*Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.*

Theodoric of Freiberg shifted the focus from the prior century's fascination of mechanics to optics when he studied prisms and discovered that rainbows are the result of the reflection/refraction of light. These findings were published in De Iride in 1310. He uncovered this by experimenting with different light angles as well as blocking out selective light and even trying different types of materials such as prisms and containers with water to represent raindrops. It was this last field that gave him the leap he needed.

Imagine each raindrop as a part of a prism. With enough of them in a vicinity, you can get a rainbow to form. He found this to be true after he experimented with the height of each container and found he could get different colors. He tried to explain all those colors but his methods and geometry were not sufficient to accomplish that, but he was able to talk about secondary rainbows as well (Wallace 34, 36; Magruder).

Thomas Bradwardine, a fellow of Norton College, wrote Treatise on the Ratios of Velocities in Motion, in which he used speculative arithmetic and geometry to examine said topic and see how it extended to relations between forces, velocities, and resistance to motion. He was spurred to work on this after discovering a problem in Aristotle’s work where he claimed velocity was directly proportional to force and inversely proportional to resistance of motion (or v = kF/R). Aristotle had then claimed that velocity was zero when the force was less than or equal to the resistance of motion (thus being unable to overcome the inherent resistance). Thus, v is a finite number expect for when the force is zero or when the resistance is infinite.

This did not jive well with Thomas, so he developed the “ratio of ratios" to solve what he felt was a philosophical problem (for how can anything be unmovable). His “ratio of ratios” eventually led to the (not right) idea that velocity is proportional to the log of the ratios, or that v=k*log(F/r). Our buddy Newton would show this is just plain wrong, and even Thomas offers no justification for its existence other than it removes the aforementioned case of the finite/infinite dichotomy because of logarithm properties pertaining to log(0).

He most likely didn't have access to the necessary gear to test out his theory, but some of Thomas’ footnotes discuss the calculations of his equation and hints at the idea of an instantaneous change, an important bedrock of calculus, versus an average change and how they approach one another as the differences shrink. He even hinted at the idea of taking a bit of infinity and still having infinity. Richard Swinehead, a contemporary of Bradwardine, even went through 50 variations of the theory and in said work also has those hints of calculus (Wallace 37-8, Thakker 25-6, Freely 153-7).

John of Dumbleton also made headway into the field of physics, when he wrote Summa logical et philosophiae naturalis. In it, rates of change, motion, and how to relate them to scale were all discussed. Dumbleton was also one of the first to use graphs as a means of visualizing data. He called his longitudinal axis the extension and the latitudinal axis the intensity, making the velocity the intensity of motion based off the extension of time. He used these graphs to provide evidence for the direct relation between strength of a shining object and the distance one is from it and also as evidence for an indirect relation between "the density of the medium and the distance of action (Freely 159)."

Even thermodynamics was given the time of day for research during this time period. People such as William of Heytesbury, Dumbleton, and Swineshead all looked at how heating non-uniformly affected the heated object (Wallace 38-9).

All the aforementioned people were members of Merton College, and it is from there that others worked on the mean-speed theorem (or the Merton rule, after Heytesbury’s work on the subject was heavily read), which was first developed in the early 1330s and worked on by said group in the 1350s. This theorem is also wordy but gives us a glance into their thought process. They found that a

“uniformly accelerated motion is equivalent, so far as the space traversed in a given time is concerned, to a uniform motion whose velocity is equal throughout to the instantaneous velocity of the uniformly accelerating body at the middle instant of the period of acceleration.”

That is, if you are accelerating at the same rate throughout a given period, then your average speed is simply how fast you were going at the midpoint of your journey. The Mertonians, however, failed to consider the application of this with a falling object nor were they able to come up with what we would consider a real-life application of this. But, to a student of calculus this finding is critical (Wallace 39-40, Thakker 25, Freely 158-9).

Another Mertonian piece of work was impetus, which would eventually evolve into what we call inertia. Biblically, impetus meant a push towards one goal and some of that meaning stayed with the word. Many Arabs had used the term to talk about projectile motion and the Mertonians worked with it in the same context. Franciscus de Marcha talked about impetus as a lingering force on projectiles caused by its launch. Interestingly, he says the projectile leaves behind a force as it is launched, then said force catches up to the projectile and gives it impetus. He even extends inputs when referencing how sky objects move in a circular fashion (Wallace 41).

John Buridan took a different viewpoint in his Questions on Aristotle's Physics and Metaphysics, feeling that impetus was an inherent part of the projectile and not something exterior to it. Impetus, he claimed, was directly proportional to velocity as well as the matter in motion and was a "quantity of matter" times velocity, aka momentum as we know it today. In fact, impetus would be an everlasting quantity if it weren’t for other objects impeding the path of the projectile, a major component of Newton's 1st law. John also realized that if the mass was constant then the force acting on an object had to be related to a changing velocity, essentially discovering Newton's 2nd law. Two out of the three big motion laws attributed to Newton had their roots here. Finally, John argued for impetus being responsible for falling objects and therefore gravity as well, stacking up in its full effect (Wallace 41-2, Freely 160-3).

In a follow-up, Nicole Oresine, one of Buridan's students, found that impetus was not a permanent fixture of the projectile but instead is a quantity that is used up as the object moves. In fact, Nicole postulated that acceleration was somehow connected to impetus and not at all to uniform motion.

In his Fractus de Configurationibus Quantitatum et Motuum, Oresine gave a geometrical proof for the mean speed theorem that Galileo ended up using also. He employed a graph where velocity was the vertical axis and time on the horizontal. This gives us slopes values of acceleration. If that slope is constant, we can make a triangle for a given time interval. If acceleration is zero, we could instead have a rectangle. Where the two meet is the location of our mean speed, and we can take the upper triangle we have just created and past it below to fill in that empty space. This was further evidence for him that velocity and time were indeed proportional.

Additional work by him established falling objects tend to fall onto a sphere, another precursor to Newton. He was able to calculate the spin rate of the Earth rather well but didn't readily release the results because of his fears for contradicting doctrine. He even pioneered mathematics, with a "proportional parts to infinity" summation happening, aka converging and diverging series (Wallace 41-2, Freely 167-71)!

But others studied falling objects and had their own theories as well. Albert of Saxony, another student of Buridan, found that the velocity of a falling object was directly proportional to the distance of the fall and also to the time of the fall. That, dear audience, is the basis of kinematics, but the reason why Albert isn’t remembered is because his work defended the claim that distance was an independent quantity and so it was therefore not a valid finding. Instead, he tried to break up little bits of velocity and see if it could be attributed to a set time interval, set distance, or set space amount. He did predict correctly that an object, if given a horizontal motion, should continue in that direction until the impetus of gravity overcomes the vertical distance required to get to the ground state (Wallace 42, 95; Freely 166).

Okay, so we have talked about the concepts people were thinking of, but how did they notate it? Confusingly. Bradwardine, Heytesbury, and Swinehead (our Mertonians) used something akin to function notation, with:

- -U(x) = constant velocity over a distance x
- -U(t) = constant velocity over a time interval t
- -D(x) = changing velocity over a distance x
- -D(t) = changing velocity over a time interval t
- -UD(x) = uniform change over a distance x
- -DD(x) = difform change over a distance x
- -UD(t) = uniform change over a time interval t
- -DD(t) = difform change over a time interval t
- -UDacc(t) = uniform accelerated motion over a time interval t
- -DDacc(t) = deform accelerated motion over a time interval t
- -UDdec(t) = uniform decelerated motion over a time interval t
- -DDdec(t) = difform decelerated motion over a time interval t

Yikes! Rather than realize a sign convention would result in familiar kinematic concepts, we have under the Mertonian system 12 terms! (Wallace 92, Freely 158)

This all sets up the stage for the wonders of the Renaissance, where science as we know it was born, but that is for another article....

## Works Cited

Freely, John. Before Galileo. Overlook Duckworth, New York. 2012. Print. 153-63, 166-171.

Magruder, Kerry. “Theodoric of Freiberg: Optics of the Rainbow.” *Kvmagruder.net*. University of Oklahoma, 2014. Web. 12 Sept. 2017.

Thakker, Mark. “The Oxford Calculators.” Oxford Today 2007: 25-6. Print.

Wallace, William A. Prelude to Galileo. E. Reidel Publishing Co., Netherlands: 1981. Print. 34, 36-42, 92, 95.

**© 2021 Leonard Kelley**