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Minneapolis: University of Minnesota Press, , pp. New York: Marcel Dekker, Pieces of the Action. New York: Morrow, New York: The Science Press, A Century of Matheamtics in America. Providence: American Mathematical Society, — Emmy Noether, — Chicago: University of Chicago Press, The Intellectual Migration: Europe and America, — Perspectives in American History Mary F. Partners in Science: Foundations and Natural Scientists — David E. Rowe and John McCleary.

Number Theory Books, 1996

Boston: Academic Press, , pp. Moore: Mathematician and Teacher. Donald Albers, et al. HMATH 8. New York: Springer-Verlag, Science in America: A Documentary History — Degree and Mathematical Research. 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. The first printed edition appeared in based on Campanus of Novara 's edition , [10] and since then it has been translated into many languages and published in about a thousand different editions.

Theon's Greek edition was recovered in 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. The Elements is still considered a masterpiece in the application of logic to mathematics.

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In historical context, it has proven enormously influential in many areas of science. Mathematicians and philosophers, such as Thomas Hobbes , Baruch Spinoza , Alfred North Whitehead , and Bertrand Russell , have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced.

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.

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Abraham Lincoln kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight". 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!

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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". The success of the Elements is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid.

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Much of the material is not original to him, although many of the proofs are his. However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the Elements , encouraged its use as a textbook for about 2, years.

The Elements still influences modern geometry books. Further, its logical axiomatic approach and rigorous proofs remain the cornerstone of mathematics. Euclid's axiomatic approach and constructive methods were widely influential. Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using a compass and straightedge.

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His constructive approach appears even in his geometry's postulates, as the first and third postulates stating the existence of a line and circle are constructive. Instead of stating that lines and circles exist per his prior definitions, he states that it is possible to 'construct' a line and circle. It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition.

For example, he proves the Pythagorean theorem by first inscribing a square on the sides of a right triangle, but only after constructing a square on a given line one proposition earlier. As was common in ancient mathematical texts, when a proposition needed proof in several different cases, Euclid often proved only one of them often the most difficult , leaving the others to the reader. Later editors such as Theon often interpolated their own proofs of these cases. Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places.

For example, there was no notion of an angle greater than two right angles, [17] the number 1 was sometimes treated separately from other positive integers, and as multiplication was treated geometrically he did not use the product of more than 3 different numbers. The geometrical treatment of number theory may have been because the alternative would have been the extremely awkward Alexandrian system of numerals.

The presentation of each result is given in a stylized form, which, although not invented by Euclid, is recognized as typically classical. It has six different parts: First is the 'enunciation', which states the result in general terms i. Then comes the 'setting-out', which gives the figure and denotes particular geometrical objects by letters. Next comes the 'definition' or 'specification', which restates the enunciation in terms of the particular figure. Then the 'construction' or 'machinery' follows.

Here, the original figure is extended to forward the proof. Then, the 'proof' itself follows. Finally, the 'conclusion' connects the proof to the enunciation by stating the specific conclusions drawn in the proof, in the general terms of the enunciation. No indication is given of the method of reasoning that led to the result, although the Data does provide instruction about how to approach the types of problems encountered in the first four books of the Elements. However, Euclid's original proof of this proposition, is general, valid, and does not depend on the figure used as an example to illustrate one given configuration.

Euclid's list of axioms in the Elements was not exhaustive, but represented the principles that were the most important. His proofs often invoke axiomatic notions which were not originally presented in his list of axioms. Later editors have interpolated Euclid's implicit axiomatic assumptions in the list of formal axioms. For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points.

If superposition is to be considered a valid method of geometric proof, all of geometry would be full of such proofs. For example, propositions I. Mathematician and historian W. Rouse Ball put the criticisms in perspective, remarking that "the fact that for two thousand years [the Elements ] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose. It was not uncommon in ancient time to attribute to celebrated authors works that were not written by them.

The book continues Euclid's comparison of regular solids inscribed in spheres, with the chief result being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being.

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This book covers topics such as counting the number of edges and solid angles in the regular solids, and finding the measure of dihedral angles of faces that meet at an edge. From Wikipedia, the free encyclopedia. Mathematical treatise by Euclid.

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Perhaps no book other than the Bible can boast so many editions, and certainly no mathematical work has had an influence comparable with that of Euclid's Elements ". There, the Elements became the foundation of mathematical education. More than editions of the Elements are known.

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In all probability, it is, next to the Bible , the most widely spread book in the civilization of the Western world. It is thought that this book may have been composed by Hypsicles on the basis of a treatise now lost by Apollonius comparing the dodecahedron and icosahedron. This book also deals with the regular solids, counting the number of edges and solid angles in the solids, and finding the measures of the dihedral angles of faces meeting at an edge. Reynolds and Nigel G. Wilson, Scribes and Scholars 2nd.