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order of perception, the ordering of basic categories: substance, quantity, quality, was assumed to have a conceptual necessity. This seemed to preclude, as it had with the Greeks and Arabs, any quantification of qualities as a reversal of a necessary ordering. Yet, accepted theology demanded some quantification of qualities (See Crombie, 1959, pp. 85-119). The accepted tradition, which supplies the basic structure for Dante's Divine Comedy, is that rank in heaven or hell depends on the degree of sanctifying grace (or of its absence) that the soul possess at death. Thomas Aquinas seems to have been the first to speak of the quantity of a quality.4  He distinguished between quantity per se, or bulk quantity, and quantity per accidens, or virtual quantity.  A quality can have magnitude by reason of the subject in which it inheres, as a bigger wall has more whiteness than a small wall, or by reason of the effect of its form. A secondary effect of a form is manifested through activity. Thus a man with greater strength can lift heavier rocks. This was within the Aristotelian tradition, where the change in the intensity of a quality involved the loss of one species of a quality and the acquisition of another. Subsequent discussions of the quantity of qualities concentrated, not surprisingly, on ontological issues. How is quality to be understood as a determination of a substance? How is a change in the degree of a quality to be understood in terms of causes5?

A significant modification of the Aristotelian sharp separation of categories centered on discussions of the intensification and remission of qualities. The fourteenth-century nominalists, especially William of Ockham, attempted to eliminate the ontological discussions by focusing on the question of when a word admits of the adjectives 'strong' and 'weak' and when it can be combined with the terms 'large' and 'small'. When the problem was put in these terms, then changes in local motion, rather than degrees of charity, emerged as the prime example of intensification and remission. The mathematization of motion was developed chiefly by the 'Calculators' of Merton College in Oxford and later by Nicole Oresme and others at the University of Paris. These calculations were based more on abstract considerations rooted in the system of categories than on empirical data.

Oresme effectively introduced the geometric representation of quantities that culminated in Descartes' analytic geometry:

"Therefore, every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the insensible thing, e.g., a quality. For whatever ratio is found to exist between intensity and intensity, in relating intensities of the same kind, a similar ratio is found to exist between line and line, and vice versa.' (citation from Clagett ,1968),  p. 165).

In these works there is no mention of actual measurements or units or, until the sixteenth century, any application to falling bodies. However, this treatment of the intensification and remission of qualities introduced a conceptual language that made discussions of measurement possible, and stimulated mathematical analyses of functional relations and of the representation of varying qualities. (Murdoch, 1974)

In addition to the abstractness of this treatment, there were two other factors that distanced this burgeoning language of mathematical physics from the lived world of ordinary language. First, teaching and texts were in scholastic Latin. This was the

4 St. Thomas Aquinas, Summa Theologiae, 1, q. 42, a. 1, ad 1. The relation of such quantitative reasoning to the foundations of Aristotelian science are treated in his In Librum Boethii de Trinitate.

5 Such questions continued to be discussed well into the scientific revolution. See Solère (2001)

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