Significance and Applications


Influenced by Euclid's Elements

So why does Elements still stands today as one of the most successful and influential books ever written? Why do we still teach its logical structures in schools all around the world? It is because of the way he implemented the axiomatic system. Euclid's ability to take centuries of mathematical knowledge and break it down into simple definitions, postulates, axioms (or common notions), and build a structure in which propositions could be tested and simplified. Proclus again stated "Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus' and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors."(15) His methods of deduction influenced scientist like Copernicus, Kepler, Galileo, and Newton, in their efforts to organize and prove their own theorems. Because of the spread of the publication of Elements philosophers such as Thomas Hobbes, Baruch Spinoza, and previously mentioned Bertrand Russel adopted the axiomatic deductive structure in their own works. Abraham Lincoln kept a copy close to him and stated, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies."(16) Even Einstein called is copy of Elements the "holy little geometry book." Along with a magnetic compass he said that they were the two gifts that had the greatest influence on him as a boy.(17) It would be an intensive undertaking to try and list all of the influences Elements had on how we collect and process knowledge in the modern world. The next section gives just a few branches of education and science that have been influenced by Euclid. (Below are two videos that provide a little more about Elements influnce.)

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Section Description
Video Presentation This video presents Euclid's Elements and other history.(g)
Lincoln This is a scene from the movie Lincoln.(h)


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Application and other Branches of Proof

The key to understanding Euclid's Elements influence is to recognize the axiomatic system and the way in which it allows a person to logically derive theorems. We have already mentioned that without this structure it would have been impossible to discover the realm of non-Euclidean Geometry. With this cornerstone of math basic sets of conventions can be logically structured in a systematic way so that conjectures can be proven to exist. This method doesn't rely on trial and error, or intuition. To Euclid, the creation of mathematics needed to build upon "self-evident truths" and that experimentation and specific example could never be the foundation of proofs. In building other branches of mathematics it is essential to have assumptions upon which the system can be built. An axiom needed to be simple, specific, and independent of other axioms. "Because Euclid's axioms rely heavily on his own understanding of plane geometry, he failed to explicitly state some of his most basic assumptions, which resulted in his axiom set being incomplete."(18) It is from the axiomatic structure that other branches of philosophy and mathematics could be arranged, hopefully avoiding of the logical fallacy. This helped to further the understanding Logic.

Logic is a more precise science of formal principles of reasoning or correct inference. Aristotle is believed to have been the instigator of logical thought. Mathematical logic can be divided into the fields of set theory, model, theory, recursion theory and proof theory. It is through logic that many theoretical computer science applications have been advanced. Logic and deductive reasoning not only branches through geometry but also into arithmetic and analysis. The purpose of logic is to ensure that consistency, validity, completeness, and soundness of an argument.(19)

Logic and the use of mathematical truth tables is specifically connected to Boolean algebra. These tables are used to correctly organize statements or Boolean variables and check the consistency when binary logical operators are implemented. This method works well in helping to take simple mathematical ideas and checking for the soundness of arguments. This not only branches out and is used in analysis of set theory but is also effective in discrete mathematics.

The field of logical structures branch into Semantics or the study of meaning this steps away from the strict mathematical world and looks at the relationship of words, symbols, and phrases. In this structure linguistics are broken down into simple rules and principles for organizing language and sentence structure. The proper arrangement of words and phrases are known as syntax. Syntax refers to the grammatical structure or code of language whereas semantics deals with the meaning assigned to the symbols, characters and words. Obviously this structure is recognized not only in study of oral languages but also in the field of computers.(20)

One last way of looking at how Euclid'ss Elements has influenced other fields of learning would be the modern scientific method. Although many sciences rely on experimentation and inductive reasoning. The organization, formal and simplistic model has influenced scientists throughout the Scientific Revolution. The recovery of ancient science texts such as Euclid's Elements and Optics inspired more modern scientist to test theorems, create new terminology, and formally organize their results.(21)


Section Description
History and Background This section of the wepage introduces Euclid, who influenced him, Elements, and questions that arise from Elements.
Significance and Application This section addresses the influnce of Elements and applications into other fields of study.
Explanation of Mathematics This section looks at how Elements is organized, explains inductive and deductive principles, and provides a few examples of proofs
References This section provides the references cited in this webpage.


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