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Moritz Pasch
Moritz Pasch (8 November 1843, Breslau, Prussia (now Wrocław, Poland) – 20 September 1930, Bad Homburg, Germany) was a German mathematician of Jewish ancestry specializing in the foundations of geometry. He completed his Ph.D. at the University of Breslau at only 22 years of age. He taught at the University of Giessen, where he is known to have supervised 30 doctorates. In 1882, Pasch published a book, ''Vorlesungen über neuere Geometrie'', calling for the grounding of Euclidean geometry in more precise primitive notions and axioms, and for greater care in the deductive methods employed to develop the subject. He drew attention to a number of heretofore unnoted tacit assumptions in Euclid's '' Elements''. He then argued that mathematical reasoning should not invoke the physical interpretation of the primitive terms, but should instead rely solely on formal manipulations justified by axioms. This book is the point of departure for: *Similarly concerned Italians: Peano, Mario ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moritz Pasch (mathematician)
Moritz Pasch (8 November 1843, Breslau, Prussia (now Wrocław, Poland) – 20 September 1930, Bad Homburg, Germany) was a German mathematician of Jewish ancestry specializing in the foundations of geometry. He completed his Ph.D. at the University of Breslau at only 22 years of age. He taught at the University of Giessen, where he is known to have supervised 30 doctorates. In 1882, Pasch published a book, ''Vorlesungen über neuere Geometrie'', calling for the grounding of Euclidean geometry in more precise primitive notions and axioms, and for greater care in the deductive methods employed to develop the subject. He drew attention to a number of heretofore unnoted tacit assumptions in Euclid's '' Elements''. He then argued that mathematical reasoning should not invoke the physical interpretation of the primitive terms, but should instead rely solely on formal manipulations justified by axioms. This book is the point of departure for: *Similarly concerned Italians: Peano, Mar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. 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 and influential 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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19th-century German Mathematicians
The 19th (nineteenth) century began on 1 January 1801 ( MDCCCI), and ended on 31 December 1900 ( MCM). The 19th century was the ninth century of the 2nd millennium. The 19th century was characterized by vast social upheaval. Slavery was abolished in much of Europe and the Americas. The First Industrial Revolution, though it began in the late 18th century, expanding beyond its British homeland for the first time during this century, particularly remaking the economies and societies of the Low Countries, the Rhineland, Northern Italy, and the Northeastern United States. A few decades later, the Second Industrial Revolution led to ever more massive urbanization and much higher levels of productivity, profit, and prosperity, a pattern that continued into the 20th century. The Islamic gunpowder empires fell into decline and European imperialism brought much of South Asia, Southeast Asia, and almost all of Africa under colonial rule. It was also marked by the collapse of the large S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19th-century German Jews
The 19th (nineteenth) century began on 1 January 1801 ( MDCCCI), and ended on 31 December 1900 ( MCM). The 19th century was the ninth century of the 2nd millennium. The 19th century was characterized by vast social upheaval. Slavery was abolished in much of Europe and the Americas. The First Industrial Revolution, though it began in the late 18th century, expanding beyond its British homeland for the first time during this century, particularly remaking the economies and societies of the Low Countries, the Rhineland, Northern Italy, and the Northeastern United States. A few decades later, the Second Industrial Revolution led to ever more massive urbanization and much higher levels of productivity, profit, and prosperity, a pattern that continued into the 20th century. The Islamic gunpowder empires fell into decline and European imperialism brought much of South Asia, Southeast Asia, and almost all of Africa under colonial rule. It was also marked by the collapse of the large ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1930 Deaths
Year 193 ( CXCIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Sosius and Ericius (or, less frequently, year 946 ''Ab urbe condita''). The denomination 193 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Roman Empire * January 1 – Year of the Five Emperors: The Roman Senate chooses Publius Helvius Pertinax, against his will, to succeed the late Commodus as Emperor. Pertinax is forced to reorganize the handling of finances, which were wrecked under Commodus, to reestablish discipline in the Roman army, and to suspend the food programs established by Trajan, provoking the ire of the Praetorian Guard. * March 28 – Pertinax is assassinated by members of the Praetorian Guard, who storm the imperial palace. The Empire is auctioned of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1843 Births
Events January–March * January ** Serial publication of Charles Dickens's novel ''Martin Chuzzlewit'' begins in London; in the July chapters, he lands his hero in the United States. ** Edgar Allan Poe's short story "The Tell-Tale Heart" is published in a Boston magazine. ** The Quaker magazine '' The Friend'' is first published in London. * January 3 – The ''Illustrated Treatise on the Maritime Kingdoms'' (海國圖志, ''Hǎiguó Túzhì'') compiled by Wei Yuan and others, the first significant Chinese work on the West, is published in China. * January 6 – Antarctic explorer James Clark Ross discovers Snow Hill Island. * January 20 – Honório Hermeto Carneiro Leão, Marquis of Paraná, becomes ''de facto'' first prime minister of the Empire of Brazil. * February – Shaikh Ali bin Khalifa Al-Khalifa captures the fort and town of Riffa after the rival branch of the family fails to gain control of the Riffa Fort and flees to Manama. Shaikh Mohamed bin Ahmed is kille ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Geometry
Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective geometry). History Moritz Pasch first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904). Euclid anticipated Pasch's approach in definition 4 of ''The Elements'': "a straight line is a line which lies evenly with the points on itself". Primitive concepts The only primitive notions in ordered geometry are points ''A'', ''B'', ''C'', ... and the ternary relation of intermediacy 'ABC''which can be read as "''B'' is between ''A'' and ''C''". Definitions The ''segment'' ''AB'' is the set of points ''P'' such that 'APB'' The ''interval'' ''AB'' is the segment ''AB'' and its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pasch Hypergraph
In geometry, a truncated projective plane (TPP), also known as a dual affine plane, is a special kind of a hypergraph or geometric configuration that is constructed in the following way. * Take a finite projective plane. * Remove one of the points (vertices) in the plane. * Remove all lines (edges) containing that point. These objects have been studied in many different settings, often independent of one another, and so, many terminologies have been developed. Also, different areas tend to ask different types of questions about these objects and are interested in different aspects of the same objects. Example: the Pasch hypergraph Consider the Fano plane, which is the projective plane of order 2. It has 7 vertices and 7 edges . It can be truncated e.g. by removing the vertex 7 and the edges containing it. The remaining hypergraph is the TPP of order 2. It has 6 vertices and 4 edges . It is a tripartite hypergraph with sides ,, (which are exactly the neighbors of the removed ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pasch's Theorem
In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch, is a result in plane geometry which cannot be derived from Euclid's postulates. Statement The statement is as follows: ere, for example, (, , ) means that point lies between points and . See also *Ordered geometry Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affi ... * Pasch's axiom Notes References * * External links * Euclidean plane geometry Foundations of geometry Order theory Theorems in plane geometry {{elementary-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pasch's Axiom
In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882. Statement The axiom states that, The fact that segments ''AC'' and ''BC'' are not both intersected by the line is proved in Supplement I,1, which was written by P. Bernays. A more modern version of this axiom is as follows: (In case the third side is parallel to our line, we count an "intersection at infinity" as external.) A more informal version of the axiom is often seen: History Pasch published this axiom in 1882, and showed that Euclid's axioms were incomplete. The axiom was part of Pasch's approach to introducing the concept of order into plane geometry. Equivalences In other treatments of elementary geometry, using different sets of axioms, Pasch's axiom can be proved as a theorem; it is a consequence of the plane separation axiom when that is taken ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic. Life Early life and edu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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