Euclid's Elements

Proclus holds that Geometry, just like other sciences, has certain defined principles from which it demonstrates what follows. Therefore, it is necessary to discuss the principles separately from the things that flow from them. One should render no account of the principles themselves, but one should confirm by arguments what follows from these principles. For no science demonstrates its own principles; rather, it establishes trust in them through themselves, since they are more evident than what is derived from them. It knows the former through themselves, but the latter through the former. Likewise, the natural philosopher produces arguments from a determined principle, positing that motion exists; so too does the physician, and the expert in other sciences and arts. If anyone mixes principles with things that flow from them, he disturbs the entire cognition, and he binds together things that in no way agree with one another. Therefore, one must distinguish the principles first, and then the things that follow. Indeed, Euclid observed this in each of his books, for before every treatment he exposes the common principles of this science and divides them into suppositions—or definitions—postulates, and axioms. These all differ from one another, and the axiom, the postulate, and the supposition are not the same, as Aristotle asserts. When a listener accepts a proposition immediately, without a teacher, as true and gives it the most certain belief, this is called an Axiom a self-evident truth, such as "things equal to the same thing are also equal to each other." But when someone hears a statement and does not yet have a notion of what is being said that would create belief by itself—yet they suppose it and assent to it while using it—that is a supposition an assumption accepted for the sake of argument; for example, that a circle is a figure of such a kind, we do not perceive this by any common notion, but we approve it upon hearing it without any demonstration. But when what is being said is unknown to the learner, and yet it is assumed with his assent, then we call it a postulate a requirement or demand, such as "all right angles are equal." Those things that arise from the principles are either Problems or Theorems. A problem a task to be performed is that in which something not yet existing is proposed to be found and constructed. A theorem a statement to be proved, however, is that in which it is demonstrated that something in a figure already established is or is not such and such. In this elementary instruction, therefore, who would not admire Euclid most highly for the order and selection of the theorems and problems he distributes throughout the elements? He did not assume everything he could have said, but only those things he could deliver in an elementary order. Furthermore, he employed various modes of syllogisms logical arguments: some receiving belief from causes, others proceeding from signs—all necessary, certain, and adapted to science. Moreover, he used all the dialectical ways and methods: dividing in the inventions of forms, defining in essential reasons, demonstrating in the progressions that occur from principles to the things sought, and finally resolving in the regressions that occur from the things sought to the principles. Indeed, one may observe in this treatise various species of conversions, both simple and compound; what can be converted as a whole to a whole, what as a whole to its parts, and vice versa, and what as parts to parts. Finally, there is an admirable disposition of all things, in the order and coherence of preceding and following, so that nothing at all seems able to be added or taken away.

In the first book, therefore, he treats of rectilinear figures, namely of triangles and parallelograms. And first he hands down the origin and properties of triangles, comparing them among themselves both according to angles and according to sides. Then, interjecting the properties of parallels, he passes to parallelograms, declares their origin, and demonstrates the properties that exist within them. Afterward, he shows the communication of triangles and parallelograms, and in what manner a parallelogram is made equal to a triangle. Finally, he discusses the squares described from the sides in right-angled triangles, and what proportion the square described from the side subtending the right angle has to those described from the sides containing it.

In the second book, the rectangular parallelogram and the gnomon the part of a parallelogram remaining after a smaller parallelogram is taken from a corner are defined. Then the proportions of rectangular parallelograms and squares, which are made from the sections of straight lines, are declared. Afterward, it discusses the squares that are described from the sides of obtuse-angled and acute-angled triangles, and what proportion the squares made from the sides subtending the obtuse and acute angle have to those described from the sides containing them. Finally, it addresses in what manner a square equal to a given rectilinear figure may be constructed.

In the third book, it treats of the things that happen to circles, and of straight lines drawn in a circle or to a circle, and likewise of the angles that consist at the centers or circumferences of circles.

In the fourth book, it deals with the inscriptions and circumscriptions of plane figures.

In the fifth, it deals with Analogies proportions.