The Mean Speed Theorem

Did you know students at Oxford in 1335 were solving problems about objects moving with constant acceleration? This blew my mind.

Medieval scientists were deeply confused about the connection between force and velocity: it took Newton to realize force is proportional to acceleration. But in the early 1300s, a group of researchers called the Oxford Calculators made huge progress in understanding objects that move with changing velocity. That was an incredibly important step.

You see, Aristotle had only defined velocity over an interval of time, as the change in position divided by the change in time. But the Oxford Calculators developed the concept of instantaneous velocity, allowing them to discuss objects with changing velocity—and even the concept of acceleration!

That’s really cool. But it gets better. The Oxford Calculators made their students write essays on ‘sophisms‘, meaning puzzles or paradoxes. Some of these were logical or philosophical, others physical. And in 1335, one of Oxford Calculators named William Heytesbury wrote a book called Rules for Solving Sophisms. This gives us a hint as to what these puzzles were like—though it’s more of a theoretical monograph than a practical how-to book, so we can’t really tell what the students were expected to do.

This book has a very interesting section on physics. Here Heytesbury states something called the Mean Speed Theorem. This says that if an object’s velocity is changing at a constant rate over some period of time, it goes just as far as if it were moving uniformly with the velocity it had at the middle instant of its motion!

We can think of this as a step toward integrating linear functions. Later in the 1300s, in Paris, Nicole Oresme, gave a clear picture proof of the Mean Speed Theorem. For example, he pointed out that the triangle ACG below has the same area as the rectangle ACDF:

This handles the Mean Speed Theorem for an object starting with zero velocity—an important special case which Heytesbury also found interesting.

Heytesbury’s Rules for Solving Sophisms also has some pictures:

But when it comes to the Mean Speed Theorem I haven’t seen him giving a picture proof. Instead, he gives a complex purely verbal argument. It’s so complicated I haven’t managed to follow it yet. I tried, but I gave up: it involves comparing the motion of four different objects. You can see the argument here, starting on page 122:

• Curtis Wilson, William Heytesbury: Medieval Logic and the Rise of Mathematical Physics, University of Wisconsin Press, Madison, Wisconsin, 1956.

I managed to find this book in the bowels of my university library, in one of those electronically movable stacks that makes me worry some evil denizen of the nether reaches is going to crush me:

But I’m pleased to report that the tireless criminals of Library Genesis have made this book available to the world in electronic form, so you don’t have to take such risks. If you can understand Heytesbury’s argument, please explain it to me!

Who were the Oxford Calculators, exactly? They were a group of thinkers associated to Merton College in Oxford. Thus, they’re also called the Merton School. Here are the four most important:

• Thomas Bradwardine (c. 1300 – 26 August 1349) was an English cleric, scholar, mathematician, physicist, and—for just 4 months before he died of the bubonic plague—Archbishop of Canterbury. The Pope called him Doctor Profundus, and the nickname stuck. He was very interested in logic and wrote about the Liar Paradox. His Treatise on the Proportions of Velocities in Movements started the interest in motion at Merton College. As I recently explained here, he came up with a really lousy rule relating force to velocity, called Bradwardine’s Rule. But the bright side is that in the process, he began clarifying the concept of instantaneous velocity!

• John Dumbleton (c. 1310 – c. 1349) studied in Paris for a brief time before becoming a fellow at Merton College. His Summa Logica et Philosophiae Naturalis, never finished, seems to be in part a reworking of ideas from Aristotle. But it also has new ideas on physics: for example, he argues that the brightness of a light source does not decrease in inverse proportion to the distance—though not, alas, that it goes with the inverse square of distance.

• Richard Swineshead (flourished c. 1340 – 1354) has been called “in many ways the subtlest and most able of the four”. In his fragment On Motion he defined the instantaneous velocity of an object as the distance it would have traveled in a certain time, divided by that time, if it had continued moving at the same velocity. His The Book of Calculations: Rules on Local Motion gives an argument for the Mean Speed Theorem which again I don’t understand. He also argues that an object starting at rest, whose velocity increases at a constant rate over some time, will travel 3 times as much in the second half of that time as in the first.

• William Heytesbury (c. 1313–1372/3) had a degree in theology, and near the end of his life became chancellor of Oxford University. But he’s most famous for his popular textbook Rules for Solving Sophisms. The Mean Speed Theorem is stated but not proved there. A proof appears in a book Probationes Conclusionem which may or may not have been written by Heytebury.

If you want to dig deeper into this subject, the place to go is here:

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

It’s a sourcebook, with a lot of important texts translated into English, but it also has long sections explaining the history and the ideas. Note that both books I recommended today were published by the University of Wisconsin in the late 1950s and early 1960s. Maybe they were a powerhouse in medieval physics at that time?

Anyway, I find this stuff fascinating. One obvious question is: why did the Oxford Calculators sort of fizzle out after 1350? Why did it take until 1589 for Galileo to announce that falling bodies actually move in a way described by the Mean Speed Theorem?

There are a lot of possible reasons, but one is hinted at by Thomas Bradwardine’s death in 1349. The bubonic plague killed about one third of the population of Europe from 1346 to 1353!