Euclid
(330 B.C. – 275 B.C.) Greek mathematician. Together with Archimedes and Apollonius of Perga, after him, Euclid was soon included in the triad of the great mathematicians of Antiquity. However, in light of the immense influence that his work would exert throughout history, it must also be considered as one of the most illustrious of all times.
Although he made contributions and relief corrections, Euclid has sometimes been seen as a mere compiler of Greek mathematical knowledge.
In reality, Euclid’s great merit lies in his work of systematization: starting from a series of definitions, postulates and axioms, he established by rigorous logical deduction the entire harmonious building of Greek geometry.
Judged not without reason as one of the highest products of human reason and admired as a finished and perfect system, Euclidean geometry would remain valid for more than twenty centuries, until the appearance, already in the nineteenth century, of the so-called non-Euclidean geometries.
Little is known for sure about Euclid’s biography, despite being the most famous mathematician of antiquity.
It is probable that he was educated in Athens, which would allow him to explain his good knowledge of the geometry elaborated in Plato’s school, although it does not seem that he was familiar with Aristotle’s works.
Euclid taught in Alexandria, where he opened a school that would end up being the most important in the Hellenic world, and achieved great prestige in the exercise of his teaching during the reign of Ptolemy I Sóter, founder of the Ptolemaic dynasty that would govern Egypt from the death of Alexander the Great until the Roman occupation.
It is said that the king required him to show an abbreviated procedure to access knowledge of mathematics, to which Euclid replied that there was no royal way to reach geometry.
This epigram, however, is also attributed to the mathematician Menecmo, as a replica of a similar demand by Alexander the Great.
Tradition has preserved an image of Euclid as a man of remarkable kindness and modesty, and has also transmitted an anecdote relating to his teaching, collected by Juan Estobeo: a young beginner in the study of geometry asked him what he would gain from his apprenticeship.
Euclid explained to him that the acquisition of knowledge is always valuable in itself; and since the boy had the pretense of obtaining some benefit from his studies, he ordered a servant to give him some coins.
The Elements of Euclid
Euclid was the author of several treatises, but his name is mainly associated with one of them, «The Elements», which rivals for its dissemination with the most famous works of universal literature, such as the Bible or Don Quixote.
It is, in essence, a compilation of works by previous authors (including Hippocrates of Chios), which was immediately surpassed by its general plan and the magnitude of its purpose.
Of the thirteen books that compose it, the first six correspond to what is still understood as flat or elementary geometry. In them Euclides collects the geometric techniques used in Pythagoras‘ school to solve what today are considered examples of linear and quadratic equations; it also includes the general theory of proportion, traditionally attributed to Eudoxo.
The books from the seventh to the tenth deal with numerical questions: the main properties of number theory (divisibility, prime numbers), the concepts of commensurability of segments to their squares and the questions related to the transformations of double radicals. The remaining three deal with the geometry of solids, culminating in the construction of the five regular polyhedrons and their circumscribed spheres, which had already been studied by Teeteto.
Of the remaining works by Euclides, we only have references or brief summaries from later commentators. The treatises on the Surface Places and the Conics already contained, apparently, some of the results subsequently presented by Apolonio de Perga.
In the Prisms the geometric theorems currently called projective theorems are developed; only the summary drawn by Pappo of Alexandria is preserved from this work. In Optics and Catoptrics we studied the laws of perspective, the propagation of light and the phenomena of reflection and refraction.
Two thousand years in force
The later influence of the Elements of Euclid was decisive; after its appearance, it was immediately adopted as an exemplary textbook in the initial teaching of mathematics, thus fulfilling the purpose that must have inspired Euclid. After the fall of the Roman Empire, his work was preserved by the Arabs and again widely disseminated from the Renaissance.
Beyond even the strictly mathematical field, Euclid was taken as a model, in his method and exhibition, by authors such as Galeno, for medicine, or Spinoza, for ethics. Not to mention the multitude of philosophers and scientists of all ages who, in their search for universally valid explanatory systems, had in mind the admirable logical rigour of Euclid’s geometry.
In fact, Euclid established what, from his contribution, was to be the classical form of a mathematical proposition: a statement logically deduced from previously accepted principles. In the case of the Elements, the principles taken as a starting point are twenty-three definitions, five postulates and five axioms or common notions.
The nature and scope of these principles have been the subject of frequent discussion throughout history, especially with regard to the postulates and, in particular, the fifth postulate, known as the parallel postulates.
According to this postulate, a point outside a line can only be drawn parallel to that line. Its different condition from the other postulates was already perceived from Antiquity itself, and there were several attempts to demonstrate the fifth postulate as a theorem.
The efforts to find a demonstration were unsuccessful and continued until the 19th century, when some unpublished works by Carl Friedrich Gauss (1777-1855) and research by the Russian mathematician Nikolai Lobachevski (1792-1856) showed that it was possible to define a perfectly consistent geometry (hyperbolic geometry) in which the fifth postulate was not fulfilled.
Thus began the development of non-Euclidean geometries, of which the elliptical geometry of the German mathematician Bernhard Riemann (1826-1866) stands out, judged by Albert Einstein as the one that best represents the relativistic space-time model.
Source: Biographies and lives
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