Lecture 10 Oresme's 'Configuration of Qualities and Motions' |
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From 'A treatise on the configuration
of qualities and motions', 1356
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Nicole
Oresme (1320--1382), Bishop of Lisieux and a counsellor at court of
Charles V of France was one of the earliest scholars who tried to model
the concept of motion mathematically. Imagine you are trying to describe
the flight of an arrow, the fall of a heavy object or the course of a running
deer but you have had no previous introduction to functions or graphs.
This is the background of Oresme and his readers.
He was probably the inventor of the idea of functional relationship between two variables, and also of graphical representation of such a relationship. The philosophically difficult problem of the nature of velocity is this: if a point moves from A to B in time t, then its average speed is (B - A)/t . But what is its velocity at B? Oresme associated with a moving point its longitude, representing an instant of time, and its latitude or intensity, representing its velocity at the time; ie he had the notion of a functional relationship. He represented longitude as a horizontal line (no scale) and latitude as a series of vertical segments. The tops of these segments trace out a line, representing uniform motion ie constant velocity, unformly difform motion ie constant acceleration, or difformly difform motion, ie variable acceleration. Though no scale is attached, we can recognize the beginning of graphical representation of motion, instantaneous rate of change, and continuous functions, for example he considers the question: What constant speed produces the same motion (displacement) as a uniformly difform motion? This suggests the idea that area under the curve, ie the integral of velocity, represents distance. Oresme's most famous work is entitled 'Tractatus de configurationibus qualitatum et motuum' which translates as 'A treatise on the configuration of qualities and motions'. For an image of the actual manuscript of 'de Latitudinibus' (Latitude of forms) click here. |
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From Part 1, Chapter 1 of the
'Treatise'
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Every
measurable thing except numbers is imagined in the manner of continuous
quantity. Therefore, for the mensuration of such a thing, it is necessary
that points, lines, and surfaces, or their properties, be imagined. For
in them (i.e. the geometrical entities), as the Philosopher has it, measure
or ratio is initially found, while in other things it is recognized by
similarity as they are being referred by the intellect to them (i.e., to
geometrical entities). Although indivisible points, or lines, are nonexistent,
still it is necessary to feign them mathematically for the measures of
things and for the understanding of their ratios. 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 intensible
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. For just as one
line is commensurable to another line and incommensurable to still another,
so similarly in regard to intensities certain ones are mutually commensurable
and others incommensurable in any way because of their [property of] continuity.
Therefore, the measure of intensities can be fittingly imagined as the
measure of lines, since an intensity could be imagined as being infinitely
decreased or infinitely increased in the same way as a line.
Again, intensity is that according to which something is said to be "more such and such," as "more white" or "more swift." Since intensity, or rather the intensity of a point, is infinitely divisible in the manner of a continuum in only one way, therefore there is no more fitting way for it to be imagined than by that species of a continuum which is initially divisible and only in one way, namely by a line. And since the quantity or ratio of lines is better known and is more readily conceived by us--nay the line is in the first species of continua, therefore such intensity ought to be imagined by lines and most fittingly by those lines which are erected perpendicularly to the subject. The consideration of these lines naturally helps and leads to the knowledge of any intensity, as will be more fully apparent in chapter four below. Therefore, equal intensities are designated by equal lines, a double intensity by a double line, and always in the same way if one proceeds proportionally. And this is to be understood universally in regard to every intensity that is divisible in the imagination, whether it be an active or non-active quality, a sensible or nonsensible subject, object, or medium. For example, it is to be understood in regard to the light of the body of the sun, to the illumination of a medium, or to a species in the medium, to a diffused influence or power, and similarly to others, with the possible exception of curvature, concerning which we shall speak in a limited way p in chapters twenty and twenty-one of this part [of our work]. Of course, the line of intensity of which we have just spoken is not actually extended outside of the point or subject but is only so extended in the imagination, and it could be extended in any direction whatever except that it is more fitting to imagine it standing up perpendicularly on the subject informed with the quality.
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I.ii On the latitude of qualities |
Every intensity designated by the aforesaid line ought properly to be called the longitude of the quality. This is primarily because in continuous alteration succession according to the extension or parts of the subject is not [essentially] demanded, [for the whole subject can begin to be altered simultaneously], but succession according to intensity is required there. Therefore, just as in local motion that dimension according to which succession is demanded is called length of space or path, so similarly intensity of this sort according to which succession is required should be called the longitude of this quality. Also, just as velocity in local motion is measured according to length of space, so velocity in alteration is a function of intensity. Therefore, such intensity should be called longitude. Also, no quality acquirable by alteration can be imagined without intensity or divisibility according to intensity, but it can well be imagined without extension. Nay, a quality of an indivisible subject, such as a soul or an angel, does not have extension. Since, therefore, length is imagined mathematically without breadth but not conversely, and since intensity ought to be referred to some dimension, as is evident in the preceding chapter, it (intensity) ought to be referred to length not to breadth, and it more properly ought to be called by the name of longitude rather than by that of latitude. Thus it is clear that the quality of an indivisible subject does not properly have latitude. But many theologians speak improperly of the "latitude of charity"--for if by "latitude" they understand intensity then breadth would be found without length, and so their transference of the meaning would seem unfitting. Be that as it may, I shall caII intensity of this kind the "latitude" of quality, as I shall declare more fully in the chapter that immediately follows. | . |
Background of the work |
A long
period from 400--1100 following the conquest of Greece by Rome and Rome
by the 'Barbarians' and the spread of Christianity in Europe saw the centre
of mathematical developments in the West shift to the Arabic world. There
was no mathematical development in Europe except for a few translations
of Greek works into Latin, especially Aristotle and Plato, who were the
only pagan philosophers accepted by the Christian church, especially Aristotle's
' Logic'.
The knowledge and transmission of Arabic works, which had inherited the Greek tradition, began with the Crusades (1000--1200) and were carried on by travellaers and traders. In 1150 Robert of Chester translated Euclid from Arabic to Latin. Leonardo of Pisa = Fibonacci (1170--1240), a merchant, travelled in N. Africa, and brought back Hindu--Arabic numerals. He wrote new algebra and number theory books. Jordanus of Nemore (1220) Used ' literal' symbols in algebraic problems. This period saw the first Universities--Bologna, Paris, Oxford 1180--1220. Bradwardine at Oxford studied motion using Aristotle's ideas of the connection between force, resistance and velocity, and the concept of natural force (gravity). Oresme himself was connected with the University of Paris. He wrote several works commenting on Euclid and giving own ideas. For example, he considered a fraction as a numerator (= how many?) over denominator (=what type?). He discussed fractional powers, eg 43/2 = 8 and gave rules for manipulating exponents. He suggested that since 24 = 16 and 25 = 32 , every number from 16 to 32 should be 2x for some x. He expressed ideas on real exponents, logarithms, continuous functions, convergent and divergent series defined numerically and geometrically.
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To return to Table of Contents, click here.
For next Lecture , click
here. For more on Oresme,
see here.
For more on 'The Latitude of Forms', see here.
Last update: 1 August 2000
Author: Phill Schultz, schultz@maths.uwa.edu.au
