- NICOLE ORESME

Question 10

Consequently it is sought whether some quadrangular surface is uniformly difform in altitude.

It is argued in the negative: for no altitude is difformly uniform, therefore no altitude is uniformly difform. The consequence holds by analogy. The antecedent is evident, for it, that which is uniform or equal there is no difformity or inequality.

The opposite is argued: there is some uniform altitude, therefore there is some uniformly difform altitude.

In the first place we must consider the question under inquiry. Then secondly we must apply it to the matter as concerned with mean qualities.

In connection with the first, it is to be known that the altitude of a surface is measured by a perpendicular line lying directly upon the base, as can be evident in a figure [see Fig. 55.1]. Secondly, it is to be noted that a surface is said to be uniformly and equally high when all the lines by which the altitude is measured are equal; it is said to be difformly high when they are unequal and they rise to a line which is not parallel to the base. Thirdly, it is to be noted that an altitude is said to be uniformly difform when any three or more of the lines which are at equal distances apart exceed one another according to arithmetic proportion, i.e., by the amount that one line exceeds the second, so the second exceeds the third [see Fig. 55.2]. From this it is evident that the upper line limiting them [i.e., the perpendiculars] is a straight line not parallel to the base.
Fourthly, it is to be noted that an altitude is said to be difformly difform when the [perpendicular] lines do not exceed one another in this manner; and in such a case the line crossing through their summits is not a straight line [see Fig. 55.3). And the difformity in altitude varies according to the variation of such a [summit] line.

As for the second part, namely the mathematical mean which is in qualities and velocities, it is to be noted firstly that in quality two things are to be imagined, namely intensity according to degrees and extension through the subject; and therefore such a quality is imagined to have two dimensions.² Accordingly we sometimes say that it has "latitude," understanding by this, "intensity," on the ground that we understand its "extension" by the term "longitude." [Hence every latitude presupposes longitude.] Secondly it is to be noted that a quality can be imagined to reside in a point, or in an indivisible subject like a soul. It can also be imagined to be in a line, as well as in a surface or in a body.

1. Hence let this be the first conclusion, that the quality of a point or an indivisible subject is to be imagined as a line, for it has only one dimension, namely intensity. From this it follows that such a quality, like knowledge or virtue, ought not to be described as either "uniform" or "difform," just as a line is not properly said to he "uniform" or "difform." It follows also that one speaks improperly of a latitude of knowledge or virtue since no longitude is to be imagined there and every latitude presupposes longitude.

2. The second conclusion is that the quality of a line is imagined as a surface whose longitude is the rectilinear extension of the subject and whose latitude is its intensity which is imagined by lines perpendicular to the line which is the subject.

3. The third conclusion is that the quality of a surface is to be imagined, using a similar imagery, by means of a body whose longitude and latitude constitute the extension of the subject and whose depth is the intensity of the quality. And by like reasoning the quality of a whole body would have to be imagined as a body whose longitude and latitude would be the extent of the whole body and the depth its intensity.⁴ But someone may raise a doubt: if the quality of a point is imagined as a line, the quality of a line as a surface, and that of a surface as a body having three dimensions, therefore the quality of a body will be imagined to have four dimensions and be in another genus of quantity. I answer that such is not necessary, for just as a flowing point imaginatively produces a line. a line a surface, a surface a body, so if a body were imagined to flow it is not necessary for it to produce a fourth kind of quantity but in fact only a body. And it is on this account that Aristotle says in the first book of the On the Heaven⁵ that from this, i.e., from a body, no passage to another genus of quantity takes place by this method of imagining. One ought to speak similarly in the matter at hand. Hence one ought to speak [thus] of the quality of this line and similarly of the quality of a surface and of a body.

4. The fourth conclusion is that a uniform linear quality is to be imagined by a rectangle that is uniformly high, so that the extension is imagined by the base⁶ and the intensity is measured by a [summit] line parallel to it, as is evident in the figure [see Fig. 55.4] [and it is obvious, for just as any line which would be erected on the given [base] line would be equal to another, so any point there would be imagined as equally intense]. But a quality uniformly difform is to be imagined by a surface which would be uniformly difformly high, so that the line of altitude [i.e., the summit line,] would not be parallel to the base, as is evident in the figure [see Fig. 55.5]; still it would be a straight line. This can be proved.
The ratio is intensity of [any] points would be as the ratio in altitude of the perpendicular lines on these points. And this can be in two ways just as a surface uniformly difform in altitude can exist in two ways. In one way such a quality is terminated at no degree [i.e., zero] and then it is like a surface uniformly difformly high [terminated in one extreme] at no degree, i.e., like a triangle. Or [it is terminated] on both sides at a degree; in this case it is like a quadrangle whose [line of] altitude [i.e., summit line] would be a straight line not parallel to the base [see Fig. 55.6].

5. The penultimate [conclusion] is that from this latter together with the aforesaid it can be proved that a quality uniformly difform is equal to the middle degree, i.e., that it would be just as great in quantity as if it were uniform at the middle degree. And this can be proved as for a surface [see Fig. 55.7].

6. The last conclusion is that a quality difformly difform is to be imagined as a surface whose subject line would be the base and whose altitude [i.e., summit line,] would be a line which is neither straight nor parallel to the base. From this it is evident that such difformity can be imagined in almost an infinitude of ways according as this line of altitude [i.e., summit line,] can be multiply varied, as is evident in the figure [see Fig. 55.8].

But one might say: "Master, it is not necessary for it to be so imagined." I answer that the imagination [i.e., imagery,] is a good one. This is evident by Aristotle who imagines time by means of a line. Similarly in perspective it is expressly imagined that active force is to be imagined by means of triangular surfaces. Further, following this imagination I can more easily understand those things which are said about qualities uniformly difform and so on. Therefore, I say that the imagination is a good one.