how many axioms is Euclid’s construction built?
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Euclid’s axiom refers to one of the fundamental principles on which Euclidean geometry is based, formulated by the Greek mathematician Euclid around 300 BC in his monumental work, The Elements. In this work, Euclid establishes the logical structure of geometry based on a small number of self-evident truths, which he calls axioms or postulates. Euclid’s construction is thus based on five main axioms, which form the foundations of all classical geometry.
Before Euclid, geometry already existed in a practical form, used by the Egyptians and Babylonians to measure land or build monuments, but without a rigorous theoretical foundation. Euclid, a professor in Alexandria, had the innovative idea of constructing geometry as a logical system: he laid down a few basic principles considered to be self-evident, then deduced all other geometric properties from them using proofs.
This approach marked the birth of the axiomatic method, which would become a model for the mathematical sciences for more than two millennia.
Euclid’s five axioms are simple statements, considered to be intuitive truths about space. Here they are in their most well-known formulation:
Axiom 1 – A straight line can be drawn connecting any two points
This axiom establishes the possibility of connecting two points with a straight line, the most fundamental concept in geometry.
Axiom 2 – A finite straight line can be extended indefinitely
This means that a line has no limits: it can be extended in both directions to infinity.
Axiom 3 – A circle can be drawn from a given center and radius.
This postulate introduces the concept of a circle and establishes the possibility of constructing it from a central point and a segment.
Axiom 4 – All right angles are equal to each other
This axiom establishes a standard for comparing right angles, thus defining perpendicularity as a universal concept.
Axiom 5 – If a line intersecting two other lines forms interior angles whose sum is less than two right angles, then these two lines intersect on the side where these angles are located
This is the famous parallel postulate. It states that there is only one line parallel to another passing through a given point. This axiom, more complex and less intuitive than the others, would later become the subject of mathematical debates that led to the birth of non-Euclidean geometries.
Euclid’s five axioms dominated geometric thinking for nearly 2,000 years. Based on these axioms, Euclid constructed a coherent system comprising more than 450 proven propositions, ranging from the properties of triangles to the theory of regular solids (the famous Platonic solids).
This work profoundly influenced logic, philosophy, and science. The idea that all knowledge could be deduced from first principles inspired thinkers such as Descartes, Spinoza, and Newton.
However, the fifth axiom, that of parallels, has long posed a problem. Less obvious than the others, it has given rise to centuries of attempts to prove it based on the first four. These efforts failed, ultimately leading, in the 19th century, to the discovery of non-Euclidean geometries by Gauss, Lobachevsky, and Riemann, in which the parallel postulate is replaced by other hypotheses. These new geometries paved the way for Einstein’s general relativity, revolutionizing our conception of space and time.
Euclid’s construction is based on five fundamental axioms, formulated in The Elements. These simple principles served as the basis for all classical geometry and shaped Western scientific thought for centuries. Euclid’s axiom, particularly that of parallels, remains a cornerstone of mathematical thought and an emblematic example of the power of deductive logic.
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how many axioms is Euclid's construction built?
Answer
Euclid's geometric construction is based on five axioms, or postulates, set out in his major work The Elements, written around 300 BC.