Nicole Oresme

From __A Source Book
in Medieval Science__, pp. 251 – 253

Here Begins The Third Part [Of This Treatise]: On The Acquisition And Measure Of Qualities And Velocities

*III.i How the Acquisition of Quality Is To Be Imagined *

Succession
in the acquisition of quality can take place in two ways: (1) according to
extension, (2) according to intensity, as was stated in the fourth chapter of
the second part. And so extensive acquisition of a linear quality ought to be
imagined by the motion of a point flowing over the subject line in such a way
that the part traversed has received the quality and the part not yet traversed
has not received the quality. An example of this occurs if point *c *were
moved over line *AB* so that any part traversed by it would be white and
any part not yet traversed would not yet be white.

c

A ----------------------------- B

Further the extensive acquisition of a surface quality ought to be imagined by the motion of a line dividing that part of the surface that has been altered from the part not yet altered. And the extensive acquisition of a corporeal quality in a similar way is to be imagined by the motion of the surface dividing the part altered from the part not yet altered.

The intensive acquisition of punctual quality is to be imagined by the motion of a point continually ascending over a subject point and by its motion describing a perpendicular line imagined [as erected] on that same subject point. But the intensive acquisition of a linear quality is to be imagined by the motion of a line perpendicularly ascending over the subject line and in its flux or ascent leaving behind a surface by which the acquired quality is designated. For example,

C D

A B

let *AB* be the subject line. I say, therefore, that
the intension of point *A *is imagined by the motion, or by the
perpendicular ascent, of point *C*, and the intension of line *AB*,
or the acquisition of the intensity, is imagined by the ascent of line *CD*.
Further, the intensive acquisition of a surface quality is in a similar way to
be imagined by the ascent of a surface, which (by its motion) leaves behind a
body by means of which that quality is designated. And similarly the intensive
acquisition of a corporeal quality is imagined by the motion of a surface
because a surface by its imagined flux leaves behind a body, and one does not
have to pose a fourth dimension, as has been said in the fourth chapter of the
first part.

One should speak and conceive of the loss of quality in the same way that we have now spoken of its acquisition, whether that loss is of extension or intensity. For such loss is imagined by movements which are the opposite of the movements described before. Furthermore, one ought to speak of the acquisition or loss of velocity, both in extension and intensity, in the same way we have just spoken of the acquisition or loss of quality.

*III.vii On the Measure of Difform Qualities and
Velocities*

Every
quality, if it is uniformly difform, is of the same quantity as would be the
quality of the same or equal subject that is uniform according to the degree of
the middle point of the same subject. I understand this to hold if the quality
is linear. If it is a surface quality, [then its quantity is equal to that of a
quality of the same subject which is uniform] according to the degree of the
middle line; if corporeal, according to the degree of the middle surface,
always understanding [these concepts] in a conformable way. This will be
demonstrated first for a linear quality. Hence let there be a quality
imaginable by Δ*ABC*, the quality being uniformly difform and
terminated at no degree in point *B*.

C

E

F G

A D B

And let *D* be the middle point of the subject line.
The degree of this point, or its intensity, is imagined by line *DE*.
Therefore, the quality which would be uniform throughout the whole subject at
degree *DE* is imaginable by rectangle *AFGB, *as is evident by the
tenth chapter of the first part. Therefore, it is evident by the 26^{th}
[proposition] of [Book] I [of the __Elements__] of Euclid that the two small
triangles *EFC* and *EGB* are equal. Therefore, the larger Δ*BAC*,
which designates the uniformly difform quality, and the rectangle *AFGB*,
which designates the quality uniform in the degree of the middle point, are
equal. And this is what has been proposed.

In the same
way it can be argued for a quality uniformly difform terminated in both
extremes at a certain degree, as would be the quality imaginable by quadrangle *ABCD*.
For let line *DE* be drawn parallel to the subject base and Δ*CED *would
be formed. The let line *FG* be drawn through the degree of the idle point
which is equal and parallel to the subject base. Also, let line *GD* be
drawn. Then, as before, it will be proved that Δ *CED *= ٱ*EFGD*.
Therefore, with the common rectangle *AEDB *added to both of them, the two
total areas are equal, namely quadrangle *ACDB*, which designates the
uniformly difform quality, and the rectangle *AFGB*, which would designate
the quality uniform at the degree of the middle point of the subject *AB*.
Therefore, by chapter ten of the first part, the qualities designatable by
quadrangles of this kind are equal.

It can be argued in the same way regarding a surface quality and also regarding a corporeal quality. Now one should speak of velocity in completely the same fashion as linear quality, so long as the middle instant of the time measuring a velocity of this kind is taken in place of the middle point [of the subject]. And so it is clear to which uniform quality or velocity a quality or velocity uniformly difform is equated. Moreover, the ratio of uniformly difform qualities and velocities is as the ratio of the simply uniform qualities or velocities to which they are equated. And we have spoken of the measure and ration of these uniform [qualities and velocities] in the preceding chapter.

Further, if a quality or velocity is difformly difform, and if it is composed of uniform or uniformly difform parts, it can be measured by its parts, whose measure has been discussed before. Now, if the quality is difform in some other way, e.g. with the difformity designated by a curve, then it is necessary to have recourse to the mutual mensuration of the curved figures, or to [the mensuration of] these [curved figures] with rectilinear figures; and this is another kind of speculation. Therefore what has been stated is sufficient.