Commentary on the First Book of Euclid's Elements

the principles to them, and the many come to exist around it and are referenced back to it. For let the geometer say that when four magnitudes are proportional, the alternating relation the alternating proportion (alternando) will also hold, and let him demonstrate it from the proper principles, which the arithmetician would never use. And again, let the arithmetician say that when four numbers are proportional, the alternating relation will also hold, and let him do this from the principles of his own science. Who, then, is the one who knows "the alternating relation" in itself, whether in magnitudes or in numbers, and the division of composed magnitudes or numbers, and likewise the composition of divided ones? For it is not the case that there are sciences and knowledge of divisible things, while we have no science of those that are immaterial and placed closer to intellectual contemplation. Rather, the knowledge of the latter is science by a much greater priority; from it, the many receive the common logoi rational principles, and the ascent of knowledge from more particular to more holistic things proceeds until we run back to the science of Being itself, insofar as it is Being. For this science does not deign to examine what belongs to numbers in themselves, nor what is common to all quantities, but contemplates the one and only substance and existence of all things; for this reason, it is the most comprehensive of all sciences, and all others receive their principles from it. For those above always provide the first hypotheses of the proofs to those below them, and the most perfect of the sciences grants from itself to all...