This is the last of our lectures on Greek mathematics, so we'll finally start talking about astronomy. As we said, astronomy has been one of the dominant moving forces behind the development of mathematics, and in this lecture we're going to look at how astronomy led to the development of trigonometry.

Most people think of trigonometry as something that's used for land measurement, and certainly it is used in surveying, but that's not where we get it. Trigonometry actually comes to us from problems in astronomy that we'll talk about in this lecture. Its applications to problems in surveying actually didn't come about until quite recently. The earliest reference we have for the use of trigonometry for the purposes of surveying, is a book written in 1595 CE. For most of the history of the development of trigonometry, it came from astronomy and was being used exclusively in order to study astronomical phenomena.

We also think of trigonometry as being defined in terms of the ratio of the sides of a right angle triangle, but again, that's not the way people have thought about the trigonometric functions of sine, cosine, and the tangent, throughout most of the history of mathematics. We're going to see that those functions come out of the problem of trying to determine the length of a chord of a circle, of finding the straight line distance between two points on a circle. This idea of defining the trigonometric functions in terms of ratios of sides of a right triangle, only comes about in the 18th century.

Trigonometry is extremely important, not just for the trigonometric functions and the role they're going to play in mathematics, but in many different ways. It turns out to be one of those abstractions that is extremely important in many different contexts. Yet they also really begin to introduce the idea of a function, which we'll talk more about later in this lecture and the next. This is the idea of having a process where you can take an input that can be any real number whatsoever, and will produce a well defined output, no matter what is input into the function.

We need to start by thinking about the solar system, and most of us have a standard picture of a bunch of balls, with the bright yellow one at center that represents the sun. A small blue ball represents earth, and a little red one is Mars. All of these balls lie out there, with nice elliptical paths that are marked out which the planets follow, going around the sun.

Some people wonder how the ancients could be so confused that they didn't realize that the planets go around the sun. Of course this traditional picture of the solar system that we have is something that nobody has ever seen. Nobody has ever put a satellite into that position and taken a picture of the solar system. Even if they did, that's not what they would have seen. The planets are much too small to show up as nice round balls, and simply would have been dots of light. Planets also don't leave vapor trails as they travel around the sun. You don't get these nice elliptical paths coming out.

What the astronomers have seen is simply the night sky, so it's important to begin by thinking about what you can see in the night sky and how the astronomers began this process. The first thing you observe by looking at the night sky is that the relative position of the stars is fixed. This is why we create the constellations, why we are able to identify them as certain groupings of stars which always stay in the same relationship.

Yet even though the relative position of the stars stays fixed, the stars themselves are moving in the sky throughout the night. In fact, if you take a time lapse photograph, pointed at the north pole, you'll see the stars actually moving over time in their tracks. The one star that stays fixed is the north star or polar star. That does not move, but all of the others are turning in a great celestial sphere, around this north star.

One of the things that was observed in antiquity, far into pre-history, so that we have no idea when it was first observed, was the fact that the sun does not have a fixed position against this dome of the stars, but is constantly changing its position. One can see that when the first stars you see on the horizon at sunset, or the last stars you see on the horizon at sunrise, are going to be different constellations at different times of the year.

It was quickly realized that the position of the sun against the dome of the night sky is very important. Knowing where the sun is located, tells you where you are in the process of the year. So it tells you when to expect the rains, when to expect spring to come, when to plant, and when to expect that the harvest is ready to be brought in.

So the position of the sun against the heavenly dome is extremely important thing to be aware of, and the actual path of the sun is what we call the ecliptic. That comes from the means that astronomers found for determining where the sun is located. It's very difficult to see where the sun is against the stars, because when the sun is out, it's so bright that you can't see the stars around it.

Yet ancient astronomers figured out the path of the sun by using lunar eclipses. When the sun, earth, and moon, are all lined up, the moon is at the exact point in the sky, opposite form where the sun is located. So the moon, during a lunar eclipse, is located where the sun will be exactly six months in the future.

By realizing this, astronomers were then able to gradually plot out the points where the sun would be located on the sky, at different times of the year. Once you begin to collect data on lunar eclipses over a period of some 700 years, you get a very accurate track of where the sun is. The position of the sun at different times in the year, then tells you what to expect from the seasons. What we get are the signs of the zodiac, which are the constellations that give a rough idea of where the sun is located.

There's a nice illustration of this in a frontispiece from Ptolemy's great work, the Almagest, that we'll talk about a little bit later in this lecture. What we see in this frontispiece is what's called an armillary sphere, an object which shows the relative position of the sun and various stars. The earth is located in the center of the object, and you see the band of the ecliptic, the path the sun moves along, with the signs of the zodiac on it.

There are a number of problems that come up with the ideal situation for what's happening with the motion of the sun and stars. One of the things that was observed fairly early on, was that in addition to the fixed stars, there were stars that moved. In Greek they became known as wanderers or planetes from which we get our word planet. So the planets are the stars that move, and it was observed that these starts that moved, also followed the ecliptic.

To the ancient astronomers, astrologers, or mathematicians, all these terns really being synonymous, they observed that these points of light followed the same path as the sun. If the position of the sun was so important for what's happening on earth, than the position of these smaller lights must also be important for what's happening on the earth. They are much smaller than the sun, so that one would expect their influence to be much more subtle. Yet nevertheless they should have an important influence, and this is why astrologers/astronomers started tracking the position of the planets in order to figure out what's going on.

This was Aristotle's view of an earth-centered universe, with a great sphere of the stars, with planets traveling on this path of the ecliptic. Yet it ran into some problems, the first of which was the problem of retrograde motion. The fact that the planets do not keep moving along this great curve of the ecliptic, but every once in a while they will slow down and back up. This was a problem that Aristotle put out to the scientists of his time, the mathematicians and astronomers, to explain what's going on. If in fact we have circular motion going on out in the heavens, why do the planets occasionally stop, back up, stop again, and then start going in the original direction?

We understand this phenomenon today, because we realize that a planet such as Mars is circling the sun, not the earth. The earth is also circling the sun and is closer to it, so that it takes less time for the earth to go around the sun. So every once in awhile, the earth is going to lap Mars. When earth comes closest to Mars, it's actually going to pass it and make it appear from earth, the Mars is backing up. Yet when the earth is far enough along, it will again look like Mars has begun moving in its normal direction. In the case of this animation, it's going around the sun, or as it looks like from the earth, Mars is again going around the earth.

This problem of trying to explain retrograde motion, this backing up motion of the planets, was something that Aristotle posed and first solved by a person named Aristarchus of Samos 310-230 BCE. He suggested that maybe retrograde motion was coming from the fact that Mars does not go around the earth, but maybe Mars and earth both go around the sun. He actually came up with the correct solution.

Yet of course nobody believed him, because his solution was totally preposterous. There's no possible way that the earth could be circling the sun. We would have to be traveling at incredible speeds, such as that of earth around the sun itself, some 67,000 miles per hour. How could we possibly be traveling around the sun at that kind of speed, and not be aware of it? So people began to look for other solutions.

The person who would eventually come up with a solution to retrograde motion that would generally be adopted, was Appolonius of Perga 269-190 BCE. This is the same Appolonius we met in the last lecture, famous for his work on comets. He's the one who suggested that's what really going on is that as Mars circles the earth, Mars itself is actually located on a little circle who's center circles the earth. So what happens is that we have Mars going around and around a little circle, and the center of this little circle is going around the earth.

So Mars actually executes a kind of looping motion, as it travels around this little circle called an epicycle, or outer circle. It goes around this little circle, the epicycle, as the center of the epicycle goes around the earth. This preserved one of the foundational Aristotelian viewpoints about the nature of the heavens, which was that everything that goes on in the heaves must be built out of circles.

Now there are other problems with trying to explain what goes on in the heavens. One such problem observed by Aristotle was finally solved by Hipparchus of Rhodes 170-126 BCE, was the problem that if you look at the length of the seasons, they are not equal. The ancient astronomers knew where the sun was located at the summer solstice, as well as the winter solstice. These two points are exactly opposite along this great circle of the ecliptic.

One then looks at right angles from this line connecting the summer and winter solstices, and that gives you the vernal or spring equinox, and the autumnal equinox. The seasons are marked by the time from the winter solstice to the spring equinox, which gives you winter. The spring equinox to summer solstice, gives you spring. Summer solstice to autumnal equinox gives you summer, and so on. One would expect that if the sun were just traveling in a nice big circle with the sun at the center, each of the seasons would be the same length, yet their not.

It may not seem this way, but actually winter is the shortest season, and summer is the longest. The difference between the two is not subtle. Winter is about 89 days long, from the winter solstice to the spring equinox, while summer is more than 4 days longer, at 93.625 days (about 93 and 5/8 of a day). So it's more than 4 days longer from the summer solstice to the autumnal equinox.

It was Hipparchus of Rhodes who came up with the idea to explain this. It could be explained if rather than putting the earth at the center of the universe, you had the earth slightly off-center. So if the earth is slightly off center, it's closer to where the sun would be in winter, which means you would get a smaller arc that the sun has to travel during winter. You're going to get a longer arc than the one the sun has to travel during summer.

Of course this immediately begins to upset the Aristotelian world view, because it moves the earth off the center of the universe, yet that seems to be the way to do it. Hipparchus was looking at this problem and trying to figure out how off-center the earth had to be. That's where we get to the problem of calculating the length of a chord. In order to do this, Hipparchus essentially invented trigonometry.

What he did was invent a way of determining the length of a chord that connects two points on the arc of a circle. So this would become the basic problem of trigonometry, where you have an arc of a circle, and you want to figure out the length of the straight line that connect the two points at the end of that arc.

So first of all, one has to decide how to measure that arc of circle, such as in degrees? This has been inherited from the Babylonians who measured arcs or circles in this way. People often do not understand that today we use degrees to measure how much something has turned. We say that something is turned 90 degrees, which means it's made a quarter turn.

That is very recent interpretation of degrees, coming in the 18th century. Up until then, degrees were a measure of arc length, of how far it had traveled around the outer edge of a circle. The Babylonians divided it into 360 degrees, almost certainly because it's approximately the number of days it takes the sun to travel completely around the earth, or the earth around the sun.

Interestingly the Chinese divided a circle into 365 and 1/4 degrees, which is nice in the respect that the sun moves exactly one degree everyday. Yet working with a circle of 365.25 degrees is very difficult to do. So the Babylonians made a compromise by picking an easier number to work with, 360, which means that the sun does not move exactly one degree across the ecliptic each day, but a little bit less than that.

These degrees of arc length then, would be subdivided according to the Babylonian system into 60ths of a degree, which we call a minute. This word actually comes from the Latin "pars minuta" (small part). The minute is divided up, in turn, into 60 seconds of arc, so that a 60th of a 60th of a degree is called a second. This word comes from the Latin phrase "pars minute secunda" (second small part).

So we're measuring the arc length in degrees, minutes, and seconds. Given any particular arc length, we want to find the length of the chord that connects the two end points of that arc. Now that's going to depend on the value of the radius. So as we make that radius larger, that chord length is going to change. So the chord length is always described in terms of the radius.

If we take an arc of 90 degrees, a quarter of a circle, and if we know the radius as simply the length of the diagonal (1), then that chord of 90 degrees of arc is going to be the radius multiplied by root 2 (r√2). If we take a chord which corresponds to an arc of 60 degrees, then we simply get an equilateral triangle formed by the two radial lines that go out to the circumference, so that the chord of 60 degrees is exactly one radius (1). The Greeks were also able to figure out the exact value of a chord of 72 degrees, which turns out to to be √(5-√5)/2.

We've said that this problem of finding chord lengths is where we get trigonometry, which in fact does correspond to these chords. So we now think about taking an arc and then taking the chord which connects the two endpoints of the arc, using the terminology of bows and arrows. That arc, or bow, has the chord or string which connects the two endpoints of that arc, and we then refer to the line from the center of the circle out to the midpoint of our arc, as the arrow.

We can now connect the radial lines that go from the center of the circle, down to the bottom of the arc, and the center of the circle up to the top of the arc, which really does look like a bow and arrow at this point. If you consider not the full chord length, but only half of it, where it attaches at the top and then down to the arrow, we have precisely the sine.

Now the Greeks did not work with the sine, but with what today we would think of as twice the sine, the chord. The idea of working with half that quantity, working with the sine, is something that would come out of India. So we'll talk about that in the next lecture, how Indian astronomers picked up the ideas of Hipparchus and the other Greek astronomers, developing them into the kind of trigonometry we understand today. Instead of working with a full chord, they worked with a half chord, what they called the sine. Then you also get the other trigonometric functions of cosine and tangent, the latter being invented by Islamic astronomers.

In the remaining time we'll talk about the greatest astronomer of the Hellenistic period, and also one of the greatest mathematicians of this time, Ptolemy of Alexandria 100-170 CE. We've run across a number of Ptolemys before, Ptolemy I, II, and III, all rulers of Egypt in Alexandria. This is a scientist who happened to share the same name, and lived during the second century CE.

Ptolemy took the ideas of Hipparchus and put them into an incredible work on astronomy, one that runs to 13 books. It's so comprehensive and does such a great job of explaining astronomical phenomena, that just as when Euclid's Elements came out, and people threw away any mathematical works they had before Euclid, the same was true of Ptolemy. When his astronomical work came out, everybody threw away any other astronomical texts out there. So none of the astronomical texts before Ptolemy exist anymore. All we have are a few references to what was contained in them. It's Ptolemy's own references to Hipparchus which tell us what Hipparchus had been doing.

This great work of Ptolemy's is known as the Mathematiki Syntaxis (Mathematical Collection), and would come to be known as the Megisti Syntaxis (Great Collection). The Islamic astronomers would then translate this into Arabic, then coming to be known simply as al-Magisti (Great Work), taken from Megisti Syntaxis, the al-Magisti. When this then was translated from the Arabic into Latin, for European astronomers, and European scientists, they took the Arabic term, the al-Magisti, and they called the work the Almagest. So that's the term by which it is commonly known as today.

One of the great results that Ptolemy was able to do in this book, in addition to all the astronomy and mathematics in it, was something we'll focus on now. This is the part in which Ptolemy constructs a table of values where one could put in the arc length they were interested in, and then read off what the chord length would need to be.

As we've said, he started with certain chord lengths that were fairly easy to determine, and of course one of the things that you have to decide as you come into this, is what your radius is going to be. The circumference of a circle is 360 degrees. It's going to be easiest to work with chord lengths if we measure them in the same units that we use to measure arc length. So we really want to measure the chord length in degrees.

Well if we have a circle of circumference 360, that means that we're going to need a radius which is 360/(2pi), which is a little bit more than 57 degrees. It's about 57 degrees, 17 minutes, and 45 seconds. That's very difficult to work with, so what Ptolemy and the other early astronomers usually did, was to convert everything to minutes. So the total circumference of the circle, would be 360x60=21,600 minutes. Then the radius in minutes is approximately 3,438.

Ptolemy was then able to take the values of chord length that he knew, for example 72 degrees and 60 degrees, and then work out how the chord length of a given arc could be determined if you knew the chord lengths of two arcs that added up to that. Or equivalently, if you knew the chord length of one arc, and the chord length for a smaller arc, he found the formula for finding the chord length of the difference.

This is equivalent to our modern formulas for finding the sine of a sum of angles, or the sine of a difference of angles:

sin(a-b) = sina x √(1-sin²b) - sinb x √(1-sin²a)

So by using these he was able to find an exact value for the chord length of 12 degrees:

chord 12° = 2sin6° = 2sin(36°-30°)

Then you can also use this formula if you know the chord length for a given arc. You can find the chord length for half of that arc, so once you know the chord length for 12°, you can find the chord length for 6°, 3°, 3/2°, or 3/4°:

sin3° = √(1-√(1-sin²6°))/2

sin3/2° = √(1-√(1-sin²3°))/2

sin3/4° = √(1-√(1-sin²(3/2°)))/2

Yet that doesn't give you the chord length for 1 degree, and this was something Ptolemy really wanted to work out, to quite a bit of accuracy. One of the advantages of using the same unit of measure to measure the arc length and chord length, is that as we take smaller and smaller arcs, the ratio between the chord length and the arc length should approach one.

So for very very small arcs, we can assume that they're approximately equal. That means that the 3/4 degree arc, divided by the chord of 3/4, should about equal the arc of one degree divided by the chord of one degree. That tells you that the chord of one degree is very close to the reciprocal of 3/4, which if 4/3, times the chord of 3/4:

chord 1° ≈ 4/3 x chord 3/4°

He knew the exact value of a chord of 3/4, and could then use that in order to find a very very close approximation to the chord of one degree, and once he had the chord of one degree, he could find the chord of half a degree. Once he had that, and he had this formula, if you know the chord of one arc length and another, you can add them together. So he was able to build up an entire table of chord lengths, in intervals of half a degree.

If you know the chord length for half a degree, that corresponds to the sine of a quarter of a degree. Effectively what Ptolemy was able to construct, was a table of values of the sine, down to just a quarter of a degree at a time.

Now Ptolemy was 2nd century CE, and as we've said, Greek mathematics really ends at about 400 CE. Fortunately though, this was not at the end of the developments in astronomy, because these astronomical works that came out of the Eastern Mediterranean would be imported into India, where the astronomers would pick up this idea, carrying it much further, also bringing in many new ideas in mathematics. In particular, one of the things we'll see in the next lecture, is how this work on trigonometry would lead to a general appreciation for polynomials.