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According to [K. Simonyi, p. 79],

Archimedes was the first to forge a synthesis between mathematics and physics, liberating this connection from Pythagorean mysticism.

Ironically, it seems to me that the oldest physical law - Archimedes' Law of Buoyancy - is still often shrouded in mystery.

The law itself admits a pretty simple formulation. Simonyi quotes Proposition 16 from Archimedes' work On Floating Bodies:

Any body that is less dense than water attempts, on being completely immersed, to rise with a force that is equal to the difference in the weight of the water displaced by the body and the weight of the body itself. If a body is denser than the water, it will sink with a force that is the difference between the weight of the body and the weight of the water displaced.

In my edition of the works of Archimedes, this is a combination of Propositions 6 and 7 in Part I of On Floating Bodies. The formulations there are less anthropomorphic. For example, Proposition 6, is almost the same as above but with a subtle difference:

If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards by a force equal to the difference between its weight and the weight of the water displaced.

Archimedes goes to a great length to immerse (a pan intended) those statements in an axiomatic theory. He starts by stipulating homogeneity and continuity of fluids, but Proposition 1 is abstractedly geometric:

If a surface be cut by a plane always passing through a certain point, and if the section be always a circuference [of a circle] whose centre is the aforesaid point, the surface is that of a sphere.

Which sphere Archimedes is planning to refer to in deriving his law of buoyancy becomes clear from Proposition 2:

The surface of any fluid at rest is the surface of a sphere whose centre is the same as that of the earth.

He then considers the forces that act on a submerged pyramid (or a frustum) with the vertex at the center of the earth. This explains why it is unlikely for a present day school physics text to follow in Archimedes' footsteps. I'd be surprised to come across a book that does. What do they offer instead? Judging by what I found on the Web, the validity of the Law is commonly not explained, although the Law is practically self-explanatory.

What it takes is a small shot at abstraction. (It's an abstraction of the same sort that helps finding the same curve on two vastly disparate surfaces.)

Every body has a shape. Bodies may have the same or different shapes. We visualize a shape as separate from any connection to any physical body. Shapes have volume.

There are various forces that act on a physical body: gravity, air pressure, surface tension, forces due to the possible interaction of the surface of the body with the surrounding environment. For a derivation of Archimedes' law, we make an assumption that, excluding the weight, the force total that acts on a body in a given environment depends solely on the shape of the body. This is a reasonable assumption, especially because we shall be only interested in the bodies of the same shape that occupy a fixed volume, i.e., the bodies that may occupy the same physical space.

Give a shape, we can think of it as bounding a volume of water. By our assumption, there are two forces that act on the body of water inside that (imaginary) volume: weight of the water inside the volume (W) and the force (E) exerted by the environment. Since that shape/volume exists only in our imagination, the whole of the water is at rest, such that W + E = 0. The two forces cancel each other, implying that they act in different directions. Since W acts downwards, E acts upwards.

Similarly, there are two forces that act on a submerged physical body, say, V and E, the latter - by our assumption - being common to all the bodies of the same shape, i.e. occupying the same volume. In particular, for the body of water in the same volume, W + E = 0. E is an upward force; W and V act downwards. If V = W, the body is in equilibrium. If V < W, the body is lighter than water, so that E dominates the sum and the body is pushed upwards. If V > W, the body is heavier than water, so that V dominates the sum and the body is pushed downwards.

Reference

1. Great Books of the Western World, v. II, The University of Chicago, 1952
2. K. Simonyi, A Cultural History of Physics, A K Peters, 2012