The Moon is distant from the Sea
And yet, with Amber Hands
She leads Him – docile as a Boy –
Along appointed Sands
He never misses a Degree
I love these lines from Emily Dickinson, for several reasons. First, as a feminist, I am pleased to see that the leader here is a woman and the follower a man (even more radical in the nineteenth century). Secondly, Dickinson subtly affirms the idea of a lawful nature with “He never misses a degree.” The tides are precise. The physical cosmos does not unwind whilly nilly, but follows definite and predictable patterns.
It is perhaps remarkable that nature should be lawful, that every effect has a cause, and that we can quantify that relationship. If it were otherwise, the universe would be a frightening place. Wheelbarrows might certainly begin floating in mid air, day might turn into night and back again at random moments, dead people might suddenly reappear. Some scientists think that the physical universe could not exist at all unless it were orderly, logical, lawful. Otherwise, the universe would be self contradictory, as if two plus two equaled four and two plus two equaled five.
As both a scientist and a humanist, I am interested not only in the lawfulness of the cosmos, but also in what that lawfulness tells us about ourselves. Why are we attracted to logic, predictability, cause and effect? Most of the time. On the subject of rules and patterns, I think we human begins are absolutely schizophrenic. We are drawn to the symmetry of a snowflake, and we also marvel at the amorphous shape of a cloud floating high in the sky. We appreciate the regular features of animals of a pure breed, and we are also fascinated by hybrids and mongrels that do not fit into any classification scheme. We honor those people who have lived upright and sensible lives, and we also esteem the mavericks who have broken the mold. In some perplexing and ill-understood manner, we human beings with our oversized craniums seem to have a fondness both for the predictable and the unpredictable, the rational and the irrational, regularity and irregularity. Yes, we are certainly a difficult mess of self contradictions, like a school arithmetic book with two plus two equaled four and two plus two equaled five.
The concept of a law goes back at least four thousand years, to the ancient Assyrians and their Code of Ur-Nammu. Those first laws were, of course, rules for behavior in human society. Quantifiable only in the number of shekels of silver owed or of quarts of salt poured into the mouth for each specified infraction. For example: “If a man proceeded by force and deflowered the virgin slavewoman of another man, that man must pay five shekels of silver.” (I would imagine that Ms. Dickinson might have recommended a different kind of punishment.) Our ancient ancestors knew also about rules of geometry. The Babylonians understood that the ratio of a circle’s circumference to its diameter is a universal number (which we denote by π, pronounced “pi.”) Any circle of any size, drawn anywhere, obeyed this relation. These were the antecedents of “laws.”
One of the first human beings to formulate a law of the physical world was Archimedes (288 BC – 212 BC). Here is his “law of floating bodies” (ca 250 BC):
Any solid lighter [less dense] than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced.
We can speculate on how Archimedes arrived at his law. At the time, balance scales were available for weighing goods in the market. The scientist could have first weighed an object, then placed it in a rectangular container of water and measured the rise in height of the water. The area of the container multiplied by the height of the rise would give the volume of water displaced. Finally, that volume of water could be placed in another container and weighed. Undoubtedly, Archimedes would have performed this exercise many times with different objects before devising the law. He probably also performed the experiment with other liquids, like mercury, to discover the generality of the law.
All laws of the physical world are like Archimedes’ law. They are precise. They are quantitative. And they are general, applying to a large range of phenomena.
Let’s explore the laws of nature ourselves. Let’s make a pendulum, requiring only a string, a key, and a watch. Cut off a length of string, attach a key at the end of it to provide some weight, and let the key swing back and forth, as shown in this video:
Cut off a length of string, attach a key at the end of it to provide some weight, and let the key swing back and forth, as shown in this video: [Video] Now, let’s see if there is any definite relationship between the length of the string and the time it takes the pendulum to make one complete swing, called the period. I suggest that you let the pendulum swing back and forth 4 times, measure how long that takes, and then divide by 4 to get the time for one swing, the period. Letting the pendulum go several times averages out any small mistakes in starting and stopping your stopwatch. Such a recognition of small errors and compensating for them is part of the “scientific method.”
If you have a good supply of string and a pair of scissors, you can make pendulums of different lengths. For a string length of 1 foot, you should get a period of about 1.1 seconds; for a length of 2 feet, you should get a period of about 1.6 seconds; for a length of 4 feet, you should get a period of about 2.2 seconds. But don’t take my word for it. Make your own pendulums and do the experiments. If you want, you can plot your data, as in the figure:
If we draw a curve through our data points, it looks like this:
The more data points we have, the smoother the curve. In fact, you can now use this curve to predict the period of a pendulum even before you have made it and timed it. Try that out. Make a new pendulum at a length you have not made before, use the curve to predict its period before letting it swing, then measure its period. Voila! You can make a successful prediction with the law you have found for pendulums. The curve is the law. That curve is called a parabola.
Mathematically, the relationship is:
Period = 1.1 seconds x √length
where √ is the symbol for “square root” and can be found on most calculators. This law for pendulums was first found by Galileo around the year 1590. It is not true because we believe everything the great Galileo says, or because we read it in some book, or because we want it to be true. It is true because nature tells us that’s the way she operates. Nature is lawful. And you have just discovered a law of nature yourself.
There are a couple more things I would like to say about the pendulum law and about the parabola. We did indeed discover the law experimentally. But with a bit of college physics, using very general equations for force and acceleration not particular to pendulums, we could also have theoretically derived the law, even without ever making any pendulums. When we do that, we find that the 1.1 seconds in our formula actually comes from the number 2 x π / √g , where g stands for the acceleration of gravity. On earth, g is 32 feet per second per second. On the Moon, the acceleration of gravity is about 5.3 feet per second per second, so the 1.1 seconds in the above law for earthly pendulums would be replaced by 2.73 seconds for the moon. The rest of the formula is the same. This kind of calculation is a great testament to the lawfulness of nature, and also to the human mind’s ability to discover those laws. Not only can we make find patterns and laws for pendulums on earth, and make predictions based on those patterns, but we can then make predictions for pendulums elsewhere in the universe. Scientists believe that the laws of nature are universal, holding everywhere and at all times in the universe.
The second thing I would like to point out about the parabola – which hardly needs pointing out – is that it is a beautiful shape. It appeals to us aesthetically. Of course, beauty is in the eye of the beholder, so it is we human beings who judge a thing to be beautiful or not. Nature itself just is what it is. It happens that many phenomena in nature take the beautiful shape of a parabola, like the trajectory of a ball thrown sideways. And some architectural stuctures as well. Bridges, for example.
It turns out that a parabolic suspension is often used in the construction of bridges because that particular curve distributes the weight equally. So there are engineering reasons as well as aesthetic reasons to find parabolas in our world, both the natural world and the human-built world.
In sum, as far as we know, the physical cosmos is a lawful place. In my next posting, I will further explore how we human beings relate to those laws.
Notes
“The moon is distant from the sea” Emily Dickson, poem 429. My copy of Dickinson’s poems is The Complete Poems of Emily Dickinson (Boston: Little Brown, 1961).
“If a man proceeded by force . . . “ O.R. Gurney and S.N. Kramer, “Two Fragments of Sumerian Laws,” Assyriological Studies, 1965, April 21, no. 16, pgs. 13–19. [See also http://en.wikipedia.org/wiki/Code_of_Ur-Nammu]
“Any solid lighter [less dense] than a fluid . . . ” “On Floating Bodies” (ca 250 BC) in The Works Archimedes, ed. T.L. Heath (Cambridge UK: Cambridge University Press, 1897) Book I, Prop 5.
Suggested Readings
The Character of Physical Law, by Richard Feynman (Cambridge MA: MIT Press, 1965)
Re our fascination with order & disorder, the rational & irrational: I am amused by the lawful ratio being a universal number that is: never ending. Nothing is quite quite completely fixed..
I’m amused that, with our love of logic and illogic, rationality and irrationality, the ratio’s” universal number ” is– never-ending.
Re our desire for logic and illogic, the rational and irrational: I’m amused that the circle ratio, the “universal number” is– never-ending!