Six Geometrical Exercises (Exercitationes Geometricae Sex)

V.

But since there are always threads in a web, and always leaves in books in a finite number—for they have some thickness—we must assume, in both methods, that lines in plane figures, and indeed planes in solids, are in an indefinite number, as being devoid of all thickness. We use these, however, with a distinction; for in the former method we consider them as collectively [compared], but in the latter, as distributively [compared].

VI.

For let there be, for example, any two plane figures, A B C D and E F G H, established between the same parallels, I K and L M; and let one of them, such as L M, be taken as the rule of the parallels capable of being drawn in an indefinite number within these same figures, some of which in the figure A B C D are N O, B D, P Q, etc., and in the figure E F G H are R S, F H, T V, etc. Now, therefore, we can compare the lines of the figure A B C D to the lines of the figure E F G H in two ways: namely, either collectively, that is, by comparing the aggregate to the aggregate; or distributively, that is, by comparing individually each straight line of the figure A B C D to each straight line of the figure E F G H that exists directly in line with it. According to the former rationale, the prior method proceeds, for it compares to one another the aggregates of all the lines of plane figures, and the aggregates of all the planes of solids, however many those may be. But according to the latter, the posterior method behaves; for it compares single lines to single lines, and single planes to single planes, established directly in line with the same. Both, however, deliver their own general rule for comparing the measurement of figures, of which the former offers such as this.