- NICOLE ORESME

On Quadrangular Quality

A certain quality is imaginable by a rectangle- in fact by any such rectangle constructed on the same base - and by no other type of figure can it be designated. This last part is made clear by means of the prior statements in chapter six.

And so let there be rectangle ABCD [see Fig.  54.1(a)]. Therefore, it is possible that the quality of line AB be proportional in intensity to this rectangle in altitude. Therefore, it will be proportional to any rectangle constructed on AB, because all such rectangles are of proportional, although unequal, altitude. Therefore, by chapter seven, this quality is imaginable by rectangle ABCD and similarly by rectangle ABEF which is greater and also by one that is less. Moreover, any such quality is said to be "uniform" or 'of equal intensity" in all of its parts.

Again it ought to be known that some quality is imaginable by a quadrangle having two right angles on the base and the other two angles unequal, e.g., by quadrangle ABCD [see Fig. 54.1(b)] and by every quadrangle constructed on base AB which is of proportional altitude, whether it be greater or less, as is clear in chapter seven. Moreover, any such quality is spoken of as "uniformly difform terminated in both extremes at some degree," so that the more intense extreme is designated in the acute angle C and the more remiss in the obtuse angle D. The superior line, e.g., line CD, is called "the line of summit," or in relation to quality it can be called "the line of intensity" because the intensity varies according to its variation.

And so every uniform quality is imagined by a rectangle and every quality uniformly difform terminated at no degree is imaginable by a right triangle. Further, every quality uniformly difform terminated in both extremes at some degree is to be imagined by a quadrangle having right angles on its base and the other two angles unequal. Now every other linear quality is said to be "difformly difform" and is imaginable by means of figures otherwise disposed according to manifold variation. Some modes of the "difformly difform" will be examined later. The aforesaid differences of intensities cannot be known any better, more clearly, or more easily than by such mental images and relations to figures, although certain other descriptions or points of knowledge could be given which also become known by imagining figures of this sort: as if it were said that a uniform quality is one which is equally intense in all parts of the subject, while a quality uniformly difform is one in which if any three points [of the subject line] are taken, the ratio of the distance between the first and the second to the distance between the second and the third is as the ratio of the excess in intensity of the first point over that of the second point to the excess of that of the second point over that of the third point, calling the first of those three points the one of greatest intensity.

Let us clarify this first with respect to a quality uniformly difform which is terminated at no degree and which is designated or imagined by TABC [see Fig. 54.2(a)]. With the three perpendicular lines BC, FG, and DE erected, then let HE be drawn parallel to line DE and similarly GK parallel to line FB. Therefore, the two small triangles CKG and GHE are formed and they are equiangular. Hence, by [proposition] VI.4 of [the Elements of] Euclid, GK/EH =CK/GH, CK and GH being excesses. And since GK=FB and similarly EH =DF, so FB/DF= CK/GH, FB and DF being the distances on the base of the three points and CK and GH being the excesses of altitude proportional to the intensity of these same points. Since, therefore, the quality of line AB is such that the ratio of the intensities of the points of the line is as the ratio of the altitudes of the lines perpendicularly erected on those same points, that which has been proposed is evidently clear, namely that the ratio of the excess in intensity of the first point over the second to the excess of the second over the third is the same as the ratio of the distance between the first and second points to the distance between the second and the third, and similarly for any other three points. Hence what we have premised in regard to a qual ity difform in this way is quite fitting, and so it (this quality) was well designated by such a triangle.

By the same method the aforesaid description of property can be demonstrated for a quality uniformly difform terminated in both extremes at [some] degree, and thus for one which we let be imagined by quadrangle ABCD in which line DE is drawn parallel to base AB forming ADEC [see Fig. 54.2(b)]. Then let lines of altitude be drawn in the quadrangle and also transversals parallel to the base in this triangle, thus forming small triangles. And then one can easily argue concerning the excesses and the distances in this triangle just as was argued in the other one. This will be easily apparent to one who is observant.

Further, every quality which is disposed in [any] other way than those described earlier is said to be "difformly difform." It can be described negatively as a quality which is not equally intense in all parts of the subject nor in which, when any three points of it are taken, the ratio of the excess of the first over the second to the excess of the second over the third is equal to the ratio of their distances.