Hitherto Unpublished Works, Fascicle 4

[The ratio] is composed of two proportionals which are two equalities of the halves of the diameter to the semidiameter; but that same ratio is composed of the ratio of the diameter to the side doubled. Therefore, the ratio of the diameter to the side is one of those equals, 94 b 1. and thus the ratio of the diameter to the side will be subduplicate. In this way, the followers of Democritus original: "Democritei"; the followers of the Greek philosopher Democritus. They believed all matter was composed of indivisible atoms, which led them to argue that all lines must be commensurable because they would be made of a finite number of points. oppose the incommensurability incommensurabilitas: the mathematical property where two lengths, like the diagonal and side of a square, have no common unit of measurement. of the diameter to the side upon which their adversaries are based. This can be opposed in many other ways, namely, through the ninth and nineteenth [propositions] of the sixth book and through the twelfth of the same, and in other similar ways. But because it is not appropriate to explain these things here, I pass them over. I refer the pure natural philosopher to those things which I have explained regarding the seventh [proposition] of the tenth book of Euclid's Elements. i.e., in the Communia Mathematice.

Chapter 3. .

If, therefore, the parts of the world are finite and dissimilar, as was maintained before, and it has been proved that there are at least two, namely, the body moving with rectilinear motion Movement in a straight line, which Aristotle and Bacon attributed to the four earthly elements. and the body moving with circular motion, O. 6 a. now, thirdly, it must be considered that there must be four bodies moving with rectilinear motion which we call earth, water, air, and fire. Concerning three of them, it is manifest to the sense. This is evident regarding the earth and the water of the sea, which are placed before the sight and at which the sight is terminated. Air, however, is the medium in sight which we perceive by means of the sense. We know it exists because after it there occurs a dense object at which the sight is terminated. And since the sight is not terminated before it hits the dense object, we judge that between that object and us there is a rare body, since a vacuum cannot exist. For thus we perceive the transparent through the dense, as the authors of Perspective original: "autores Perspective"; the study of optics and vision. teach. i.e., Euclid, Ptolemy, and Alhazen. Likewise, earth and water are necessary for the generation of animals, plants, and minerals. For it is necessary that all things naturally consist of earth and water. Air is indeed necessary for breathing animals, because they could not live without it. But if these are the parts of the world, then a fourth element is necessarily...