Chapter 4
Medieval Europe: somebody invents the graph
The Oxford Calculators
At Merton College, Oxford, in the 1330s and 1340s, a group of logicians produced the most important kinematic result of the Middle Ages, and they did it inside a logic textbook.
The mean speed theorem says a body moving with uniformly changing speed covers the same distance as a body moving steadily at the average of its starting and ending speeds. William Heytesbury states it in 1335, in a book about sophismata, the trick sentences logic students were trained to dismantle (Source 101, Jung). The single most important kinematic result of the Middle Ages arrives as a rule for settling a brainteaser. Physics teachers do not look in chapter six of a logic textbook.
The Calculators' method, the wildly artificial thought experiment, gets defended in one line that licenses the whole later tradition of frictionless planes and massless ropes: the scenario does not have to be physically real, only conceivable, and "for purposes of the sophisma, that is enough" (Source 102, Jung).
Guess before you read on
Merton College gets the mean speed theorem, and the line you will read in most histories is that Merton discovered it and proved it. Name the half of that Merton did not supply.
I have a guess
The proof. Heytesbury states the theorem in 1335 and never proves it. The proof belongs to Nicole Oresme, in Paris, with a picture. "Stated at Merton, proved at Paris" is the accurate line, and Clifford Truesdell's often-quoted claim that it was discovered and proved at Merton overreaches (Source 101, Jung; Source 104, Lovell-Read).
If you assumed the people who stated it also proved it, you are with Truesdell, whose version is the one that gets quoted. Stating and proving are two jobs, and this is the chapter where they get done in two countries.
Heytesbury stated it. He did not prove it. The proof belongs to Nicole Oresme in Paris. "Stated at Merton, proved at Paris" is the accurate line, and Clifford Truesdell's often-quoted claim that it was discovered and proved at Merton overreaches (Source 101, Jung; Source 104, Lovell-Read).
The Calculators also died the way medieval people died. Thomas Bradwardine was made Archbishop of Canterbury on 10 July 1349, in the middle of the Black Death, and was dead of plague by 26 August, a primacy of about seven weeks (Source 103, O'Connor). John Dumbleton also died in 1349. Dumbleton is the most under-documented person in this whole document: for a long time no usable source could be found for him at all. Edith Sylla's DSB entry turned him up: a ten-part Summa surviving in more than twenty manuscripts, nearly forty percent of it about living things and minds (Source 480, Sylla). He was a bestseller, not a footnote.
Richard Kilvington prompted the group and Bradwardine founded it (Source 480, Sylla, conflict C-M002).
Oresme draws the picture
Nicole Oresme (c. 1320 to 1382), in Tractatus de configurationibus qualitatum et motuum, does something nobody had done: he draws time along the bottom, speed up the side, and then says the thing that makes calculus possible. The area of the figure is the total effect.
Marshall Clagett, whose critical edition is the source read here, calls chapter III.vii "the most significant chapter of the work, historically speaking" (Source 107, Oresme).
This is a Riemann-integral idea in the 1350s, and a student can reproduce the entire proof with a ruler in ninety seconds. Cut the triangle at the halfway height, rotate the top piece, get the rectangle. Done.
Oresme also proved the harmonic series diverges, by grouping terms into blocks that each sum to at least one half:
Read this equation in words
The sum from k equals one to infinity of one over k equals one, plus one half, plus a group of one third plus one quarter, which is at least one half, plus a group of one fifth through one eighth, which is also at least one half, and so on. So the harmonic sum through two to the m is at least one plus m over two.
The bound holds at every power of two I tested up to 2¹⁴ (Appendix E.5). The series diverges, and it does so at a speed that surprises people: to pass 10 you need more than 12,000 terms.
He was also a king's man. Charles V paid him to translate Aristotle into French, and the by-product was a French scientific vocabulary that had not existed before. He argued at length and brilliantly that no available argument disproves the Earth's daily rotation, and then said he believed it stood still anyway, because Scripture (Source 100, Kirschner). His argument against astrology is mathematical: if celestial ratios are almost certainly irrational, no exact conjunction ever repeats, so astrological prediction fails in principle (Source 108, Caroti). That is one of the earliest uses of "most probable" as an argument in European science.
One question before you go
Oresme put time along the bottom and speed up the side. Say what he claimed the area of that figure measures.
Show the answer
The total effect. Distance, when the thing up the side is speed (Source 107, Oresme). Clagett calls the chapter it sits in the most significant of the work, and the proof takes ninety seconds with a ruler: cut the triangle at half height, rotate the top piece, get the rectangle.
That sentence is the theorem behind Integrals 1.5, the Net Change Theorem. It was written around 1350, roughly three hundred years before anyone had a sign for an integral.