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.
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,
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 ΔABC, the quality being uniformly difform and terminated at no degree in point B.
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 26th [proposition] of [Book] I [of the Elements] of Euclid that the two small triangles EFC and EGB are equal. Therefore, the larger ΔBAC, 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 ΔCED 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 = ٱEFGD. 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.