Rebirth of Reason


Archimedes and the Principle of Derivation
by Adam Reed

One of the most glorious traditions of the classical European curriculum is the teaching of Archimedes' Principle in the fifth or sixth grade of Middle School. According to the account of the first century BC Roman architect Vitruvius, Archimedes was asked by Hiero II of Syracuse to determine, without damaging the object, whether a golden crown commissioned by Hiero was indeed made of the pure gold that the king gave to the artisan for the task. Vitruvius wrote that Archimedes "on getting into a tub observed that the more his body sank into it the more water ran out over the tub. As this pointed out the way to explain the case in question, he jumped out of the tub and rushed home naked, crying with a loud voice that he had found what he was seeking; for he as he ran he shouted repeatedly in Greek, "Eureka! Eureka!" meaning "I found it! I found it!"

The story was not told merely to exploit a tale of absent-minded nudity as a tool for getting the attention of middle-school children. The attention was used to tell the story of the Principle of Derivation, a crucial epistemological and ontological lesson implicit in Archimedes' solution to the Problem of the Golden Crown.

Imagine Archimedes' tub as large enough to float in, and full to the rim. Imagine placing the tub on a pan to contain the displaced water, and the pan on a scale, to measure the combined weight of the tub and the water and the pan. Finally, imagine Archimedes suspended from a second scale, like this, to measure his weight. Now when Archimedes gets in the tub, spilling some water into the pan, the total of the weights on the two scales must be the same before and after, because it is the total weight of the same things.

Of the downward forces on the scale below, the force exerted by the tub is the weight of the tub itself, which does not change, plus the force exerted on the tub by the water it contains. But the force exerted by water pressure on any point of the tub's inner surface depends only on the height of the water column above that point - and the depth of the water has not changed. So the total weight exerted on the lower scale, by the tub and its content, stays constant. The only change in the weight measured by the lower scale is the additional weight of the displaced water that spills from the tub into the containment pan. But the total weight on the two scales must stay constant. Therefore the weight of Archimedes, as measured by the upper (suspension) scale, must decrease by exactly the weight of the liquid that his body displaced.

What is causing Archimedes' weight to decrease? The pressure of water at a given depth is the weight of the column of water that would be supported by a horizontal surface placed under the water at this depth. Imagine an object with vertical side surfaces H units high, and with horizontal top and bottom surfaces of area A, at depth D from the surface of the water to the top of the object. If water has a weight W per unit volume, the weight pressing down on the top surface of the object will be D times A times W: DAW. Being a liquid, water distributes pressure equally on all surfaces at a given depth, so that the upward force of water pressure on the bottom surface of the object is exactly the same as the pressure on any surface, even an up-facing surface, of the same area at at the same depth. That is, the upward pressure from the bottom surface at depth (D+H) will be (D+H)AW. The net force of buoyancy on the object, (D+H)AW upward minus DAW downward, is HAW. Height H times the top/bottom area A is the volume of the object, so HAW is the weight of the water displaced by the object's volume. We can fill any three-dimensional shape with little slicy objects of this kind, so this result will hold for submerged objects of any shape.

And so, Archimedes was able to determine whether the golden crown's volume was the volume of gold corresponding to its weight. As laws of physics go, Archimedes' Principle is astoundingly simple: the buoyant force is equal and opposite to the weight of the displaced fluid. What makes it important is its derivation. Before Archimedes, Greek science worked pretty much along the lines of pre-modern (and, unfortunately, also post-modern) empiricism: this is what we observe in the world, and the world just happens to be that way. What Archimedes demonstrated is that the world does NOT "just happen to be this way." When one understands, with logic and math, why the things one observes are the way they are, one also understands that the laws of physics do not just "happen to be" the way they are - that they must be the way they are, by necessity, and that they could not have been otherwise. This is Archimedes' second, unstated principle: the Principle of Derivation. It is the more important of the two. There are not, after all, a great many situations in which one's life depends on knowing that the buoyant force is exactly equal to the weight of the displaced water. But knowing that behind every regularity of empirical observation is an operation of natural law, identifiable with reason, known with certainty when identified, and deriving its necessity from the nature of reality itself - that is a truly foundational principle of Life Qua Man.

Full disclosure: my own first commitment to the principles at the core of Objectivism did not come from Ayn Rand. It came from learning the derivation of Archimedes' Principle in the fifth-grade science classroom of an Israeli school. I would later learn from Rand and others an ocean of implications and corollaries, but I still look to Archimedes' Principle of Derivation as the Source.

Some years back, as an elected member of a suburban school board in New Jersey, I was sent a sample middle-school physical science textbook by a textbook publisher looking to extend their market. The publisher was best known for printing, on demand from "Red-State" Christianist school boards, textbooks that present "Creation Science" as an "equally valid" interpretation of the evolutionary record. But this was a textbook on physical, not biological sciences, and I felt obliged to give it a fair reading. Eventually the text got to Archimedes. The golden crown was there. The overflowing bathtub was there. "Eureka, Eureka" was there. The law of buoyant force was there. Its derivation was not.

I later learned that the same publisher pioneered a new generation of school "science" textbooks promoting something called "Intelligent Design." "Intelligent Design" is the Creationists' second line of defense, after the massive evidence for Evolution breaks through the first. The defense of the doctrine of Divine Creation is fundamental to the politics of repression. If there is a God who created the universe - if there is a God who created you - then you are God's property, and you can have no rights that are not trumped by God's Will. "Intelligent Design" is the idea that the universe, if it were not created by an Intelligence, could have turned out any old way, including may ways that would have excluded the eventual emergence of human life. So if the universe is such that human life did emerge, this must be the result of it having been "Intelligently Designed" for the emergence of human life.

The foundation of "Intelligent Design" is the post-modern (and also pre-modern, pre-Archimedean) idea that what we call the Laws of Physics, are just arbitrary conventions, imposed on a happenstance world by Masculinist, order-seeking, Patriarchal minds. A middle-school student exposed to the derivation of Archimedes' Principle, may also understand the Principle of Derivation itself - and its corollary: that the physical laws of the universe must be what they are, by unavoidable necessity. Archimedes' Principle of Derivation excludes the idea that the physical laws of the universe could have been different, and that the universe needed some kind of divine intervention in order to be what it is.

If you or any of your children did not learn the Principle of Derivation, learn it and teach it now. If you have a child in, or headed for, Middle School, check the school's physical science textbook and make sure that the derivation of Archimedes' Principle, and not just the story of the golden crown, is in it. If it isn't, your children need you to confront your school board. Tell them Archimedes sent you.
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