Bradwardine, Thomas

« velocity is a self-contradiction, a vacuum is impossible. In De proportionibus , Bradwardine first laid out the theory of ratios familiar from the theory of musical ratios as found in Boethius .

He used this understanding of operations on ratios to reinterpret Aristotle 's theory.

Velocities vary, he said, as the ratio of force to resistance.

When the ratio of force to resistance is ‘doubled' the velocity is doubled, when the ratio is ‘tripled' the velocity is tripled and so on, with the understanding that ‘double' the ratio 3:1 is the ratio 9:1 (its square), and that ‘triple' the ratio 2:1 is the ratio 8:1 (its cube).

This new understanding of Aristotle 's theory had the advantage of being expressed in similar words while avoiding a serious weakness of the theory itself, namely that it could not explain the mathematical relationships of forces and resistances in very slow motions.

Everyone agreed that for motion to occur at all the force must be greater than the resistance.

Suppose, then, that the ratio 2:1 produces a certain velocity.

It follows on Aristotle 's theory that the ratio of 1:1 should produce half that velocity, but in fact it produces no velocity at all, since the force is not greater than the resistance.

Thus Aristotle 's theory cannot account for any velocity smaller than half the velocity produced by the ratio 2:1.

By contrast, Bradwardine's function provided values of the ratio of force to resistance greater than 1:1 for any velocity down to zero, since any root of a ratio greater than 1:1 is always a ratio greater than 1:1. Bradwardine's proposed new function won immediate acceptance in universities all over Europe, beginning at Oxford where the so-called Oxford Calculators took it up.

It also inspired people to quantify motions in every way possible.

Bradwardine's function was assumed to hold for rotations as well as rectilinear motions, and for alterations, augmentations and diminutions as well as local motions.

In Paris, Nicole Oresme and Albert of Saxony wrote treatises on ratios, building on Bradwardine's rule.

Oresme used the theory to argue that since most ratios are incommensurable with one another, when ratios of ratios are understood in the Bradwardinian sense, the velocities of motion of the planets caused by ratios of force to resistance will also likely be incommensurable, meaning that planets will never return to exact conjunctions in the same location in the sky.

Astrological predictions based on such repetitions of positions will therefore be impossible. Taking the ratio of force to resistance as the measure of motion ‘as if with respect to cause' (tanquam penes causam ), philosophers then asked about the measure ‘as if with respect to effect' (tanquam penes effectum ).

For instance, should a rotation be measured by the average velocity of the points of the rotating body or by the distance traversed by the fastest moved point, as Bradwardine had argued in the last book of De proportionibus ? Learning how to operate with ratios as Bradwardine did became a standard part of the arts curriculum in many later medieval universities, and subsequently became part of the theology curriculum as well. 3 Other works In contrast to the novelty of Bradwardine's De proportionibus , his De continuo argued for what were at the time well-established conclusions.

As Aristotle had argued, no continuum is composed of indivisibles.

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