User:Janbcb/Oxford Calculators

Science
The advances these men made were initially purely mathematical but later became relevant to mechanics. Using Aristotelian logic and physics, they studied and attempted to quantify physical and observable characteristics such as: heat, force, color, density, and light. Aristotle believed that only length and motion were able to be quantified. But they used his philosophy and proved it untrue by being able to calculate things such as temperature and power. They developed Al-Battani's work on trigonometry and their most famous work was the development of the mean speed theorem, (though it was later credited to Galileo) which is known as "The Law of Falling Bodies". Although they attempted to quantify these observable characteristics, their interests lay more in the philosophical and logical aspects than in natural world. They used numbers to disagree philosophically and prove the reasoning of "why" something worked the way it did and not only "how" something functioned the way that it did.

The Oxford Calculators distinguished kinematics from dynamics, emphasizing kinematics, and investigating instantaneous velocity. It is through their understanding of geometry and how different shapes could be used to represent a body in motion. The Calculators related these bodies in relative motion to geometrical shapes and also understood that a right triangle's area would be equivalent to a rectangle's if the rectangle's height was half of the triangle's. This is what led to the formulating of what is known as the mean speed theorem. A basic definition of the mean speed theorem is; a body moving with constant speed will travel the same distance as an accelerated body in the same period of time as long as the body with constant speed travels at half of the sum of initial and final velocities for the accelerated body. Relative motion, also referred to as local motion, can be defined as motion relative to another object where the values for acceleration, velocity, and position are dependent upon a predetermined reference point.

The mathematical physicist and historian of science Clifford Truesdell, wrote:"The now published sources prove to us, beyond contention, that the main kinematical properties of uniformly accelerated motions, still attributed to Galileo by the physics texts, were discovered and proved by scholars of Merton college.... In principle, the qualities of Greek physics were replaced, at least for motions, by the numerical quantities that have ruled Western science ever since. The work was quickly diffused into France, Italy, and other parts of Europe. Almost immediately, Giovanni di Casale and Nicole Oresme found how to represent the results by geometrical graphs, introducing the connection between geometry and the physical world that became a second characteristic habit of Western thought ..."In Tractatus de proportionibus (1328), Bradwardine extended the theory of proportions of Eudoxus to anticipate the concept of exponential growth, later developed by the Bernoulli and Euler, with compound interest as a special case. Arguments for the mean speed theorem (above) require the modern concept of limit, so Bradwardine had to use arguments of his day. Mathematician and mathematical historian Carl Benjamin Boyer writes, "Bradwardine developed the Boethian theory of double or triple or, more generally, what we would call 'n-tuple' proportion".

Boyer also writes that "the works of Bradwardine had contained some fundamentals of trigonometry". Yet "Bradwardine and his Oxford colleagues did not quite make the breakthrough to modern science." The most essential missing tool was algebra.

Lawrence M. Principe wrote:"A group known as the Oxford Calculators had begun applying mathematics to motion in the 1300s; in fact, Galileo begins his exposition of kinematics in the Two New Sciences with a theorem they enunciated. But Galileo went much further by linking mathematical abstraction tightly with experimental observation."Lindberg and Shank wrote:"Like Bradwardine's theorem, the methods and results of the other Oxford Calculators spread to the continent over the next generation, appearing most notably at the Univeristy of Paris in the works of Albert of Saxony, Nichole Oreseme, and Marsilius of Inghen."

Science
The advances these men made were initially purely mathematical but later became relevant to mechanics. Using Aristotelian logic and physics, they studied and attempted to quantify physical and observable characteristics such as: heat, force, color, density, and light. Aristotle believed that only length and motion were able to be quantified. But they used his philosophy and proved it untrue by being able to calculate things such as temperature and power. Although they attempted to quantify these observable characteristics, their interests lay more in the philosophical and logical aspects than in natural world. They used numbers to disagree philosophically and prove the reasoning of "why" something worked the way it did and not only "how" something functioned the way that it did.

Historian David C. Lindberg and professor Michael H. Shank in their 2013 book, Cambridge History of Science, Volume 2: Medieval Science, wrote:"Like Bradwardine's theorem, the methods and results of the other Oxford calculators spread to the continent over the next generation, appearing most notably at the University of Paris in the works of Albert of Saxony, Nicole Oresme, and Marsilius of Inghen."

Mean Speed Theorem
The Oxford Calculators distinguished kinematics from dynamics, emphasizing kinematics, and investigating instantaneous velocity. It is through their understanding of geometry and how different shapes could be used to represent a body in motion. The Calculators related these bodies in relative motion to geometrical shapes and also understood that a right triangle's area would be equivalent to a rectangle's if the rectangle's height was half of the triangle's. This, and developing Al-Battani's work on trigonometry is what led to the formulating of the mean speed theorem (though it was later credited to Galileo) which is also known as "The Law of Falling Bodies". A basic definition of the mean speed theorem is; a body moving with constant speed will travel the same distance as an accelerated body in the same period of time as long as the body with constant speed travels at half of the sum of initial and final velocities for the accelerated body. Relative motion, also referred to as local motion, can be defined as motion relative to another object where the values for acceleration, velocity, and position are dependent upon a predetermined reference point.

The mathematical physicist and historian of science Clifford Truesdell, wrote:"The now published sources prove to us, beyond contention, that the main kinematical properties of uniformly accelerated motions, still attributed to Galileo by the physics texts, were discovered and proved by scholars of Merton college.... In principle, the qualities of Greek physics were replaced, at least for motions, by the numerical quantities that have ruled Western science ever since. The work was quickly diffused into France, Italy, and other parts of Europe. Almost immediately, Giovanni di Casale and Nicole Oresme found how to represent the results by geometrical graphs, introducing the connection between geometry and the physical world that became a second characteristic habit of Western thought ..."

Boethian Theory
In Tractatus de proportionibus (1328), Bradwardine extended the theory of proportions of Eudoxus to anticipate the concept of exponential growth, later developed by the Bernoulli and Euler, with compound interest as a special case. Arguments for the mean speed theorem (above) require the modern concept of limit, so Bradwardine had to use arguments of his day. Mathematician and mathematical historian Carl Benjamin Boyer writes, "Bradwardine developed the Boethian theory of double or triple or, more generally, what we would call 'n-tuple' proportion".

Boyer also writes that "the works of Bradwardine had contained some fundamentals of trigonometry". Yet "Bradwardine and his Oxford colleagues did not quite make the breakthrough to modern science." The most essential missing tool was algebra.

Bradwardine's Rule
Lindberg and Shank also wrote: "In Book VII of Physics, Aristotle had treated in general the relation between powers, moved bodies, distance, and time, but his suggestions there were sufficiently ambiguous to give rise to considerable discussion and disagreement among his medieval commentators. The most successful theory, as well as the most mathematically sophisticated, was proposed by Thomas Bradwardine in his Treatise on the Ratios of Speeds in Motions. In this tour de force of medieval natural philosophy, Bradwardine devised a single simple rule to govern the relationship between moving and resisting powers and speeds that was both a brilliant application of mathematics to motion and also a tolerable interpretation of Aristotle's text."