A History of Greece, Volume 04 of 12, originally published in 1847
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George Grote. This article includes a list of references , but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. February Learn how and when to remove this template message. Cambridge, England: At the University Press. Retrieved 13 February — via Internet Archive. In Stephen, Leslie ; Lee, Sidney eds. Dictionary of National Biography. A History of Greece: I. Legendary Greece II.
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Grecian History to the Reign of Peisistratus at Athens. London: John Murray. Retrieved 14 October — via Internet Archive. American Academy of Arts and Sciences. Retrieved 15 September I and II". Legendary Greece ; 2. The Quarterly Review.
June III and IV". March V and VI". December Croom Robertson". The books cover plane and solid Euclidean geometry , elementary number theory , and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science , and its logical rigor was not surpassed until the 19th century.
Euclid's Elements has been referred to as the most successful [a] [b] and influential [c] textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing in ,  with the number reaching well over one thousand. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read.
Geometry emerged as an indispensable part of the standard education of the English gentleman in the eighteenth century; by the Victorian period it was also becoming an important part of the education of artisans, children at Board Schools, colonial subjects and, to a rather lesser degree, women. The standard textbook for this purpose was none other than Euclid's The Elements. Scholars believe that the Elements is largely a compilation of propositions based on books by earlier Greek mathematicians.
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Proclus — AD , a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the Elements : "Euclid, who put together the Elements , collecting many of Eudoxus ' theorems, perfecting many of Theaetetus ', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors". 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. The first printed edition appeared in based on Campanus of Novara 's edition ,  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. 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. 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. 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. 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. Cambridge, Public Library, n.
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Dublin, Trinity College Library, , fol. Lorenzo, Cod. Firenze, Biblioteca Riccardiana, Ricc. Istambul, Maurogordateios Bibliotheke, Klosternburg, Bibliotheca Augustiniana, paper, XV , fol. London, British Library, Harley Madrid, Biblioteca Nacional, n. Milano, Biblioteca Ambrosiana, A 84 sup. Milano, Biblioteca Ambrosiana, C 30 inf. Milano, Biblioteca Ambrosiana, E 16 sup. Milano, Biblioteca Ambrosiana, E 20 sup. Milano, Biblioteca Ambrosiana, E 64 sup. Milano, Biblioteca Ambrosiana, F sup. Milano, Biblioteca Ambrosiana, G 76 inf. Milano, Biblioteca Ambrosiana, M 57 sup. Milano, Biblioteca Ambrosiana, M 83 sup.
Milano, Biblioteca Ambrosiana, Q 50 sup. Milano, Biblioteca Ambrosiana, Q 74 sup. Milano, Biblioteca Ambrosiana, S 23 sup. Milano, Biblioteca Ambrosiana, Z 75 sup. Sebald Bibl. Oxford, Bodleian Library, Ashmol. Oxford, Bodleian Library, Barrocc. Oxford, Bodleian Library, Digbeian. Oxford, Corpus Christi College Library, n. Oxford, Jesus College Library, Oxford, Jesus College Library, 65, fol. Patmos, Moni Agiou Ioannou tou theologou, Patmos, Moni Agiou Ioannou tou theologou, , fol.
Roma, Biblioteca Angelica, A. Roma, Biblioteca Vallicelliana, E 40 n. Roma, Biblioteca Vallicelliana, Gr. Salisbury, Library of the Cathedral Church, 61, fol. San Daniele del Friuli, Biblioteca Guarneriana, 87, fol. II, 77 paper; XVI , fol. II, 90 paper; XVI , fol. Venezia, Biblioteca del monastero di S. Michele, Unknown, apud P. Achelis, H. Hippolytstudien , Texte und Untersuchungen Bardy, G. Beyer, G. Burgon, J. Cameron, A.
Eusebius, Life of Constantine. Introduction, translation and commentary , Clarendon Ancient History Series.
Carriker, A. Leiden and Boston. Caubet Iturbe, F. Studi e testi, Vatican City. Cavallo, G. Cavallo, ed. Courcelle, P. Cramer, J. Catenae in evangelia s. Matthaei et s. Marci, ad fidem codd. Lucae et s.
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Joannis, ad fidem codd. Corderius, B. Catena sexaginta quinque Graecorum patrum in S. Dorival, G. XV, Texte und Untersuchungen , E. Livingstone, ed. Graf, G. Geschichte der christlichen arabischen Literatur, I. Grafton, A and M. Christianity and the Transformation of the Book. Origen, Eusebius and the Library of Caesarea.
Guignard, C. Hollerich, M. Jacob, C. Volgers, A. Zamagni, 25— Kannengiesser, C. Handbook of Patristic Exegesis, II. Krikonis, Ch. Labriolle, P.
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Lagarde, P. Catenae in Evangelia Aegyptiacae quae supersunt. Mittheilungen, II. Layton, B. Catalogue of Coptic literary manuscripts in the British Library acquired since the year Lightfoot, J. Smith, W. Wace, — Mai, A. Scriptorum veterum nova collectio, e Vaticanis codicibus edita, tomus I [pars prior], Rome. Reprinted in and republished in PG Novae patrum bibliothecae tomus IV. Republished in PG Norelli, E. Pouderon, B. Duval, 1— Papadoyannakis, Y. Dynamism, Didacticism, Classicism , S.
F Johnson, ed. Pearse, R. Eusebius of Caesarea, Gospel Problems and Solutions. Miller Greek, Latin , A. McCollum Syriac, Arabic , C. Downer Coptic , and others. Perrone, L. Alle origini di un genere letterario. Presupposti e sviluppi del genere letterario fino al IV sec. A Retrospective after Two Millennia , ed. Raban, A. Holum, — Preuschen, E. Preuschen , —