Euclid

Summary

Euclid (Greek Εὐκλείδης, Eukleidēs, Latin Euclīdēs) was a Greek mathematician and geometrician (ca. 325 BC – ca. 265 BC). ca. 265 BC).He is known as “the father of geometry”.He was an active in Alexandria (ancient Egypt) in the time of Ptolemy I Soter (323 – 283 BC).He was the founder of the city”s school of mathematics.

His most famous work was the Elements, often considered the most successful textbook in the history of mathematics. The properties of geometric objects and natural numbers are deduced from a small set of axioms. This work, one of the oldest known treatises presenting in a systematic way, with proofs, a large set of theorems on geometry and theoretical arithmetic, has known hundreds of editions in all languages, and its topics remain at the basis of the teaching of mathematics at the secondary level in many countries. From Euclid”s name, he derived Euclid”s algorithm, Euclidean geometry (and non-Euclidean geometry), and Euclidean division. He also wrote on perspective, conic sections, spherical geometry and number theory.

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 claim that Euclid was born in Tyre and lived in Damascus. Certain Arab authors claim that Euclid was born in Tyre and lived in Damascus. There is no direct source on Euclid”s life: no letter, no autobiographical indication (even in the form of a preface in a work), no official document, and not even any allusion by one of his contemporaries. As summarized by the mathematical historian Peter Schreiber, “about Euclid”s life, 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:

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 constitute a valuable source of information on the history of Greek mathematics. Thus we know, for example, that Euclid gathered contributions from Eudoxus of Cnidus in relation to the theory of proportion, and from Theaetetus on regular polyhedra.

Precisely, the oldest known writing concerning the life of Euclid appears in a summary on the history of geometry written in the 5th century A.D. 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, and evokes in irrefutable demonstrations that which 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 thus 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, around 300 BC.

No document contradicts these few sentences, nor truly confirms them. Euclid”s direct mention of Archimedes” works comes from a passage considered doubtful.

Archimedes refers to some results of the Elements and an ostrachus, found on the island of Elephantine and dated III B.C.: it deals with figures studied in book XIII of the Elements, such as the decagon and the icosahedron, but without reproducing exactly the Euclidean statements; they could, therefore, come from sources prior to Euclid. The approximate date of 300 before our era is, however, considered compatible with the analysis of the content of the Euclidean work and is the one adopted by historians of mathematics.

On the other hand, an allusion of the mathematician of the IV of our era Papo of Alexandria, which suggests that students of Euclid would have taught in Alexandria. Some authors have associated on this basis Euclid with the Museion of Alexandria, but it does not appear in any official document. The adjective often associated with Euclid in antiquity is simply Stoitxeiotes, the author of the Elements.

Several anecdotes circulate about Euclid, but as they also appear for other mathematicians, they are not considered as real: thus, the famous one, explained by Proclus, according to which Euclid would have answered Ptolemy – who wanted an easier way than those of the Elements – that there were no real ways in geometry; a variant of the same anecdote is also attributed to Menecmus 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 a contradictory manner. Some authors thus make Euclid born in Tyre, others in Gela; various genealogies, particular masters, different dates of birth and death are attributed to him, in order to respect the rules of the genre, or to favor some interpretations. In the Middle Ages and at the beginning of the Renaissance, the mathematician Euclid is often confused with a contemporary philosopher of Plato, Euclid of Megara.

Mentions of works attributed to Euclid appear in several authors, in particular in the Mathematical Collection of Pappus (usually dated to the 3rd or 4th century) and in the Commentary on Euclid”s Elements by Proclus. Only a part of these works has survived to our days.

The works that have come down to us are five: Data, On Divisions, Catoptrics, Appearances of the Sky and Optics. From Arabic sources, several treatises on mechanics are attributed to Euclid. On the Heavy and the Light contains, in nine definitions and five propositions, the Aristotelian notions of motion of bodies and the concept of specific gravity. On equilibrium deals with the theory of the lever also in an axiomatic manner, with one definition, two axioms and four propositions. A third fragment, on the circles described by the ends of a movable 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 Elements is one of the best known scientific productions in the world and was a compilation of the knowledge taught in the academic world at the time. The Elements was not, as is sometimes thought, a compendium of all geometrical knowledge, but rather an introductory text covering all elementary mathematics, i.e. 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 the last three mainly on geometry of solids.

In the books devoted to geometry, the study of the properties of lines and planes, circles and spheres, triangles and cones, etc., i.e., of regular forms, is presented in a formal manner, starting from only five postulates. 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 writing The Elements, since he presents a large number of definitions that are not used, such as that of an oblong, a rhombus, and a rhomboid. Euclid”s theorems are the ones generally learned in modern school. To cite some of the best known:

In books VII, VIII and IX of The Elements the theory of divisibility is studied. It considers the connection between perfect numbers and Mersenne primes (known as the Euclid-Euler theorem), the infinity of prime numbers (Euclid”s Theorem), Euclid”s lemma on factorization (which leads to the fundamental theorem of arithmetic on the uniqueness of factorizations of primes) and Euclid”s algorithm for finding the greatest common divisor of two numbers.

Euclid”s geometry, besides being a powerful tool for deductive reasoning, has been extremely useful in many fields of knowledge; for example, in physics, astronomy, chemistry and various engineering fields. It is certainly very useful in mathematics. Inspired by the harmony of Euclid”s presentation, in the second century the Ptolemaic theory of the universe was formulated, 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, i.e., circles and combinations of circles. However, Euclid”s ideas constitute 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, and so on. Since a point, according to Euclid, has no size, it is assigned a dimension of zero. A line has only length, so it acquires a dimension equal to one. A surface has no thickness, no height, so it has dimension two: width and length. Finally, a solid body, such as 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 axiom of parallels seemed less evident. Several mathematicians tried unsuccessfully 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 succeeding in

Finally, some authors created new geometries based on invalidating or substituting the axiom of parallels, giving rise to “non-Euclidean geometries”. The main characteristic of these geometries is that by 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 of Euclid that deals with geometry and of which a Greek version is extant (it is, for example, in the X manuscript discovered by Peyrard). It is also described in detail in Book VII of Papo”s Mathematical Collection, the “Treasury 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. Data is placed in the framework of plane geometry and is considered by historians as a complement to the Elements, in a form more suited 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.

About the divisions

This work (there are pieces in Latin (De divisionibus), but above all a manuscript in Arabic discovered in the 19th century is preserved, which contains 36 propositions, four of which are demonstrated.

It deals with the division of geometric figures into two or more equal parts or parts of given proportions. It is similar to a third century A.D. work by Heron of Alexandria. In this work he tries to construct lines that divide given figures into given proportions and shapes. For example, it is asked, given a triangle and a point inside the triangle, to construct a line passing through the point and cutting the triangle into two figures of equal area; or, given a circle, to construct two parallel lines, so that the portion of the circle they limit makes one third of the area 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 the latter, the work was intended to accustom beginners to detect false reasoning, in particular those that imitate deductive reasoning and thus have the appearance of truth. He gave examples of parallelogisms.

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” 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” conics quickly replaced the original work, and by Papo”s time, Euclid”s work had been lost.

Three books of porisms

Three books of porisms (Πορισμάτων Βιβλία) could have been an extension of his work on conic sections, but the meaning of the title is not quite clear. It is a work that is lost. The work is evoked in two passages of Proclus and, above all, is the subject of a long presentation in Book VII of the Pappus Collection, the “Treasury of Analysis,” as a significant and far-reaching example of the analytic approach. The word porisma has several uses: according to Papo, it would designate here a statement of an intermediary type between theorems and problems. Euclid”s work would have contained 171 such statements and 38 lemmas. Pappos gives examples, such as “if, starting from two given points, straight lines intersecting on a given straight line are drawn, and if one of these carves on a given straight line a segment, the other will do the same on another straight line, with a fixed relation between the two segments cut. Interpreting the exact meaning of what is a porism, and eventually restoring all or part of the statements of Euclid”s work, from the information left by Pappus, has occupied many mathematicians: the best known attempts are those of Pierre Fermat in the seventeenth century, Robert Simson in the eighteenth century, and especially Michel Chasles in the nineteenth century. If Chasles” reconstruction is not taken seriously as such by today”s historians, it gave the mathematician the opportunity to develop the notion of anharmonic relation.

Two books on geometric places

Τόπων Ἐπιπέδων Βιβλία Β” was about geometric places on surfaces or geometric places that were themselves surfaces. In a later interpretation, it is hypothesized that the work could have dealt with quadric surfaces. It is also a lost work, of two books, mentioned in the Treasury of Papo”s analysis. The indications given in Proclus or Pappus about these places of Euclid are ambiguous and the exact question 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 locus could be surfaces of revolution, spheres, cones or cylinders.

Appearances of the sky

Appearances of the sky or Phenomena (

Optics

Optics (Ὀπτικά) is the oldest surviving Greek treatise, in several versions, devoted to problems that we would now say of perspective and apparently intended for use in astronomy, takes the form of Elements: it is a continuation of 58 propositions of which the 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 angles.

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 an inferior surface in the middle of the sphere, an even smaller proportion as the sphere is near, even if the surface seen seems larger, and the outline of the one seen is a circle. The treatise, in particular, contradicts an opinion defended in some schools of thought, according to which the real size of objects (in particular of celestial bodies) is their apparent size, that which is seen.

Papo considered these results to be important in astronomy and included Euclid”s Optics, together with his Phenomena, in a compendium of minor works to be studied before Claudi Ptolemeu”s Almagest.

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, figures 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 are their possible links with the Elements. The two writings (a Section of the canon on musical intervals and a harmonic Introduction) are, on the other hand, considered to be contradictory, and the second, at least, is now considered by specialists to be by another author.

Works falsely attributed to Euclid

Catoptrics (Κατοητρικά) deals with the mathematical theory of mirrors, in particular of images formed in planar concave and spherical mirrors. Its attribution to Euclid is doubtful; its author may 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, Catoptricus, long published as a continuation of Optics in ancient editions, is no longer attributed to Euclid; it is considered a later compilation.

Euclid is also mentioned as the author of fragments related to mechanics, specifically in texts on the lever and the balance, in some Latin or Arabic manuscripts. The attribution is now considered doubtful.

Other references

Sources

  1. Euclides
  2. Euclid