Euclid, also called Euclid of Alexandria (b. 323 BC – d. 285 BC, approximate dates) was a Greek mathematician. His life coincided with the end of the Hellenistic age, being part of the next generation after Aristotle. There are not many historical references about the life of the mathematician, but it is possible that Euclid studied at the famous and important School of Plato, established a century earlier.
Euclid lived and taught mathematics in the Egyptian city of Alexandria during the reign of Ptolemy I (b. 323 BC – d. 285 BC) who is said to have asked Euclid to show him a way much easier to learn geometry, a moment that Euclid would have approached with fine humor answering Ptolemy I that “in geometry there are no special roads for kings”.
Also, without proven authenticity, it is said that Euclid was very fond of mathematics but in a non-materialistic way. Moreover, in the writings that have been preserved about him, Euclid is described as a modest man, very gentle and especially devoted to the study of geometry.
The Elements – the work that influenced the entire evolution of European mathematics
Euclid’s most important work is “Stihia” (in Romanian, “Elements”) a book that has been translated into over 300 foreign languages. The Elements was composed of 13 parts (called books) and laid the foundations of arithmetic and plane and spatial geometry.
In the first six parts, Euclid presented and synthesized the theorems of plane geometry, in the next three he expounded number theory (the connection between perfect numbers and Mersenne primes) – a perfect number is an integer equal to the sum of its divisors, from which the number itself is excluded.
The tenth part of the work Elements is a continuation of the study of irrational numbers, a study begun by Eudoxus (astronomer, mathematician and disciple of Plato), and the last three parts focused on the study of the geometry of solid bodies.
Masterpiece of how logic is applied to mathematics
Although it provides no clues as to the discovery of his proofs, Euclid’s treatise and theories contained in the Elements have stood the test of time, being studied and applied for more than 2,000 years, due to the value of the work, according to Britannica.
In the Elements (a work that is on the list of the 100 most influential books in human history), one finds definitions such as “the point is that which has no parts or size”, axioms “and the congruent are equal to each other”, “and the whole is greater than the parts”, “and two straight lines do not close a space between them”, as well as postulates (certain but unprovable truths) “from a point to any point a straight line can be drawn”, “from any center and any radius a circle can be described”, “all right angles are equal”, “a point is something that has no parts”, “the ends of a line are points”.
Euclid’s work also includes research in the field of optics
Euclid’s work also includes research in the field of optics expounded in the works “Optica” and “Catoptrica”. In Optics, for example, Euclid presents the notion of a ray of light, stating the law of rectilinear propagation of light – “Rays propagate in a straight line and go to infinity”. In “Catoptrica”, studies on the formation of images in concave, plane or spherical mirrors are presented, noting that “everything visible is seen in a straight line”.
The use of the abbreviation “qed” (quod erat demonstrandum, meaning “what was to be demonstrated”) also belongs to Euclid, this abbreviation being often mentioned at the end of a mathematical demonstration, but not only. The manuscripts of his famous work “The Elements” were written in Latin and Arabic.
He was not a first-rate mathematician
Apart from the Bible, the Elements is the most translated, published and studied book. He may not have been a first-rate mathematician, but Euclid certainly laid the foundation for deductive reasoning in geometry, a standard that has remained unchanged over time. Euclid’s theory remained valid for over 2300 years, until Janos Bolyai (Hungarian mathematician from Transylvania, 1802 – 1860) gave birth to a new theory in geometry known as non-Euclidean geometry.
Non-Euclidean geometry uses all the axioms and postulates expounded by Euclid in the Elements, minus the axiom of parallels which reads like this: through a point outside a line there is only one line parallel to it.
Euclid’s fifth postulate was not valid
In non-Euclidean hyperbolic geometry, usually called the geometry of Lobacevsky (1792 – 1856, Russian mathematician, who was awarded the rank of general by Tsar Nicholas I of Russia), through a given point at least two parallel lines can be drawn to a given line. In non-Euclidean elliptic geometry there are no parallel lines.
The creation of these non-Euclidean geometries proved that several geometric systems are logically possible.
In 1902, Henri Poincare (1854 – 1921, one of the greatest French mathematicians and physicists) proposed a simple model in which Euclid’s fifth postulate was not valid. The line is here defined by extension as the curve of the shortest path joining two points in the given space.