Freshman Math; 'The Organization of Euclid's 'Elements'

                                'The Organization of Book One of Euclid's 'Elements'

Euclid’s method of organization contributes to the consideration of the truths he is trying to prove, and gives an example of how to organize information, quite a feat when one considers all the ways to approach the large body of knowledge of geometry which he pares down in his Elements. The name ‘elements’ indicates that which is more simple, into which the composite is divided. So things that are more like principles are elements of those things related to the principles as results, as postulates are elements of theorems. Euclid founds his investigation on common notions, to make them reasonable, and moves organically from the more simple to the more complex. To do this he orders the movement through each individual proposition similarly and he orders all the propositions within the book according to his preference, and therefore do not necessarily need to be ordered so to come to the same truths. Euclid’s method of organization reflects the pattern of organization that is natural to man.

            The propositions have a general order that can be seen in some way in all of them. At the beginning of each proposition, there is the enunciation, where he says what he wants to prove. Euclid sticks to embracing theorems in general terms, and this is seen in his enunciation. For example, in Proposition 2 of Book 1, the enunciation reads, “To place at a given point (as an extremity) a straight line equal to a given straight line.” To divide his demonstration of this theorem into something more specific would make it more difficult to understand, like if he actually gave a point at a certain location instead of calling it a given point, which indicates that any point one wants to use will be compatible with the truth of the theorem. In this way the truth of his propositions may be adapted in a broader setting, and this is important when he must build upon them in later propositions.

            After his enunciation, there is the ‘setting out,’ in each proposition,

in which he says what is given and adapts it beforehand for the audience. The definition comes next, in which he tells us how the given applies to the enunciation and shows what he needs to prove specifically.

            In the construction, Euclid adds what is needed to the given to find what is sought. He often speaks in a way that is reminiscent of creation, with the words, ‘let there be…’ For example, in Proposition 2, his construction reads, “From the point A to the point B let the straight line AB be joined; and on it let the equilateral triangle DAB be constructed.” God uses the same language of creation in the book of Genesis when he creates the world – the words that pronounce them so make them so.  

            After the construction, the machination draws the inferences needed by reasoning from previously acknowledged facts. In Book 1, the previously acknowledged facts are his definition, postulates, and common notions. One must assume these to be true before any more truths can be made manifest. Of course, as the book progresses, the truths from previous theorems may contribute to this body of knowledge. Finally the proof is stated, returning to the given to confirm it.

             Besides the general ordering within the propositions, there is a different sense of ordering within Book 1 of each of the theorems or constructions. Book 1 may be likened to a novel in which the Prop 47, the Pythagorean prop, is the climax of the plot. It seems that Euclid is pressing to prove the Pythagorean prop, which is why he begins his Elements with figures instead of, for instance, magnitudes. Book 1 progresses through each general understanding of geometry needed to understand and prove prop 47, but instead of sticking strictly with the bare minimum of theorems, he is willing to veer off slightly from this ‘plot’ to prove other theorems that will be useful in later books. For example, Proposition 41 is necessary, where he proves that ‘if a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle.’ Within this theme, he also explores relationships between parallelograms and triangles in the next few propositions following 41, and discusses truths about parallel lines, equal parallel lines, and parallelograms in the seven or so props before it. He does this though they are not immediately necessary for Prop 47.

            Euclid’s ordering of Book 1 and all his books imitate the way that man orders his world naturally. He begins with general manifest notions and postulates, as well as truths about figures that are immediately present to our sense of logic, and from there he dissects and divides the truths of geometry according to the pattern that he deems best. In a similar manner, man can order the animals he encounters into different species and genus’ and this division is far more immediate to our senses than the divisions into classes and orders, which are as arbitrary as Euclid’s decision to order his first book around the Pythagorean theorem. This is why Lineaus says in his Systema Naturae that ‘the classes and orders are arbitrary; the genera and species are natural.’ Euclid’s order is logical within each proposition, naturally built upon the first and most know things, and subjectively ordered by him in each book.