Euclid's Elements Euclid's Elements (Ancient Greek: Στοιχεῖα Stoicheia) is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. The work also includes an algebraic system that has become known as geometric algebra, which is powerful enough to solve many algebraic problems,[1] including the problem of finding the square root of a number.[2] With the exception of Autolycus' On the Moving Sphere, the Elements is one of the oldest extant Greek mathematical treatises,[3] and it is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science. The name 'Elements' comes from the plural of 'element'.
The Compendious Book on Calculation by Completion and Balancing A page from the book Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة, "The Compendious Book on Calculation by Completion and Balancing"), also known under a shorter name spelled as Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala and other transliterations) is a mathematical book written in Arabic language in approximately AD 820 by the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī in Baghdad, the capital of the Abbasid Caliphate at the time. The book was translated into Latin in the mid 12th century under the title Liber Algebrae et Almucabola (with algebrae and almucabola being simply Latinized corruptions of the words in the Arabic title). Today's term algebra is derived from the term الجبر al-ğabr in the title of this book. The al-ğabr provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree.[1] Legacy[edit] R. J. The book[edit] References[edit] Barnabas B.
Mathematical Treatise in Nine Sections Mathematical Treatise in Nine Sections in The Siku Quanshu 1842 wood block printed Shu Shu Jiu Zhang surveying a round city from afar.Shu Shu Jiu Zhang The Mathematical Treatise in Nine Sections (simplified Chinese: 数书九章; traditional Chinese: 數書九章; pinyin: Shùshū Jiǔzhāng; Wade–Giles: Shushu Chiuchang) is a mathematical text written by Chinese Southern Song dynasty mathematician Qin Jiushao in the year 1247. This book contains nine chapters: Da Yan type (Indeterminate equations);Heaven phenomenaArea of land and fieldSurveyingTaxationStorage of grainsBuilding constructionMilitary mattersPrice and interest. Each chapter contains nine problems, a total of 81 problems. Like many traditional Chinese mathematical works, the text reflects a Confucian administrator's concern with more practical mathematical problems, like calendrical, mensural, and fiscal problems.
Elements of Algebra Elements of Algebra is a mathematics textbook by mathematician Leonhard Euler, originally published circa 1765. His Elements of Algebra is one of the first books to set out algebra in the modern form we would recognize today. However, it is sufficiently different from most modern approaches to the subject to be interesting for contemporary readers. The original German name is: Vollständige Anleitung zur Algebra, which literally means: Complete Instruction to Algebra. In 1771, Joseph-Louis Lagrange published a follow-up volume entitled Additions to Euler's Elements of algebra, which featured a number of important mathematical results. Disquisitiones Arithmeticae Title page of the first edition The Disquisitiones Arithmeticae (Latin: Arithmetical Investigations) is a textbook of number theory written in Latin[1] by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own. Scope[edit] The inquiries which this volume will investigate pertain to that part of Mathematics which concerns itself with integers. Contents[edit] The book is divided into seven sections, which are: Section I. Section II. Section III. Section IV. Section V. Section VI. Section VII. Sections I to III are essentially a review of previous results, including Fermat's little theorem, Wilson's theorem and the existence of primitive roots. From Section IV onwards, much of the work is original. Importance[edit] Notes[edit] References[edit]
On the Number of Primes Less Than a Given Magnitude The article "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse" (usual English translation: "On the Number of Primes Less Than a Given Magnitude") is a seminal 10-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin. Among the new definitions, ideas, and notation introduced: Among the proofs and sketches of proofs: Among the conjectures made: The Riemann hypothesis, that all (nontrivial) zeros of ζ(s) have real part 1/2. New methods and techniques used in number theory: Riemann also discussed the relationship between ζ(s) and the distribution of the prime numbers, using the function J(x) essentially as a measure for Stieltjes integration. The paper contains some peculiarities for modern readers, such as the use of Π(s − 1) instead of Γ(s), writing tt instead of t2, and using the bounds of ∞ to ∞ as to denote a contour integral. References[edit] External links[edit]
Vorlesungen über Zahlentheorie Vorlesungen über Zahlentheorie (German for Lectures on Number Theory) is a textbook of number theory written by German mathematicians Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Based on Dirichlet's number theory course at the University of Göttingen, the Vorlesungen were edited by Dedekind and published after Lejeune Dirichlet's death. Dedekind added several appendices to the Vorlesungen, in which he collected further results of Lejeune Dirichlet's and also developed his own original mathematical ideas. Scope[edit] The Vorlesungen cover topics in elementary number theory, algebraic number theory and analytic number theory, including modular arithmetic, quadratic congruences, quadratic reciprocity and binary quadratic forms. Contents[edit] The contents of Professor John Stillwell's 1999 translation of the Vorlesungen are as follows Chapter 1. Chapter 2. Chapter 3. Chapter 4. Chapter 5. Supplement I. Supplement II. Supplement III. Supplement IV. Supplement V.
Zahlbericht In mathematics, the Zahlbericht (number report) was a report on algebraic number theory by Hilbert (1897, 1998, (English translation)). History[edit] Corry (1996) and Schappacher (2005) and the English introduction to (Hilbert 1998) give detailed discussions of the history and influence of Hilbert's Zahlbericht. Some earlier reports on number theory include the report by H. J. In 1893 the German mathematical society invited Hilbert and Minkowski to write reports on the theory of numbers. Contents[edit] Part 1 covers the theory of general number fields, including ideals, discriminants, differents, units, and ideal classes. Part 2 covers Galois number fields, including in particular Hilbert's theorem 90. Part 3 covers quadratic number fields, including the theory of genera, and class numbers of quadratic fields. Part 5 covers Kummer number fields, and ends with Kummer's proof of Fermat's last theorem for regular primes. References[edit] External links[edit]
Yuktibhāṣā Contents[edit] Mathematics[edit] As per the old Indian tradition of starting off new chapters with elementary content, the first four chapters of the Yuktibhasa contain elementary mathematics, such as division, proof of Pythagorean theorem, square root determination, etc.[8] The radical ideas are not discussed until the sixth chapter on circumference of a circle. Yuktibhasa contains the derivation and proof of the power series for inverse tangent, discovered by Madhava.[2] In the text, Jyesthadeva describes Madhava's series in the following manner: This yields which further yields the theorem which he obtained from the power series expansion of the arc-tangent function. Using a rational approximation of this series, he gave values of the number π as 3.14159265359 - correct to 11 decimals; and as 3.1415926535898 - correct to 13 decimals. The text describes that he gave two methods for computing the value of π. The other method was to add a remainder term to the original series of π.
Philosophiæ Naturalis Principia Mathematica Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, in Latin, first published 5 July 1687.[1][2] After annotating and correcting his personal copy of the first edition,[3] Newton also published two further editions, in 1713 and 1726.[4] The Principia states Newton's laws of motion, forming the foundation of classical mechanics, also Newton's law of universal gravitation, and a derivation of Kepler's laws of planetary motion (which Kepler first obtained empirically). The Principia is "justly regarded as one of the most important works in the history of science".[5] The French mathematical physicist Alexis Clairaut assessed it in 1747: "The famous book of mathematical Principles of natural Philosophy marked the epoch of a great revolution in physics. Contents[edit] Expressed aim and topics covered[edit] In the preface of the Principia, Newton wrote[10]