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Before we're done with this lesson we will have learned of the important contributions of Gilbert and Bacon to the river of knowledge. We will learn Kepler's three laws of Planetary motion and use them to describe the orbits of the planets. We will learn the anatomy of the ellipse, and we will learn about the significance and implications of the laws.
Kepler's laws of planetary motion mark an important turning point in the transition from geocentrism to heliocentrism. They provide the first quantitative connection between the planets, including earth. But even more they mark a time when the important questions of the times were changing. By this time there were many intellectuals who favored the simplicity of a heliocentric system, but were unwilling to throw out the comfortable geocentrism of Ptolemy without good evidence. There remained the circular paradigm, lingering from the days of Plato. It was assumed that the heavenly motions were circular because that's the way it had always been.
The focus of the fundamental questions shifted from which system it "really" was, to what kinds of suppositions would required to justify a heliocentric reality.
What we mean is that it mattered less to keep the old theories just for the sake of keeping them. The evidence for and against both systems was being reworked and most thinkers of the time would have easily accepted heliocentrism if it was feasible.
The nagging thing that remained was the problem of motion. It's cause and its effects. Even if you accepted heliocentrism, there's still the problem of what makes the planets go in curved paths,regardless of the shapes of the orbits and the changing speeds of the planets as they move along their orbits.
Although the Copernican system made astronomical calculations easier, it was seen by most mathematicians as nothing more than another mathematical device. To nonscientists in Kepler's time, like most nonscientists today, who could care less about the mathematical simplicity, were found no serious objections to the geocentric theory and were unwilling to change.
There were two other individuals whose ideas added to the growing river of knowledge. William Gilbert was of Tycho's generation, and Francis Bacon was a contemporary of Kepler.
Gilbert was the Royal physician to Elizabeth I who was managing England's rise to world power status. He published a book, titled (De Magnete) which summarized everything known to date on the properties of magnetism and electricity. The concept which most intrigued Gilbert was the ability of the magnet to attract other magnets across empty space.
Among other things, Gilbert work was a treatise on lodestones and their use in navigation. He carved a piece of lodestone into a spherical shape, then used it as a model for earth to predict where the compass needle would point and how it would behave at different locations on a spherical earth.
Recall that, although the geocentric theory was still in favor, the flat earth idea had been lost forever when Magellan's fleet sailed around it earlier in the century.
Exactly what magnetism has to do with the planetary motions, we will get to in a little while.
Can you see a connection?
Regardless of that connection, Gilbert also made an important statement which went far in defining the new scientific paradigm that was about to bloom. In the preface to De Magnete he wrote:
Francis Bacon (not to be confused with Roger Bacon, see lesson 9) became Lord Chancellor under James I but in the same year pled guilty to accepting bribes and retired. Although his career as a statesman was tainted, his philosophical musings helped spurn the growth of experimental science in England.
His contributions to the river of knowledge were in his inductive approach to experimental science, later refined by Galileo, and in an essay titled The New Atlantis. This was a utopian society based on scientific principles. That such a society might exist was a completely new idea and helped to define what would become our modern scientific principles.
Induction means to go from the specific to the general. We will study this a couple of lessons down the road, but it's good to think ahead a little. You might want to look up "induction" in the dictionary. When you've done that, look up "deduction". Can you write a short essay comparing and contrasting the two terms, and using an example?
The modern method of trial and error in problem solving was stimulated by Bacon's statement, "Truth comes out of error more easily than out of confusion."
The laws are most simply stated in their modern form, largely because Kepler did not state them clearly. In fact, it is difficult to locate them in his writings, which ramble on and on about harmony, justification for abandoning the Ptolemaic system, and defense of Copernicus. To illustrate the style of writing and also the obscurity of the principles, I want to read you Kepler's laws in his own words.
Here's what Kepler wrote:
So everybody all together: What did Kepler say? I can't hear you. OK, seriously, this excerpt contains Kepler's first and second law. You do not have to repeat it, but you should look at this quote again after you have learned the laws and their meaning.
Can't you see why this work did not cause much stir. This is from De Harmonice Mundi, published in 1619, and is actually a restatement of the laws in a more concise form than in The New Astronomy, published ten years earlier.
It is a tribute to Newton's genius, as if he needed another, that he was able to see in this statement one piece of the solution to the gravitational puzzle.
One of the things you will note about this passage is that Kepler says it twice, very explicitly that the Sun is the source of planetary motion.
Well, I'll spare you Kepler's wording of third law, for now. But it was his favorite, and we'll come back to it after we examine the laws in more detail.
The planets, including Earth revolve around the sun in elliptical orbits. The sun is at one focus of the ellipse, the other is empty
This is such a simple statement, it is amazing that it was so difficult to produce.
In this picture we see that the planet is sometimes closer and sometimes further away from the sun. The second focus of the ellipse is a geometrical point of symmetry, but has no physical reality.
It would be useful at this point to take a closer look at the anatomy and properties of the ellipse.
The ellipse is a conic section, like the circle. We saw in the last lesson how the circle and the ellipse are related in terms of slicing or sectioning a right circular cone. Now we want to consider the properties from a different perspective.
The circle can be defined as the set of all points equidistant from a single point. In other words, all the points on the circle are the same distance from the center, which is a single point. That distance is called the radius of the circle.
What about the ellipse? The ellipse can be defined as the set of points equidistant from two points. Each of the two points is called a focus. Each focus plays the role for the ellipse that the center plays for the circle. We might make an analogy like this: The ellipse is to the circle as the rectangle is to the square. What does this mean?
It is not necessary for us to completely dissect the ellipse, as a mathematician might. But it is helpful to see how the ellipse is described and characterized as well as how it is constructed, and some of its properties.
There are two focuses, or foci, of the ellipse. The further apart the two foci, the more squashed the ellipse. The two foci are highly symmetrical; they are mirror images of each other.
Numerically, the focus is the distance from the center of the ellipse to one focus.
5.1.2. semi major axis
Unlike a circle which has a single radius, each ellipse has a long axis and a short axis. The axis is the length of a line cuts the ellipse in half. Any axis will pass through the center point of the ellipse.
The semi major axis is one half of the length of the ellipse, or the distance from the center to the furthest point on the ellipse..
5.1.3. semi minor axis
The semi minor axis is one half the width of the ellipse, or the distance from the center to the closest point on the ellipse.
Now that we have seen how the ellipse is described, let's look at the construction of an ellipse . . .
Watch the video program to see how to construct an ellipse.
But before we do that, let me remind you that you do not have to memorize and reiterate all the facts about an ellipse. The point is that the ellipse is a very Pythagorean figure. It has many interesting numerical and geometric properties. In the time of Plato the conic sections had not yet been described, and it was not known how similar the ellipse and circle really are. It wasn't until Euclid that we see the in depth study of curving plane figures such as the ellipse. Except for the circle, the classical Greek mathematicians considered mostly Polygon shapes.
It was not too much of a stretch for Kepler to consider substituting one geometric figure for another when the two were closely related. Seeing that it has these properties should help you to visualize the planetary motions and understand them better. In the same way that understanding how your car's engine works might make you a better driver, even if you can't take the engine apart and put it back together again.
You might want to try this at home. It's easy to do and it really helps to understand the ellipse.
Now that we have seen the construction of the ellipse we can look at some of its other properties. Hopefully you are beginning to understand why these shapes held so much fascination for the early mathematicians. Hopefully you are also beginning to ask questions like: Why do these shapes have mathematical properties? Were the Pythagoreans correct, is there magic in the number?
In constructing the ellipse I used the property than an ellipse is the set of all points equidistant from two points. Since the string is a fixed length and the distance between the foci is constant, then the ellipse is all of the points whose distance from the two foci is the same as the remainder of the string.
Why is this so? I don't know, and I don't think anyone else does either. It just is.
5.3.2 The Phthagorean Ellipse
The ellipse is a Pythagorean figure in more ways than one. It is not just a squashed circle, it is squashed in a very Pythagorean way.
If we call the semimajor axis a, the semiminor axis b, and the focus c, then the three numbers comprise a Pythagorean triplet, for all ellipses. You remember those? Three numbers which fit the Pythagorean relationship.
A line drawn from the focus to the point where the semiminor axis intersects the ellipse is exactly the same length as the semimajor axis. The three lengths form a Pythagorean triplet.
The measure of the degree of flattening of the ellipse is called the eccentricity. It is a number between zero and one which is focus divided by the semimajor axis.
For example, if the focus is zero then the eccentricity is zero, both foci occur at the center and the figure is a circle. So we can say a circle is really a special case of an ellipse with an eccentricity of zero.
On the other hand, if the focus is the same length as the semimajor axis, then the eccentricity is one and the figure is a straight line segment equal to the semimajor axis and the focus, and the semiminor axis is zero.
We can say the the straight line segment is a special case of an ellipse with eccentricity equal to one.
5.3.4. whispering gallery
A whispering gallery is an elliptical room with the two foci marked on the floor. The ellipse also has the property that any ray, like light or sound waves, which passes through one focus will pass through the other.
It's the equivalent of saying that if you had an elliptical pool table, then any time a ball passed over one spot (a focus) it would rebound off the bank (the ellipse) at such an angle so that it would roll over the other spot (focus).
So in the whispering gallery, a sound made by a person standing at one focus is reflected off the curved walls and focused at the other focus like sunlight through a magnifier.
As you might suspect, this is related to the constant distance of all points from the two foci.
In its motion around the sun, the line joining the planet and the sun sweeps out equal areas in equal time in all portions of its orbit.
The areas of the triangles A and B are equal everywhere in the planet's orbit.
Although A has a longer arc, its other legs are shorter. The two effects exactly counteract to give equal areas.
The area of a triangle is one half its base times its height. Although the planets move in curving arcs, for any short period of time the line is very nearly straight. So the area swept by the planet is one half the distance traveled in its arc times the distance from the planet to the sun.
Because the planet moves faster when it is closer to the sun, it moves through a larger arc in a given time. The larger arc is just offset by the shorter distance.
That the planets move at different speeds at different times is directly contrary to Plato's assertion that the speed of each planet must be constant as well as circular.
Click to see an animated movie of the ellptical orbit.
The third law is often called the harmonic law, for it is the most Pythagorean. The third law states that the planet's period and its average distance to the sun are related by the two-thirds power.
The period is the length of time for one revolution (the planet's year) and the average distance to the sun is the average of the semimajor and the semiminor axes.
There are several equivalent ways to state the third law.
a. The square of the time is proportional to the cube of the distance, where the time is the time for one period, and the distance is the average distance of the planet from the sun.
b. The ratio of the time squared to the distance cubed is the for all the planets, but the ratio for the moon is a different number.
c. The period and the average distance are related by the two-thirds power.
An easy memory device is to think of Times Square to associate the word "time" with the word "squared".
The table below contains the orbital numbers for the planets of the solar system, all of which revolve around the sun.
In this table T is the period in earth years, D is the distance from the sun in astronomical units (A.U.). The A.U. is a distance unit based on the earth's average distance from the sun. It is approximately equal to 93 million miles or 150 million kilometers.
Note that the numbers "T squared" and "D cubed" are almost identical for each of the planets. That is to say that the ratio of "T squared to "D cubed" is very nearly one for all the planets.
The numbers used are modern determinations and are somewhat more accurate than those available in Kepler's time.
Is the fact that they are not "exactly" the same, ie. the ratio is not "exactly" one, a significant contradiction to the law?
Here is a link to another site about Kepler's Laws.
As noted above, the laws are significant not only for their overthrow of the circular paradigm, although that is the sense in which we usually think of them. In terms of the continuity of ideas, just how radical was it really to break the circular paradigm? Was it really broken, or just bent?
Kepler spent much of his writing effort arguing that he was not really breaking anything. His writings contain logical arguments, similar to those of Ptolemy, but reaching the opposite conclusions in many cases.
It is interesting how, with good data to back one up, it is easy to argue away the objections to the moving earth which had been around since Aristotle's time.
The laws certainly supported the heliocentric theory, but that is the only way in which they were really Copernican. Kepler's orbits had no epicycles, no spheres within spheres, no other devices, moving or stationary. It took a total of one slightly flattened sphere per planet. Overall the elliptical orbits made for a much simpler understanding of the planetary motions, and more importantly, allowed a much easier and much more precise method of calculating their future whereabouts.
8.1.1. needed "reason" to stay in orbit
From the Scholastic perspective, the planets now needed a reason to keep moving, an way to keep them in orbit, and a way to explain how the elliptical orbits could remain stable.
8.1.2. central force concept
It is with Kepler that we see the beginning of the concept of central force, that is a force which acts continuously on the planets to keep them moving in closed, stable paths from one orbit to the next. It was apparent to Kepler that the force was directed towards the focus of the ellipse, but he could not describe the nature of the force.
8.1.3. Kepler guessed magnetism
He guessed it might be magnetism, largely because of Gilbert's De Magnete, published in 1600. Knowing that magnets can exert forces through empty space there was no reason to suppose that planets could not do likewise.
The sacred geometry of the universe is not violated. The planetary motions are describable in geometric terms, even if they were different terms that the ancients thought.
It is Pythagorean because it is harmonious.
It is Platonic because the ellipse is almost a circle so the circular paradigm is only bent, not broken.
It is Euclidian because it is a conic section, a family of shapes of which the circle is not only a member, but the exemplary member.
8.2.1. is Pythagorean
8.2.2. is Platonic
188.8.131.52. gave up circles but ellipse is "almost" a circle
184.108.40.206. doesn't violate circular paradigm, only bends it
8.2.3. is Euclidian
The mathematical relationships Kepler discovered made one further statement concerning the heliocentric/geocentric controversy. Kepler's critics could had argued that the ellipses were just one more device rather than a cosmology.
But from Kepler's third law, when we compare the dees squared and the tees cubed we find all of the planets have the same number which represents that ratio.
ALL of the planets, including earth.
Here, in the Pythagorean numbers was the proof that earth was a planet just like all the rest. You might say that the proof was in the Pythagorean pudding.
We might also note that the number representing the third law ratio is different for only one heavenly object.
Can you guess which one?
8.3.1. all planets have same constant, including Earth
8.3.2. moon is different from other planets
The last significant feature of Kepler's laws is that is was the first mathematical law which linked the motions of the planets together.
Aristotle's cosmology had claimed that the motions were linked, but it was a qualitative model, not a quantitative one. You recall that Ptolemy had given up on the concept of linking the motions because he found it unnecessary in order to calculate the motions. Well, the third law links them right back up again, but in a heliocentric framework, not a geocentric one.
8.4.1. previously math was for calculations only
Prior to Kepler's formulation of the laws, mathematics was used for calculations only, not for recognizing relationships. These laws were, in fact, the first general numerical relationships in physical science.
8.4.2. a quantitative connection requires an explanation
As far as Kepler's influences on Newton half a century later, it was the necessity for an explanation of some kind for the relationships he discovered, which stimulated Newton's curiosity and helped him to consider the motion of the planets in terms of the motion of the apple.
In this program we have summarized the influences of Gilbert on Kepler's work and we looked briefly at Francis Bacon whose preference for experiments would drive Galileo's investigation.
We saw that Kepler's laws of motion advanced a heliocentric view, but with the planets moving in elliptical rather than circular orbits, with the sun at a focus rather than at the center, sweeping out equal areas in equal times, and having all the same relationship between period and distance.
We also learned the properties of the ellipse in order to reinforce the Pythagorean nature of this conic section.