In looking back over the history of applied mathematics in my recent studies, I was reminded that the theory of statics and torques actually predates the Renaissance period considerably. Archimedes, in two of his surviving works, exhibited some sophisticated results on centers of gravity and equilibria of floating bodies circa 250 BC! His axiomatization of the "law of the lever" is the simplest of these, coming just a generation after Euclid's monumental work.

Even before one can ask about the relation between "weight" and length, it is an interesting question to ask how one extends the concepts of Euclidean geometry to account for a quantity that can not be described soley in geometric terms like volume. Clearly, the practical concept of weight is very old, and can be found in the form of scales in surviving Egyptian works cira 2000 BC. But why is it reasonable to believe weights of different substances will behavie comparably? Why is it OK to weight food using stone weights or lead weights? Clearly stone and lead are different substances, and early science provided very poor theories of what this difference actually was. Might not the differences mean that scales using stone weights and scales using lead weights were fundamentally incompatible?

When you consider this question, the mythological use of scales at the gateway to the Egyptian afterlife takes on a surprising substance. A person's heart, representative of the accumulation of good and bad deeds, would be weight on a scale against a feather, with the balance determining their fate. The scale would have been a sophisticated piece of technology 4 thousand years ago, yet one that had entered consciousness to a point where it played a central role in the most important moment in mythology. Yet the way it was employed, balancing two things that seem so different, suggests (to me) that its operation retained a mystical quality. It was important that all witnesses could see and recognize what things were being compared, and thence that the outcome was the embodiment of some supernatural will of the universal, and even the gods were subject to its will. This is far from the utilitarian tool we now perceive in our scales, and trust so implicitly in daily life that it's difficult to imagine.

Which brings me now to the main point of this post -- the axiomization of the concept of weight, in alignment with that of classical geometry. It remains a bit of a mystery to me to what extent the concept of weight is an empirical concept and to what extent it is a platonic abstraction like point or line. The modern convention is that we define mass, rather than weight, and that (atleast was until recently) mass is compared against a cubic centimeter of water at 4 degrees Celcius. But rarely does it seem to be mentioned why this convention is actually justifiable -- why is it we can adopt one universal standard for the measurement of weight (and mass). Might there not be properties in weight that can not be measured with one such comparison?

Suppose we have weights of two different substances (say, wood and stone), and a scale with two arms that can be used to test balance. These arms are not necessarily of equal length, but their lengths will be constant and we will always weigh wood on the left and stone on the right.

We can think of the scale as a mechanical implementation of a function that takes two inputs, some volume of wood on the left and some volume of stone on the right, and outputs which of the two weights more -- the wood or the stone. We can then draw a picture, of the space of inputs coloring balance tips towards wood green, and balance tips towards stone purple.

The boundary between the green and purple regions will be special because for any volume inputs on the boundary, the scale will be balanced -- the volume of wood will weight the same as the volume of stone. What shape will this boundary take?

We can imagine this boundary having some shapes but not others. For example, if the scale is tipped toward wood, and we add more wood, it should remain tipped toward wood. And if the scale is tipped toward stone, and we add more stone, it should stay tipped that way as well. These are axioms based on our real-world experiences with weight. Archimedes (and probably many before him) observed these axioms as well and used them. Also, if there is no wood, the scale can not tip that way, and if there is no stone, that scale can not tip that way either. If both sides of the scale are empty, it is balanced. It follows then that the boundary between green and purple must start at the origin, and divide the input space into exactly two pieces, one green and one purple (there cannot be any "bubbles" of color in the picture). The exact shape of the boundary, however, remains unknown. It could be a parabola, or a line, something wavey -- all of these are consistent with our elementary axioms.

To go further, we will produce a theorem, and consider its implications. Let us define the function \(F(x)\) as the volume of stone needed to balance a volume \(x\) of wood. And suppose there are volumes of wood \(a\) and \(b\) such that the sum of the volumes of stone needed to balance each of these is less than the volume of stone need to balance their sum, so \(F(a) + F(b) < F(a+b)\). Then we can show the following.

**Theorem**: Assuming we have two equal scales next to each other, and we can move weights between these scales without any work when they are tipped the same way, then we can lift as much stone up to the height of the scale as we want without doing any work.

**Proof**: First, by the continuous divisibility of volumes, there are volumes of stone \(c\), \(d\), and \(e\) such that \(F(a) < c\), \(F(b) < d\), and \(c + d + e < F(a+b)\). Now, suppose we set up two identical scales side-by-side. To start with, we weigh \(a + b\) against \(c + d\), and the scale is tipped to wood. Now, split the weights so we weigh \(a\) against \(c\) and \(b\) against \(d\). Both scales now tip in toward stone. Now, me move all back to a single scale and add the extra stone volume \(e\). The scale now tips towards wood, and we can offload the extra stone volume \(e\) as work done, return us to our original scale state, but with more work done lifting stone. We can repeat this cycle as many times as necessary to lift any designed about of stone.

This theorem is, in spirit, a perpetual motion theorem. Once the stone is lifted, it can be used to other work, and hence get things done. We then may ask ourselves, "Do we believe in perpetual motion machines?"

If you do not believe in perpetual motion, then it must be that for every pair of wood volumes \(a\) and \(b\), \(F(a) + F(b) \leq F(a+b)\). If we believe the same argument holds for stone lifting wood, then \(F(a) + F(b) \geq F(a+b)\), and hence, \(F(a) + F(b) = F(a+b)\) for all \(a\) and \(b\).

This functional equation's solution is a line, so \(F(a) = k a\) for some constant \(k\). In modern language, we'd call \(k\) the relative density of stone compared to wood.

We can repeat this argument for every other pair of substances, like lead, gold, and water, and define relative densities for each of these comparisons.

It then follows that all weights can be calculated using a standard comparison like water, and from here on, we need only consider weighting one standard against another. We can now move on to determining the law of the lever.

This seems like a useful axiomatization of the concept of weight (and mass) to me.