Euclid's Elements

...any other easier and more convenient method and way for learning Geometry. "There is no royal road, O King," he said, "that leads to Geometry." Therefore, that it is sufficient for us to have declared with the testimonies of such great men what constant diligence of mind and eager will to learn young men should bring to these studies—not only for the sake of Geometry, which is the most noble in itself, but also for the sake of all philosophy—let this suffice.

Let us now speak about the title of the work, and at the same time, about the purpose of the Author. For as often as a title is taken from the argument of a work, the explanation of one almost reveals the other. Proclus, in my judgment, seems to have read Euclidou stoicheiosis Euclid's Element-making, while the one who left this work for us from the opinion of Theon used Euclidou stoicheion bib. 15 Books of Euclid's Elements, 15. Yet the same thing is signified by both, whether it is an Elementary instruction or the fifteen books of the elements. I have said, however, not that Theon—as many believe—but a certain associate of his, a man clearly learned, whoever he may have been, allowed us to read Euclid in the way he is now held, moved by the testimony of those words ek ton Theonos synousion from the discussions of Theon. For Ioannes, surnamed Philoponus, in the commentaries he composed on Aristotle, genuinely and with an example of a grateful heart, professed that he had gathered them from the conversations and disputes of Hamonius Hermeas. I would not deny, however, that the student of Theon, when he suppressed his own name, wished us to attribute this whole fruit of labor and industry to Theon alone.

But perhaps someone might ask, and not unjustly, why the Author brought forth only this name "elements" or "elementary," which is said of many things. For since it is accustomed to be said of the principles of letters, and of natural things, and of others, it was necessary to add of what thing they were the elements, or the instruction of elements, as was done later by the Latins, who added "of geometry." To one doubting this, we would say it was omitted for the reason that it is immediately known from the first words regarding the notion of a point, or, following Hamonius, who defends the Porphyrian inscription from the same fault, we affirm this inscription is by way of kat' exochen by excellence/pre-eminence, and a certain excellence of Geometry; even if it is made from a name that is common to many, it can be understood to be of Geometric elements only. Thus, by saying "the Poet," we understand Homer or Virgil; for the study of Geometry was frequent and very famous then.

Elements are called here those theorems that have the nature of a principle. For theorems (as Proclus writes) are such that some are accustomed to be called elements, others elementary, while others are determined outside the power of these. Therefore, those are called elements whose contemplation pertains to the knowledge of others, and from which appears the solution of those things that happen to be doubted within them. For just as the principles of a written voice are primary, simple, and indivisible, to which we give the name of elements—and every diction and speech consists of these—so too there are certain principal theorems of all Geometry, having the nature of a principle to those that follow, pervading all things, and providing demonstrations for many accidents, which they call elements.

Elementary things, however, are called those which pertain to many things and have a certain simple sweetness, but not that which belongs to elements; because their contemplation is not common to every science. Finally, those things that do not have a knowledge pertaining to many, nor demonstrate any known or elegant thing, fall outside the power of the elementary. Again, "element" is said in two ways, as Menæchmus says. For that which confirms is the element of that which is confirmed, such as the first of the second in Euclid, and the fourth of the fifth; thus, many other things will be said to be elements among themselves, since one is confirmed by the other. For from the fact that the external angles of rectilinear figures are equal to four right angles, the multitude of internal right ones is shown, and conversely, the former is shown from the latter. An element of this kind is similar to a lemma. Furthermore, an element is said in another way, into which—since it is more simple—a composite thing is resolved. In this way, however, not every thing is called an element, but those which are the most principal of those established in the reason of the effected thing, just as the Postulates and Axioms are the elements of theorems.

According to this meaning of element, the elements were constructed by Euclid: some of the Geometry that concerns planes, others of that which concerns solids. Thus, many have also written elementary instructions in Arithmetics and in Astronomy. The purpose of Euclid, therefore, in these books was to hand down the elements necessary for universal Geometry—that is, the most principal, most simple, and theorems most closely related to the first principles, without which the remaining parts of this science cannot be understood. For Euclid himself, in other books, and Aristarchus, Archimedes, Apollonius, Theodosius, Autolycus, Menelaus, Ptolemy, Pappus, Serenus, and the rest, use them everywhere as most well-known for their demonstrations. Regarding the arrangement and method of geometric discourses, it must be known (as says...