He was likely born c. He is mentioned by name, though rarely, by other Greek mathematicians from Archimedes c. This biography is generally believed to be fictitious. Proclus believes that Euclid is not much younger than these, and that he must have lived during the time of Ptolemy I c. Although the apparent citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before Archimedes wrote his. However, this hypothesis is not well accepted by scholars and there is little evidence in its favor.
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Pythagoras c. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around and is the basis of modern editions. Although known to, for instance, Cicero , no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. Mentioned in T. Private collection Hector Zenil. In , John Dee provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by Henry Billingsley.
Copies of the Greek text still exist, some of which can be found in the Vatican Library and the Bodleian Library in Oxford. The manuscripts available are of variable quality, and invariably incomplete.
By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text copies of which are no longer available. Ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. Heiberg and Sir Thomas Little Heath in their editions of the text.
Also of importance are the scholia , or annotations to the text. These additions, which often distinguished themselves from the main text depending on the manuscript , gradually accumulated over time as opinions varied upon what was worthy of explanation or further study. Influence[ edit ] A page with marginalia from the first printed edition of Elements, printed by Erhard Ratdolt in The Elements is still considered a masterpiece in the application of logic to mathematics.
In historical context, it has proven enormously influential in many areas of science. The austere beauty of Euclidean geometry has been seen by many in western culture as a glimpse of an otherworldly system of perfection and certainty. Vincent Millay wrote in her sonnet " Euclid alone has looked on Beauty bare ", "O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized! Einstein recalled a copy of the Elements and a magnetic compass as two gifts that had a great influence on him as a boy, referring to the Euclid as the "holy little geometry book".
Much of the material is not original to him, although many of the proofs are his. The Elements still influences modern geometry books. Further, its logical axiomatic approach and rigorous proofs remain the cornerstone of mathematics. In modern mathematics[ edit ] One of the most notable influences of Euclid on modern mathematics is the discussion of the parallel postulate.
In Book I, Euclid lists five postulates, the fifth of which stipulates If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles , then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. The different versions of the parallel postulate result in different geometries. This postulate plagued mathematicians for centuries due to its apparent complexity compared to the other four postulates.
Many attempts were made to prove the fifth postulate based on the other four, but they never succeeded. Eventually in , mathematician Nikolai Lobachevsky published a description of acute geometry or hyperbolic geometry , a geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth postulate elliptic geometry. If one takes the fifth postulate as a given, the result is Euclidean geometry.
Contents[ edit ] Book 1 contains 5 postulates including the famous parallel postulate and 5 common notions, and covers important topics of plane geometry such as the Pythagorean theorem , equality of angles and areas , parallelism, the sum of the angles in a triangle, and the construction of various geometric figures.
Book 2 contains a number of lemmas concerning the equality of rectangles and squares, sometimes referred to as " geometric algebra ", and concludes with a construction of the golden ratio and a way of constructing a square equal in area to any rectilineal plane figure. Book 4 constructs the incircle and circumcircle of a triangle, as well as regular polygons with 4, 5, 6, and 15 sides.
Book 6 applies proportions to plane geometry, especially the construction and recognition of similar figures. Book 8 deals with the construction and existence of geometric sequences of integers. Book 9 applies the results of the preceding two books and gives the infinitude of prime numbers and the construction of all even perfect numbers.
Book 10 proves the irrationality of the square roots of non-square integers e. Euclid here introduces the term "irrational", which has a different meaning than the modern concept of irrational numbers. He also gives a formula to produce Pythagorean triples. Book 12 studies the volumes of cones , pyramids , and cylinders in detail by using the method of exhaustion , a precursor to integration , and shows, for example, that the volume of a cone is a third of the volume of the corresponding cylinder.
It concludes by showing that the volume of a sphere is proportional to the cube of its radius in modern language by approximating its volume by a union of many pyramids. Book 13 constructs the five regular Platonic solids inscribed in a sphere and compares the ratios of their edges to the radius of the sphere.