such as hexagons and octagons. They knew of several Pythagorean triples and had formalized certain rules of arithmetic, such as process of multiplication and division and the commutative and associated rules of addition.
Babylonian tablet circa 2000 B.C.E.
22.214.171.124. fractions and division by 2000 B.C.
126.96.36.199. squares, cubes, root tables
188.8.131.52. Pythagorean triples
184.108.40.206. rules of arithmetic
The Egyptian culture was significantly different from the other Mediterranean culture of the era. They were not really Mediterranean since the center of government was one thousand miles up the Nile River to the south of what is now known as Cairo. They were isolated and had only limited trade with other locales. This tended to keep their culture relatively untainted by outside ideas.
Unlike their contemporaries further to the north, the Egyptians left few records except on coffins. They were a stable, isolated society whose lives were under the control of the seasonal flooding of the Nile. Not unexpectedly, their calendar was based on the annual floods rather than on the stars.
Their calculation of pi was the most accurate of any until the heyday of Greek geometry in in 200 B.C. They did not understand the general relationship of the Pythagorean theorem, but knew of several triples and used them to construct triangle squares which were extensively used in construction of buildings.
5.3.1. Egyptians left few records, except on coffins
5.3.2. stable, isolated society, controlled by Nile seasonality
5.3.3. calendar based on flooding of Nile
5.3.4. better p (256/81 = 3.16)
5.3.5. no Pythagorean theorem but used triangle squares
220.127.116.11. several Pythagorean triples but no general law
This material will also be found in lesson 6, as we will study the Greek culture and its role in the nature of physical science in more detail in the next three lessons, as the Greek city states would become a dominant force in the Mediterranean for six hundred years, persisting well into the Roman era and having great influence indirectly into Europe to the west and into southern Asia on the east. For now we want to lay the groundwork and compare the Greeks to the others, while we are here.
5.4.1. aristocratic society with slaves
The Greeks were an aristocratic society with slaves. We can not condone slavery under any conditions, but the Greek slaves were treated well, more like servants than like slaves. The slaves performed most, if not all of the menial tasks such as cooking, cleaning, and construction. Slaves were also the artisans, making pottery, baskets, tapestries, and other goods for the rich citizens.
5.4.2. free time
The Greeks were one of the earliest cultures, at least in the West, to have large amounts of leisure time on their hands. People who are struggling to get by have little time for philosophy, politics, and critical analysis.
18.104.22.168. for sculpture, architecture, philosophy, politics, critical thinking
5.4.3. "discovered the mind" (Bruno Snell)
The historian Bruno Snell said the the Greeks 'discovered the mind', the pleasure and benefit of thinking for the sake of thinking. With the Greeks we see for the first time formalized speculation and organized thinking about the physical and spiritual universes.
The Greeks used abstraction, generalization, logic, and mathematics combined with the calendars, data, and navigational skills they inherited from the Sumerians, Babylonians, and Egyptians.
22.214.171.124. abstraction, generalization, logic, mathematics
126.96.36.199. combined with calendars, data, mathematics, navigation
188.8.131.52.1. from Sumerians, Babylonians, Egyptians
Although we tend to focus on the astronomy of the Western world, there were many cultures which had developed superior astronomies. Unfortunately, these were unknown to the western world during the development of our scientific paradigms in the early centuries of the Christian era and well into the teen centuries.
With the "discovery of the new world in the late fifteenth century and the subsequent devastation of its cultures, nearly all of the astronomical traditions, which were mostly oral, were lost forever.
Today, we can reconstruct some of the information where there are written records or artifacts. However, we will probably never know the true richness nor the extent of mysticism and astronomical skills of the new world.
The Maya, who flourished in Central America until the fourteenth century A.D. had developed and extensive astronomy, including a calendar which was much more accurate and up to date than any in the old world at the time. They built observatories which were aligned with the solstices and from them made extremely accurate observations. The civilization had been in decline for a number of years before the arrival of the Spanish conquistadors.
Many of the Native American tribes ad developed elaborate methods for keeping track of heavenly events. The Medicine Wheel, in Wyoming, is a large wheel of stones whose spokes point in the direction of significant events such as solstice sunrises and sunsets.
Calendars were carved into sticks to keep track of the passage of both the sun and the moon.
In the southwestern US were caverns into which beams of light shone only on particular days of the year. These apparently had great religious significance.
In India and China starwatching was reasonably far advanced. Calendars at least as accurate as the Mesopotamian ones had been developed and were in common use by 1500 B.C.
In the islands of Micronesia, Melanesia, and Poylnesia, navigation by the stars was at perhaps the highest level of anywhere in the world. Polynesian voyagers made multiple crossings of the Pacific Oceans guided by the stars and the directions of waves. Each island group of Polynesia had associated with a particular star or stars, so sailing to each island group involved sailing until the desired star group was directly overhead.
The extent and skills of the early transoceanic navigators was great, and it is nothing short of amazing that they were able to learn these skills while sailing the immensity of the Pacific.
Stonehenge is a group of standing stones on the Salisbury Plain in England. Enclosed by a circular ditch 300 feet in diameter, stones are arranged in four series. Two outermost form circles, the third is horseshoe shaped, the innermost is oval. Within the ovoid lies the Altar Stone. The structure was probably built between 1800 - 1400 B.C. Although the details of the purpose is not entirely clear, it is certain that it was used for astronomical purposes. From the altar stone the standing stones of the ovoid line up with solstice sunrise and sunsets. By moving stones around the two outer rings the locations of the sun and moon can be tracked, even to the extent of predicting eclipse seasons. Since there is not always an eclipse everywhere on earth during an eclipse season, it is remarkable that these motions could be tracked at all. Of course it is not known whether or not the users of the structures knew that they could predict eclipses since there would have been many predictions which did not produce eclipses.
We have now set the stage for the next step in our quest to understand the nature of physical science, which is removing the supernatural from the heavens and replacing it with a mathematical, mechanical model. That is what the Greeks did and that's the next lesson.
In this lesson we have seen how the ability of the human brain to reason abstractly allows for the use of language and mathematics, but also for symbolic and metaphorical thinking which are responsible for human culture including science as well as art, religion, and politics.
We examined some of the basics of counting and rules of arithmetic including the concepts of squares, cubes and roots.
We traced the development of mathematics, astronomy, and cosmology in Mesopotamia, Egypt and Greece through the fifth century B.C., where we will begin the next program with the presocratic Greek philosophers.
Language, mathematics, abstract reasoning: all of these are important tools for understanding the universe, and also for defining a shared physical reality. Still we see, before the Greeks, little distinction between the supernatural and natural, or between the physical and spiritual worlds.
That the physical and spiritual worlds should be so difficult to distinguish should not be surprising, taking what we have learned previously about the nature of paradigms and perceptions, and the role of culture in determining preferences..
Next time, we'll examine the Greek culture and it's contributions to The Nature of Physical Science.