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# Medieval Kinematics

## Overview

Medieval European mathematicians inherited from the works of Aristotle (384-322 b.c.) a tradition of kinematics that was more qualitative than quantitative. Without proceeding far in the direction of experiment, the medieval scholars of France and England could still use arguments that suggested new conclusions about how to describe motion. In the tradition of Aristotle, these arguments had to apply to all sorts of change. In the hands of subsequent physicists the arguments proved the basis for laying a foundation of a quantitative physics devoted to the study of motion based on experiment as well as theory.

## Background

Kinematics is the study of motion. By the end of the twelfth century the works of Aristotle on the subject were being introduced into Western Europe, frequently via translations from Arabic and Hebrew intermediaries. The transmission and preservation of the texts of Aristotle had a value in their own right, but there were clearly issues about motion that Aristotle did not resolve. The works of Aristotle made it difficult to isolate questions simply related to the movement of objects. Aristotle had looked at motion as simply another form of change, and consequently his arguments about motion had to apply equally well to changes of color or even to objects coming into existence and ceasing to be. One of the results was that Aristotle could not take a quantitative approach to issues of motion because it was unclear how one could apply numerical values to other forms of change.

One of the additional difficulties for the study of motion in the Greek tradition was the Euclidean background for the geometry that would be used to describe the motion mathematically. Euclid (330?-260? b.c.) had devoted a good deal of space in his Elements to discussion of different sorts of ratios. The treatment that was available during the Middle Ages did not allow for the possibility of taking a ratio between different sorts of quantities (like distance and time). Since fundamental notions like velocity are now expressed as a ratio of distance and time, there was not a mathematical basis for expressing the ideas of motion as they were developed subsequently.

New approaches to kinematics were formulated primarily in the universities of Paris and Oxford, which had only recently come into existence. In the thirteenth century there were arguments within the Church about the relative value of scientific and religious studies, and those who followed the scientific path had to worry about the dangers of accusations of heresy. Fortunately Albertus Magnus (1206-1280) gave the imprimatur of the Church to a variety of scientific pursuits and steered clear of doctrinal issues. In the aftermath of his leadership, scholarship within the university could look at issues discussed in Aristotle without having to treat his view as enshrined by the Church.

At Oxford, the group of mathematicians associated with Merton College looked at laws governing "local motion," as the movement of physical objects in space was called to distinguish it from other sorts of change. Aristotle had explained motion in terms of objects having natural places and seeking to return to those places when they had been moved out of them. It is perhaps characteristic that one of the chief accomplishments of the Merton school of physicists was coming up with a suitable definition for "uniform speed" and "uniformly accelerated motion." Uniform motion involved going through equal distances in equal time intervals. The slightly more complicated notion of uniform acceleration involved adding equal increments of velocity in equal time intervals. On this basis they were able to derive a relationship between the distance traveled by an object and the speed at which it was traveling. All of these notions had to be expressed in words, since there was no notation for representing the quantities in an algebraic equation.

Even more impressive was the work of Nicole Oresme (1320?-1382), a French mathematician whose work anticipated many of the developments of the next few centuries. In particular, Oresme came up with the idea of representing motion by a picture, which resembled the graphs made familiar a few centuries later by the work of René Descartes (1596-1650). Oresme looked at the general shapes of motions as represented on paper and argued that differences in the shapes of the motions indicated that there were different laws of motion being obeyed. As with the Merton school, Oresme was saddled with the Aristotelian and Euclidean traditions that made him try to apply his diagrams to changes other than motions. While the members of the Merton school had run into logical problems about the possibilities of motion, Oresme worked out his results in Euclidean terms without worrying so much about the philosophical issues that they raised.

## Impact

Perhaps the most immediate impact of the work of the Merton school in England and Oresme in France was the realization that there were many issues about motion that were not settled by Aristotle. Even though they did not verify their results through experimentation and they trusted a fundamental Aristotelian view of motion, they were careful to clarify some points that Aristotle had left obscure and to produce quantitative results. In an era inclined to take Aristotle's texts as definitive, this was an important contribution to justifying continued work in areas of science that Aristotle had tackled.

In physics the work of Oresme had a great influence on Galileo (1564-1642). It is certainly true that Galileo was in revolt against many of the ideas about science taken for granted in the educational system of his time. On the other hand, Galileo also recognized the value of Oresme's use of graphical representations for motion and built on them in his own works for his fellow scientists and the general public. Galileo's use of experimentation enabled scientists to apply to a wide variety of situations the mathematical calculations of three centuries before and to confirm that the numerical values obtained applied to objects in the physical world. With Galileo, the importance of Oresme's work was easier to recognize, as Galileo had not felt obliged to look at the broader issues of changes other than motion.

Within mathematics the works of the Merton school and of Oresme had more of an influence through the coordinate geometry of Descartes. In looking back at the texts of Oresme it is easy to see diagrams that resemble graphs with coordinate axes. As a result, there is a temptation to claim that Oresme must have invented analytic geometry long before Descartes. The problem is that Oresme does not use points as representations of two coordinates, but instead was concerned just with the general shape of the picture. The same approach has been popular with certain branches of mathematics in the twentieth century, such like catastrophe theory. Descartes's work, by contrast, used graphs in a more strictly quantitative way and laid the foundations for work that went well beyond what Oresme could have envisaged.

Most of the effects of the work of the Merton school and of Oresme did not become visible until several centuries later. One of the reasons was the limited access to texts in the days before printing rendered them more generally available. It is also the case that some of the scientific innovation in the centuries that immediately followed Oresme took place in Italy and Germany, where works from France and England were not so well known. Even though all the medieval physicists in Western Europe wrote in Latin, the transmission of texts from one part of the world to another was a painfully slow process. As a consequence, much of the work that was done in the first few centuries after the revival of Aristotle in Western Europe had to be rediscovered (either independently or when the texts became available) some centuries later.

Physics (whose very name has a Greek root referring to "nature") had received its initial formulation in the works of Aristotle. Because of self-imposed limitations in his work, Aristotle could not even ask some of the questions that are fundamental in the study of motion. The work of the Merton school and of Oresme provided clear formulations of some of the questions and even started on the path to quantitative answers. With the development of mathematical notation and the introduction of experimentation over the next few centuries, the work in medieval kinematics could serve as the basis for a revolution in the study of motions of bodies on the Earth and in the skies.

THOMAS DRUCKER