History of mathematics in the context of Geometry

Iraq, officially the Republic of Iraq, is a country in West Asia. Located within the geo-political region of the Middle East, it is bordered by Saudi Arabia to the south, Turkey to the north, Iran to the east, the Persian Gulf and Kuwait to the southeast, Jordan to the southwest, and Syria to the west. The country covers an area of 438,317 square kilometres (169,235 sq mi) and has a population of over 46 million, making it the 58th largest country by area and the 31st most populous in the world. Baghdad, home to over 8 million people, is the capital city and the largest in the country.

Starting in the 6th millennium BC, the fertile plains between Iraq's Tigris and Euphrates rivers, referred to as Mesopotamia, fostered the rise of early cities, civilisations, and empires including Sumer, Akkad, Babylonia, and Assyria. Known as the cradle of civilisation, Mesopotamia saw the invention of writing systems, mathematics, navigation, timekeeping, a calendar, astrology, the wheel, the sailboat, and a law code. After the Muslim conquest of Mesopotamia, Baghdad became the capital of the Abbasid Caliphate and a global cultural and intellectual hub during the Islamic Golden Age, home to institutions such as the House of Wisdom. Following Baghdad's destruction by the Mongols in 1258, Iraq came under successive empires and, from the 16th century until the 20th century, was governed within the Ottoman system as a defined region known administratively as ‘the Iraq Region’. Additionally, Iraq holds religious significance in Christianity, Judaism, Yazidism, and Mandaeism.

Euclid (/ˈjuːklɪd/; Ancient Greek: Εὐκλείδης; fl. 300 BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics.

Very little is known of Euclid's life, and most information comes from the scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine and early Renaissance scholars mistook him for the earlier philosopher Euclid of Megara. It is now generally accepted that he spent his career in Alexandria and lived around 300 BC, after Plato's students and before Archimedes. There is some speculation that Euclid studied at the Platonic Academy and later taught at the Musaeum; he is regarded as bridging the earlier Platonic tradition in Athens with the later tradition of Alexandria.

The Scientific Revolution was a series of events that marked the emergence of modern science during the early modern period, when developments in mathematics, physics, astronomy, biology (including human anatomy) and chemistry transformed the views of society about nature.

Gottfried Wilhelm Leibniz (or Leibnitz; 1 July 1646 [O.S. 21 June] – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic and statistics. Leibniz has been called the "last universal genius" due to his vast expertise across fields, which became a rarity after his lifetime with the coming of the Industrial Revolution and the spread of specialized labour. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history, philology, games, music, and other studies. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science.

Leibniz contributed to the field of library science, developing a cataloguing system (at the Herzog August Library in Wolfenbüttel, Germany) that came to serve as a model for many of Europe's largest libraries. His contributions to a wide range of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, French and German.

Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century.

Further progress was not made until the 14th century, when Madhava of Sangamagrama developed approximations correct to eleven and then thirteen digits. Jamshīd al-Kāshī achieved sixteen digits next. Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century (Ludolph van Ceulen), and 126 digits by the 19th century (Jurij Vega).

The Concise Book of Calculation by Restoration and Balancing (Arabic: الكتاب المختصر في حساب الجبر والمقابلة, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah; or Latin: Liber Algebræ et Almucabola), commonly abbreviated Al-Jabr or Algebra (Arabic: الجبر), is an Arabic-language mathematical treatise on algebra written in Baghdad around 820 by the Persian polymath Al-Khwarizmi. It was a landmark work in the history of mathematics, with its title being the ultimate etymology of the word "algebra" itself, later borrowed into Medieval Latin as algebrāica.

Al-Jabr provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree. It was the first text to teach elementary algebra, and the first to teach algebra for its own sake. It also introduced the fundamental concept of "reduction" and "balancing" (which the term al-jabr originally referred to), the transposition of subtracted terms to the other side of an equation, i.e. the cancellation of like terms on opposite sides of the equation. The mathematics historian Victor J. Katz regards Al-Jabr as the first true algebra text that is still extant. Translated into Latin by Robert of Chester in 1145, it was used until the sixteenth century as the principal mathematical textbook of European universities.

In mathematics, especially historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The order of the magic square is the number of integers along one side (n), and the constant sum is called the magic constant. If the array includes just the positive integers ${\displaystyle 1,2,...,n^{2}}$, the magic square is said to be normal. Many authors take magic square to mean normal magic square.

Magic squares that include repeated entries do not fall under this definition and are referred to as trivial. Some well-known examples, including the Sagrada Família magic square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant, this gives a semimagic square (sometimes called orthomagic square).

Jules Henri Poincaré (UK: /ˈpwæ̃kɑːreɪ/, US: /ˌpwæ̃kɑːˈreɪ/; French: [ɑ̃ʁi pwɛ̃kaʁe] ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. He has further been called "the Gauss of modern mathematics". Due to his success in science, along with his influence and philosophy, he has further been called "the philosopher par excellence of modern science".

As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. Poincaré is regarded as the creator of the field of algebraic topology, and is further credited with introducing automorphic forms. He also made important contributions to algebraic geometry, number theory, complex analysis and Lie theory. He famously introduced the concept of the Poincaré recurrence theorem, which states that a state will eventually return arbitrarily close to its initial state after a sufficiently long time, which has far-reaching consequences. Early in the 20th century he formulated the Poincaré conjecture, which became, over time, one of the famous unsolved problems in mathematics. It was eventually solved in 2002–2003 by Grigori Perelman. Poincaré popularized the use of non-Euclidean geometry in mathematics as well.

In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 to 40 leading mathematicians who made major contributions, about half of those being Italian. The leadership fell to the group in Rome of Guido Castelnuovo, Federigo Enriques and Francesco Severi, who were involved in some of the deepest discoveries, as well as setting the style.

Felix Christian Klein (/klaɪn/; German: [klaɪn]; 25 April 1849 – 22 June 1925) was a German mathematician, mathematics educator and historian of mathematics, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations between geometry and group theory. His 1872 Erlangen program classified geometries by their basic symmetry groups and was an influential synthesis of much of the mathematics of the time.

During his tenure at the University of Göttingen, Klein was able to turn it into a center for mathematical and scientific research through the establishment of new lectures, professorships, and institutes. His seminars covered most areas of mathematics then known as well as their applications. Klein also devoted considerable time to mathematical instruction and promoted mathematics education reform at all grade levels in Germany and abroad. He became the first president of the International Commission on Mathematical Instruction in 1908 at the Fourth International Congress of Mathematicians in Rome.

An Egyptian fraction is a finite sum of distinct unit fractions, such as$${\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{16}}.}$$That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The value of an expression of this type is a positive rational number ${\displaystyle {\tfrac {a}{b}}}$; for instance the Egyptian fraction above sums to ${\displaystyle {\tfrac {43}{48}}}$. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including ${\displaystyle {\tfrac {2}{3}}}$ and ${\displaystyle {\tfrac {3}{4}}}$ as summands, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics.

In the history of mathematics, Egyptian algebra, as that term is used in this article, refers to algebra as it was developed and used in ancient Egypt. Ancient Egyptian mathematics as discussed here spans a time period ranging from c. 3000 BCE to c. 300 BCE.

There are limited surviving examples of ancient Egyptian algebraic problems. They appear in the Moscow Mathematical Papyrus (MMP) and in the Rhind Mathematical Papyrus (RMP), among others.

St. John's College is a private liberal arts college with campuses in Annapolis, Maryland and Santa Fe, New Mexico. As the successor institution of King William's School, a preparatory school founded in 1696, St. John's is one of the oldest institutions of higher learning in the United States; the current institution received a collegiate charter in 1784. In 1937, St. John's adopted a Great Books curriculum based on discussion of works from the Western canon of philosophical, religious, historical, mathematical, scientific, and literary works.

The college grants a single bachelor's degree in liberal arts. The awarded degree is equivalent to a double major in philosophy and the history of mathematics and science, and a double minor in classical studies and comparative literature. Three master's degrees are available through the college's graduate institute: one in liberal arts, which is a modified version of the undergraduate curriculum; one in Eastern Classics, exclusive to the Santa Fe campus, which applies a Great Books curriculum to classic works from India, China, and Japan; and one in Middle Eastern Classics, also exclusive to Santa Fe, this program focuses on the Great Books from Jewish and Muslim authors written between the fall of Rome and European Renaissance.