The topic on the table is geometric as opposed to algebraic derivations.  Geometry begins with a small number of ideas that just seem true…axioms and postulates.  We then set about seeing what other truths follow from these initial assumptions.  What I think of as the geometric character of physics is just this: you take certain notions as the case, and you see what follows.

There’s a really good example of this from kinematics, the study of motion.  The idea that you can get distance travelled by multiplying speed times time was worked out a long, long time ago.  But people understood that you don’t always go at a steady rate, so they wanted to know how far you travelled if your speed varied…if you accelerated.  But they couldn’t solve this problem for centuries.  The guy who solved this problem back in the 14^th century (?), Nicholas Oresme, did not solve it by collecting data and working it out from there.  Instead he drew a rectangle.  The area of a rectangle is the product of the sides, say a x b, where a is the height and b is the base.  If we let a represent speed and b represent time, then the area of the rectangle corresponds to the distance.  Now that all by itself is neat because here you have a patch, the area of a rectangle and it corresponds to a line, the distance an object traveled.

Now the kicker.  Imagine a trapezoid.  The base is still b, but the two heights are different, a₁ is shorter, say, than a₂.  If we let a₁ represent the initial speed and a₂ the final speed, then, Oresme argued, the area of the trapezoid would represent the distance travelled while the object accelerated.  The expression for the area of a trapezoid was well-known and so you get the kinematics equation:  x = ½(v₁ + v₂)t.  This equation was true because it made sense, not because of any data.  In fact, if you did an experiment, you judged how good your apparatus was by how close you came to Oresme’s equation.

It turns out that a lot of physics expressions are derived in this way.  We start with some ‘basic’ notions and see where they take us.  Then we go to the lab, try them out and see how close things are.  Then, assuming it is pretty close, we consider the factors that might account for the differences between what we know is the right answer and the way things actually happened.  If the discrepancy between theory and data is too great, people get uncomfortable and now you have an interesting area for further study, tuning up relevant ‘basic’ notions, introducing heretofore unappreciated factors, and maybe devising some new experiment that would be a better window into the way things work.

More naïve physics next week.