Euclid
Euclid (Greek Εὐκλείδης, Eukleidēs, Latin Euclīdēs) was a Greek mathematician and geometer (ca. 325 BC- ca 265 BCE). He is known as "the father of geometry"..). He was the founder of the city's mathematics school.
His most famous work was Elements, often considered the most successful textbook in the history of mathematics. Properties of geometric objects and natural numbers are deduced from a small set of axioms. This work, one of the oldest known treatises that systematically presents, with proofs, a large set of theorems on geometry and theoretical arithmetic, has known hundreds of editions in all languages, and their subjects remain at the base of the teaching of mathematics at the secondary level in many countries. From the name Euclid derive Euclid's algorithm, Euclidean geometry (and non-Euclidean geometry) and Euclidean division. He also wrote on perspective, conic sections, spherical geometry, and number theory.
Biography
His life is little known, because he lived in Alexandria (a city located in the north of Egypt) during the reign of Ptolemy I. Certain Arab authors affirm that Euclid was born in Tire and lived in Damascus. There is no direct source about the life of Euclid: there is no letter, no autobiographical indication (even in the form of a preface to a work), no official document, and not even any allusion by one of his contemporaries. As the historian of mathematics Peter Schreiber summarizes it, "about the life of Euclid, not a single certain fact is known". Other data exist, but they are unreliable. He was the son of Naucrates and three hypotheses are considered:
- Euclides was a historical mathematician who wrote Elements and other works attributed to him.
- Euclides was the leader of a team of mathematicians working in Alexandria. All of them contributed to writing complete works of Euclideseven signing the books with the name Euclides after his death.
- The complete works of Euclides were written by a team of Alexandria mathematicians who took the name Euclides of the historical character Euclides de Mégara, who had lived a hundred years earlier.
Possibly, Euclid studied at Plato's Academy learning the basics of his knowledge.
Proclus, the last of the great Greek philosophers, who lived around 450, wrote important commentaries on Book I of the Elements. These commentaries are a valuable source of information on the history of Greek mathematics. Thus we know, for example, that Euclid gathered contributions from Eudoxus of Cnido in relation to the theory of proportion, and from Theaetetus on regular polyhedra.
Precisely, the oldest known writing related to the life of Euclid appears in a summary on the history of geometry written in the V of our era by the Neoplatonian philosopher Proclus, commentator of the first book of the Elements. Proclus himself does not give any source for his indications. He says only: «gathering his Elements , [Euclid] has co-ordinated many [...] and evokes in irrefutable demonstrations what his predecessors had taught in a relaxed manner. This man has lived, on the other hand, under the first Ptolemy, since Archimedes [...] mentions Euclid. Euclid is therefore more recent than Plato's disciples, but older than Archimedes and Eratosthenes."
If the chronology given by Proclus is accepted, Euclid lived between Plato and Archimedes, and was a contemporary of Ptolemy I, approximately around 300 BC.
No document contradicts these few sentences, nor does it truly confirm them. Euclid's direct mention of the works of Archimedes comes from a passage considered doubtful.
Archimedes refers to some results of the Elements and an ostracan, found on the island of Elephantine and dated to the III before our era deals with figures studied in book XIII of the Elements, such as the decagon and the icosahedron, but without exactly reproducing the Euclidean statements; they could, therefore, come from sources prior to Euclid. The approximate date of 300 BC is, even so, judged compatible with the analysis of the content of Euclid's work and is the one adopted by historians of mathematics.
On the other hand, an allusion to the mathematician of the IV century AD Papus of Alexandria who suggests that students of Euclid they would have taught in Alexandria. Some authors have associated Euclid with the Museion of Alexandria on this basis, but he does not appear in any official document. The qualifier often associated with Euclid in antiquity is simply Stoitxeiotes, the author of the Elements.
Several anecdotes circulate about Euclid, but as they also appear to other mathematicians, they are not considered real: thus, the famous one, explained by Proclus, according to which Euclid would have responded to Ptolemy -who wanted an easier way than those of the Elements - that there were no real paths in geometry; a variant of the same anecdote is also attributed to Menechus and Alexander the Great. Likewise, from late antiquity, various details were added to the accounts of Euclid's life, without new sources, and often in contradictory ways. Some authors have Euclid born in Tyre, others in Gela; Several genealogies, particular owners, different dates of birth and death are attributed to him, to respect the rules of the genre, or to favor some interpretations. Various examples are given, and are refuted. In the Middle Ages and early Renaissance, the mathematician Euclid is often confused with a contemporary philosopher of Plato, Euclid of Megara.
Work
Mentions of works attributed to Euclid appear in several authors, in particular in Papo's Colección matemática (usually dated in the III or IV) and in the Commentary on Euclid's Elements due to Proclus. Only a part of these works has reached our days.
There are five works that have come down to us: Data, On Divisions, Catoptrics, Appearances of the Sky and Optics. Various Arabic sources attribute Euclid to treatises on mechanics. On the heavy and the light contains, in nine definitions and five propositions, the Aristotelian notions of the movement of bodies and the concept of specific gravity. On Equilibrium deals with lever theory also in an axiomatic way, with one definition, two axioms and four propositions. A third fragment, about the circles described by the ends of a moving lever, contains four propositions. These three works complement each other in such a way that it has been suggested that they are remnants of a single treatise on mechanics written by Euclid.
The Elements
His work Elementos is one of the best-known scientific productions in the world and was a compilation of the knowledge imparted in the academic field of that time. The Elements were not, as is sometimes thought, a compendium of all geometric knowledge, but rather an introductory text covering all elementary mathematics, that is, arithmetic, synthetic geometry, and algebra..
The Elements are divided into thirteen books or chapters, of which the first half dozen are on elementary plane geometry, the next three on number theory, book X on incommensurables, and last three mainly on geometry of solids.
In books dedicated to geometry, the study of the properties of lines and planes, circles and spheres, triangles and cones, etc., is formally presented, starting only from five postulates; that is to say, of the regular forms. Probably none of the results of the Elements were first demonstrated by Euclid, but the organization of the material and its exposition are undoubtedly due to him. In fact, there is much evidence that Euclid used earlier textbooks when he wrote the Elements, since he presents a large number of definitions that are not used, such as an oblong, a rhombus and a rhomboid. Euclid's theorems are the ones that are generally learned in the modern school. To name some of the best known:
- The sum of the inner angles of any triangle is 180°.
- In a rectangle triangle, the square of the hypotenuse is equal to the sum of the squares of the catetos, which is the famous theorem of Pythagoras.
In books VII, VIII and IX of the Elements the theory of divisibility is studied. Consider the connection between the perfect numbers and the Mersenne primes (known as the Euclid-Euler theorem), the infinity of prime numbers (Euclid's Theorem), Euclid's lemma on factorization (leading to the Fundamental Theorem of arithmetic on the uniqueness of prime factorizations) and Euclid's algorithm to find the greatest common divisor of two numbers.
Euclid's geometry, in addition to being a powerful deductive reasoning instrument, has been extremely useful in many fields of knowledge; for example, in physics, astronomy, chemistry and various engineering. Of course, it is very useful in mathematics. Inspired by the harmony of Euclid's presentation, the Ptolemaic theory of the universe was formulated in the second century, according to which the Earth is the center of the universe, and the planets, the Moon and the Sun revolve around it in perfect lines., that is, circles and combinations of circles. However, Euclid's ideas are a considerable abstraction from reality. For example, he assumes that a point has no size; that a line is a set of points that has neither width nor thickness, only length; that a surface has no thickness, etc. Since the point, according to Euclid, has no size, it is assigned a null or zero dimension. A line has only length, so it has a dimension equal to one. A surface has no thickness, no height, so it has dimension two: width and length. Finally, a solid body, like a cube, has dimension three: length, width, and height. Euclid tried to summarize all mathematical knowledge in his book The Elements. Euclid's geometry was a work that lasted without variations until the 19th century.
Of the starting axioms, only the parallel one seemed less evident. Various mathematicians unsuccessfully tried to dispense with this axiom by trying to deduce it from the rest of the axioms. They tried to present it as a theorem, without achieving
Finally, some authors created new geometries based on invalidating or substituting the axiom of parallels, giving rise to "non-Euclidean geometries". These geometries have as main characteristic that when changing the axiom of parallels, the angles of a triangle no longer add up to 180 degrees.
Data
The Data (Δεδομένα) is the only other work by Euclid that deals with geometry and of which there is a version in Greek (it is, for example, in the X manuscript discovered by Peyrard). It is also described in detail in book VII of the Mathematical Collection de Papo, the «Treasure of analysis», closely related to the first four books of the Elements. It deals with the type of information given in geometric problems, and its nature. The Data is situated within the framework of flat geometry and is considered by historians as a complement to the Elements, in a form more appropriate to the analysis of problems. The work contains 15 definitions, and explains what a geometric object means, in position, in shape, in size, and 94 theorems. These explain that, if some elements of a figure are given, other relations or elements can be determined.
On divisions
This work (Περὶ διαιρέσεων Βιβλίον) is described in the Commentary of Proclus, but is lost in Greek; there are parts in Latin (De divisionibus), but above all there is a manuscript in Arabic discovered in the XIX, which contains 36 propositions, four of which are proved.
Deals with the division of geometric figures into two or more equal parts or into parts of given proportions. It is similar to a work from the III century AD. C. of Heron of Alexandria. In this work he tries to construct lines that divide given figures in given proportions and shapes. For example, given a triangle and a point inside the triangle, one is asked to construct a line passing through the point and cutting the triangle into two figures with the same area; or, given a circle, construct two parallel lines, so that the portion of the circle that they limit makes up one third of the surface of the circle.
On fallacies (Pseudaria)
On Fallacies (Περὶ Ψευδαρίων), a text on errors in reasoning, is a lost work, known only from the description given by Proclus. According to him, the objective of the work was to accustom beginners to detecting false reasoning, particularly those that imitate deductive reasoning and thus have the appearance of truth. He gave examples of paralogisms.
Four books on conic sections
Four Books on Conic Sections (Κωνικῶν Βιβλία) is currently lost. It was a work on conic sections that was expanded by Apollonius of Perga in a famous book on the same subject. It is likely that the first four books of Apollonius's work came directly from Euclid. According to Papo, "Apollonius, having completed Euclid's four books of conics, and having added four more, left eight volumes of conics." Apollonius's conics quickly superseded the original work, and by Papo's time, Euclid's work was lost.
Three nano books
Three books of porismas (Πορισμάτων Βιβλία) could have been an extension of his work on conic sections, but the meaning of the title is not entirely clear. It is a work that is lost. The work is evoked in two passages of Proclus and, above all, it is the subject of a long presentation in book VII of the Collection of Papo, the «Treasure of analysis», as a significant example and of a wide scope of the analytical approach. The word porisma has several uses: according to Papo, it would designate here an intermediary statement between theorems and problems. Euclid's work would have contained 171 statements of this type and 38 lemmas. Pappos gives examples, such as "if, from two given points, lines are drawn that intersect in a given line, and if one of these carves a segment on a given line, the other will do the same on another line, with a relation fixed between the two cut segments.» Interpreting the exact meaning of what is a porism, and finally restoring all or part of the statements of Euclid's work, based on the information left by Papo, has occupied numerous mathematicians: the attempts best known are those by Pierre Fermat in the XVII by Robert Simson in the XVIII, and especially by Michel Chasles in the XIX. If Chasles' reconstruction is not taken seriously as such by current historians, it has given the mathematician the opportunity to develop the notion of an inharmonic relation.
Two books on loci
Τόπων Ἐπιπέδων Βιβλία Β' was about loci on surfaces or loci that were surfaces themselves. In a later interpretation, it is hypothesized that the work could have dealt with quadric surfaces. It is also a lost work, consisting of two books, mentioned in Papo's Treasure of Analysis. The indications given in Proclus or Papo about these places by Euclid are ambiguous and who exactly was asked in the work is not known. In the tradition of ancient Greek mathematics, places are sets of points that verify a given property. These sets are often straight lines, or conic sections, but they can also be flat surfaces, for example. Most historians estimate that Euclid's places could be about surfaces of revolution, spheres, cones, or cylinders.
Sky Appearances
Appearances of Heaven or Phenomena (# Φαινόμενα) is a treatise on positional astronomy, preserved in Greek. It is quite similar to a work by Autolycus (On the notion of the sphere) and talks about the application of the geometry of the sphere to astronomy and has survived in Greek, in various manuscript versions, the earliest of which dates from X. This text explains what is called "little astronomy", in contrast to the themes dealt with in Ptolemy's Great Composition (the Almagest). It contains 18 propositions and is close to of the preserved works on the same subject by Autolico de Pitane.
Optics
Optics (Ὀπτικά) is the oldest surviving Greek treatise, in various versions, devoted to problems that we would now say perspective and apparently intended for use in astronomy, it takes the form of Elements: it is a continuation of 58 propositions whose proof rests on definitions and postulates stated at the beginning of the text. In his definitions, Euclid follows the Platonic tradition, which states that vision is caused by rays emanating from the eye. Euclid describes the apparent size of an object in relation to its distance from the eye, and investigates the apparent shapes of cylinders and cones when viewed from different perspectives.
Euclid shows that the apparent sizes of equal objects are not proportional to their distance from our eye (proposition 8). He explains, for example, our vision of a sphere (and other simple surfaces): the eye sees a surface lower in the middle of the sphere, an even smaller proportion as the sphere is nearer, even if the surface to be seen appears larger, and the outline of which is seen is a circle. It also details, according to the positions of the eye and the object, how a circle appears to us. The treatise, in particular, contradicts an opinion held in some schools of thought, according to which the real size of objects (particularly of celestial bodies) is their apparent size, which is seen.
Papo considered that these results were important in astronomy and included Euclid's Optics, together with his Phenomena, in a compendium of minor works that had to be studied before the Almagesto, by Claudi Ptolemeu.
Treatise on music
Proclus attributes to Euclid a Treatise on music (Εἰσαγωγὴ, Ἁρμονική), which, like astronomy, theoretical music, for example in the form of an applied theory of proportions, is listed among the mathematical sciences. Two small writings have been preserved in Greek, and have been included in ancient editions of Euclid, but their adjudication is uncertain, as well as their possible links with the Elements. The two writings (a Section of the canon on musical intervals and an Harmonic Introduction) are, on the other hand, considered to be contradictory and the second, at least, is now considered by specialists as from another author.
Works falsely attributed to Euclid
Catoptrics (Κατοητρικά) deals with the mathematical theory of mirrors, in particular the images formed in flat and spherical concave mirrors. The attribution of it to Euclid is doubtful; its author could have been Theon of Alexandria. It appears in Euclid's text on optics and in Proclus' commentary. It is now considered lost, and in particular, Catoptrics, long published as a continuation of the Optics in older editions, is no longer attributed to Euclid; it is considered to be a later compilation.
Euclid is also mentioned as the author of fragments in relation to mechanics, specifically in texts on the lever and balance, in some manuscripts in Latin or Arabic. The award is now considered doubtful.
Editions
- The first edition of the modern era of the works of Euclides in Greek is that of David Gregory, in Oxford in 1703, with a Latin translation. François Peyrard made an edition in 3 volumes and 3 languages (Greek, Latin and French) of the Elements and Date (i.e., of all the texts of Euclides of pure mathematics known in Greek) in Paris, 1814-1818.
- The reference edition of Euclides in Greek remains that of Heiberg and Menge, dated at the end of the XIX: Heiberg; Menge (1883). Teubner, ed. Euclidis operates omnia. Leipzig.
- It includes a Latin translation along with the Greek text and contains all the well-known writings (including the dubious awards), as well as several comments by ancient authors.
Acknowledgment
- The Euclides lunar crater bears this name in his memory.