Bradwardine’s Rule

I love reading about the medieval physics: you can see people struggling against mental traps, often failing, but still putting up a fight. We shouldn’t laugh at them: theoretical physicists may be stuck in their own traps today! Good new ideas often seem obvious in retrospect… but only in retrospect.

For example: Aristotle argued that a vacuum is impossible, because the velocity of an object equals the force on it divided by the resistance of the medium it’s moving through. A vacuum offers no resistance—so an object would move through it at infinite speed!

Around 1100, the medieval Arab physicist Ibn Bajja disagreed. He argued that the celestial spheres—i.e. the planets and stars — move at finite speeds even in the vacuum. So, he said, we should subtract the resistance of the medium from the object’s natural speed, not divide by it.

Averroes fought back, agreeing with Aristotle. Later, Thomas Aquinas sided with Ibn Bajja (who was known in Latin as Avempace). By the 1300s, most Western natural philosophers had sided with Aristotle.

There are definitely problems with the subtraction theory. What if the resistance exceeds the force? Does the object move backwards? But back then they didn’t know about negative numbers! So maybe proponents of the subtraction theory would say an object stands still if you push on it with insufficient force to overcome the resistance.

All this reached a pinnacle of complexity in Thomas Bradwardine’s 1328 Treatise on the Ratios of Speeds in Motions. He analyzed four theories and then proposed his own. Though he didn’t phrase it this way, it seems he argued that speed is proportional to the logarithm of the force over resistance. For example if you cube the ratio of force to resistance, you triple the object’s speed.

This theory, which came to be called ‘Bradwardine’s rule’, seems terrible to me. It doesn’t solve the problem of infinite speeds in a vacuum, and it says some force is required just to hold an object still. In fact, I hope I’m misinterpreting this theory! But anyway, it caught on: without any experimental evidence backing it up, it took root and was popular for about 100 years.

In Oxford, Richard Swineshead and John Dumbleton applied Bradwardine’s rule to solve ‘sophisms’, the logical and physical puzzles that were starting to become important at Merton College. A bit later this rule appeared in Paris, in the works of Jean Buridan and Albert of Saxony. By the middle of the 1300s it caught on in Padua and elsewhere. And Swineshead, nicknamed The Calculator, used it to study whether a body acts as a unified whole or as the sum of its parts. He imagined a long, uniform, heavy body falling in a vacuum down a tunnel through the center of the earth. Somehow he concluded that it would take an infinite time to reach the center. Don’t ask me how—I don’t know! But anyway, he rejected this conclusion as unphysical.

So, a lot of struggles!

While the theories I just explained sound pretty bad, something good was happening throughout these arguments: researchers were figuring out the concept of ‘instantaneous velocity’: the velocity of an object at an instant of time. They didn’t define it using derivatives, but it seems getting a good intuitive handle on this was a prerequisite for later work on derivatives.

And Nicole Oresme, trying to formula Bradwardine’s rule in full generality, was led to study fractional powers for the first time! Nobody knew about logarithms at the time, so Bradwardine’s rule actually said that if you have two objects feeling forces and encountering resistances , their velocities are related by

This is simple enough when the ratio is a positive integer, but Oresme extended it to fractions!

So, fundamentally misguided physics can still lead to good mathematics. (I will not attempt to draw any lessons for the present.)

I got most of this material from here:

• Walter Roy Laird, Change and motion, in Cambridge History of Science, Volume 2: Medieval Science, eds. David C. Lindberg and Michael H. Shank, Cambridge U. Press, Cambridge, 2013.

But as I dig deeper, I’m finding this very helpful:

• Michael Claggett, The Science of Mechanics in the Middle Ages, University of Wisconsin Press, Madison, Wisconsin, 1961.