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124-724: Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an ordained minister who taught mathematics and physics at the Stettin Gymnasium , where Hermann was educated. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities. Beginning in 1827, he studied theology at the University of Berlin , also taking classes in classical languages , philosophy, and literature. He does not appear to have taken courses in mathematics or physics . Although lacking university training in mathematics, it

248-543: A consecration . Many ancient sources specify that at least three bishops are necessary to consecrate another, e.g., the 13th Canon of the Council of Carthage (AD 394) states, "A bishop should not be ordained except by many bishops, but if there should be necessity he may be ordained by three," and the first of "The Canons of the Holy and Altogether August Apostles" states, "Let a bishop be ordained by two or three bishops," while

372-480: A constitutional monarchy . (This eventuated in 1871.) After writing a series of articles on constitutional law , Hermann parted company with the newspaper, finding himself increasingly at odds with its political direction. Grassmann had eleven children, seven of whom reached adulthood. A son, Hermann Ernst Grassmann, became a professor of mathematics at the University of Giessen . One of the many examinations for which Grassmann sat required that he submit an essay on

496-520: A geodesic is a generalization of the notion of a line to curved spaces . In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as

620-423: A linear space ( vector space ) [...] became widely known around 1920, when Hermann Weyl and others published formal definitions. In fact, such a definition had been given thirty years previously by Peano , who was thoroughly acquainted with Grassmann's mathematical work. Grassmann did not put down a formal definition – the language was not available – but there is no doubt that he had the concept. Beginning with

744-418: A parabola with the summation of an infinite series , and gave remarkably accurate approximations of pi . He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution . Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to

868-601: A phonological rule that exists in both Sanskrit and Greek . In his honor, this phonological rule is known as Grassmann's law . His discovery was revolutionary for historical linguistics at the time, as it challenged the widespread notion of Sanskrit as an older predecessor to other Indo-European languages. This was a widespread assumption due to Sanskrit's more agglutinative structure, which languages like Latin and Greek were thought to have passed through to reach their more "modern" synthetic structure. However, Grassman's work proved that, in at least one phonological pattern, German

992-425: A vector space and its dual space . Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics , astronomy , crystallography , and many technical fields, such as engineering , architecture , geodesy , aerodynamics , and navigation . The mandatory educational curriculum of the majority of nations includes

1116-456: A ceremony called "pastoral consecration". Jehovah's Witnesses consider an adherent's baptism to constitute ordination as a minister . Governments have generally recognized that Jehovah's Witnesses' full-time appointees (such as their " regular pioneers ") qualify as ministers regardless of sex or appointment as an elder or deacon ("ministerial servant") . The religion asserts ecclesiastical privilege only for its appointed elders, but

1240-451: A collection of 'units' e 1 , e 2 , e 3 , ..., he effectively defines the free linear space that they generate; that is to say, he considers formal linear combinations a 1 e 1 + a 2 e 2 + a 3 e 3 + ... where the a j are real numbers, defines addition and multiplication by real numbers [in what is now the usual way] and formally proves the linear space properties for these operations. ... He then develops

1364-405: A common endpoint, called the vertex of the angle. The size of an angle is formalized as an angular measure . In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry . In differential geometry and calculus ,

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1488-523: A decimal place value system with a dot for zero." Aryabhata 's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In

1612-507: A deficient form." Kummer's report ended any chance that Grassmann might obtain a university post. This episode proved the norm; time and again, leading figures of Grassmann's day failed to recognize the value of his mathematics. Starting during the political turmoil in Germany, 1848–49, Hermann and his brother Robert published a Stettin newspaper, Deutsche Wochenschrift für Staat, Kirche und Volksleben , calling for German unification under

1736-448: A geometric calculus devoid of coordinates and metric properties (what Leibniz termed analysis situs ). Grassmann's Geometrische Analyse geknüpft an die von Leibniz erfundene geometrische Charakteristik , was the winning entry (also the only entry). Möbius, as one of the judges, criticized the way Grassmann introduced abstract notions without giving the reader any intuition as to why those notions were of value. In 1853, Grassmann published

1860-440: A more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms . Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations. Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically,

1984-428: A multitude of forms, including the graphics of Leonardo da Vinci , M. C. Escher , and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is . Symmetry in classical Euclidean geometry

2108-438: A new foundation for all of mathematics, the work began with quite general definitions of a philosophical nature. Grassmann then showed that once geometry is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial dimensions ; the number of possible dimensions is in fact unbounded. Fearnley-Sander describes Grassmann's foundation of linear algebra as follows: The definition of

2232-451: A number of apparently different definitions, which are all equivalent in the most common cases. The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle , regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in

2356-444: A physical system, which has a dimension equal to the system's degrees of freedom . For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension ( Hilbert spaces , for example) and positive real numbers (in fractal geometry ). In algebraic geometry , the dimension of an algebraic variety has received

2480-528: A plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral or the Lebesgue integral . Other geometrical measures include the curvature and compactness . The concept of length or distance can be generalized, leading to

2604-602: A purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory . Wiles' proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory . One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry. Algebraic geometry has applications in many areas, including cryptography and string theory . Complex geometry studies

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2728-531: A significant topic of discussion in recent years. Texts passed down in every Buddhist tradition record that Gautama Buddha created an order of fully ordained nuns, but the tradition has died out in some Buddhist traditions such as Theravada Buddhism, while remaining strong in others such as Chinese Buddhism ( Dharmaguptaka lineage). In the Tibetan lineage , which follows the Mulasarvastivadin lineage,

2852-427: A size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert , in his work on creating

2976-458: A student's completion of a 4-year B.A. of Islamic studies or a 7–8 alim course, these ceremonies do not in any way symbolize ordination. The ordination of a rabbi within Judaism is referred to as semikhah ( Hebrew : סמיכה , "leaning [of the hands]", or semicha lerabanim Hebrew : סמיכה לרבנות , "rabbinical ordination"). The term is derived from a Hebrew word which means to "rely on", in

3100-600: A technical sense a type of transformation geometry , in which transformations are homeomorphisms . This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology , differential topology , algebraic topology and general topology . Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets , and defined as common zeros of multivariate polynomials . Algebraic geometry became an autonomous subfield of geometry c.  1900 , with

3224-518: A theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings . This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra . From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory , which allows using topological methods , including cohomology theories in

3348-400: A theory of how colors mix; his theory's four color laws are still taught, as Grassmann's laws . Grassmann's work on this subject was inconsistent with that of Helmholtz . Grassmann also wrote on crystallography , electromagnetism , and mechanics . In 1861, Grassmann laid the groundwork for Peano's axiomatization of arithmetic in his Lehrbuch der Arithmetik . In 1862, Grassmann published

3472-494: A theory of ratios that avoided the problem of incommensurable magnitudes , which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements , widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of

3596-477: A thoroughly rewritten second edition of A1 , hoping to earn belated recognition for his theory of extension, and containing the definitive exposition of his linear algebra . The result, Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet ( A2 ), fared no better than A1 , even though A2 's manner of exposition anticipates the textbooks of the 20th century. In the 1840s, mathematicians were generally unprepared to understand Grassmann's ideas. In

3720-431: A university position. Over the next 10-odd years, Grassmann wrote a variety of work applying his theory of extension, including his 1845 Neue Theorie der Elektrodynamik and several papers on algebraic curves and surfaces , in the hope that these applications would lead others to take his theory seriously. In 1846, Möbius invited Grassmann to enter a competition to solve a problem first proposed by Leibniz : to devise

3844-516: A wise and brave woman named Deborah was the fourth judge of the ancient Israelites . She was instrumental in implementing a strategic military strategy that delivered the Israelites from the oppressive Canaanite king Jabin . Likewise, Jael was courageous and primary in the Israelite victory. Her prudent actions killed the commander Sisera after he fled on foot following the battle. Within

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3968-411: Is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory . Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays , called the sides of the angle, sharing

4092-540: Is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic , one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer . Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. Originally developed to model

4216-400: Is a part of some ambient flat Euclidean space). Topology is the field concerned with the properties of continuous mappings , and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness . The field of topology, which saw massive development in the 20th century, is in

4340-413: Is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere. A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry , a differentiable manifold is a space where each neighborhood

4464-634: Is an Arabic word meaning "old man" and is used as an honorable title for a learned man; shaikhah refers to a woman learned in Islamic issues. This title is usually more prevalent in the Arabic countries. The word maulana is a title bestowed upon students who have graduated from a madrasa (Islamic theological school) throughout the Indian subcontinent region. Although different Muslim schools, universities or madrasas might follow different graduation ceremonies upon

4588-836: Is considered an essential and necessary concept for ordination in the Catholic, Orthodox, High Church Lutheran, Moravian, and Anglican traditions, with the belief that all ordained clergy are ordained by bishops who were ordained by other bishops tracing back to bishops ordained by the Apostles who were ordained by Christ , the great High Priest ( Hebrews 7:26 , Hebrews 8:2 ), who conferred his priesthood upon his Apostles ( John 20:21–23 , Matthew 28:19–20 , Mark 16:15–18 , and Acts 2:33 ). There are three ordinations in Holy Orders: deacon , presbyter , and bishop . Both bishops and presbyters are priests and have authority to celebrate

4712-409: Is defined. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. The earliest known texts on geometry are

4836-560: Is done by or under the direction of the stake or mission president. To perform a priesthood ordination, one or more authorized priesthood holders place their hands lightly on the person's head." Latter-day Saints believe in a line of priesthood authority that traces back to Jesus Christ and his apostles . LDS adherents believe the church's founder, Joseph Smith , was ordained under the hands of apostles Peter , James , and John , who appeared to Smith as angelic messengers in 1829. Muslims do not formally ordain religious leaders. Ordination

4960-430: Is in preparation for, or who is undergoing the process of ordination is sometimes called an ordinand . The liturgy used at an ordination is commonly found in a book known as an Ordinal which provides the ordo (ritual and rubrics) for celebrations. In Catholicism and Orthodoxy, ordination is one of the seven sacraments , variously called holy orders or cheirotonia (" Laying on of Hands "). Apostolic succession

5084-562: Is laid down in the Vinaya and Patimokkha or Pratimoksha scriptures. There exist three intact ordination lineages nowadays in which one can receive an ordination according to the Buddha's teachings: Saicho repeatedly requested that the Japanese government allow the construction of a Mahayana ordination platform. Permission was granted in 822 CE, seven days after Saicho died. The platform

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5208-437: Is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry , formulated by Alfred North Whitehead in 1919–1920. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given

5332-415: Is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved . Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric , which determines how distances are measured near each point) or extrinsic (where the object under study

5456-565: Is ordained by those with proper authority and ordained properly and validly; thorough records of priesthood ordination are kept by the church. Ordination is performed by the laying on of hands . Ordination to the office of priest in the Aaronic priesthood gives the ordained person the authority to: Ordination to the Melchizedek priesthood includes the authority to perform all the duties of the Aaronic priesthood, as well as ordain others to

5580-482: Is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations , geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group ' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry,

5704-405: Is viewed as a distinct aspect of other religions and is rejected. Islam does not have a formal and separated clergy. Religious leaders are usually called imams , sheikhs or maulana . The title imam (when used outside the historic Shi'ite context ) refers to someone who leads in prayer and can also be used in a linguistic sense for anyone who leads other Muslims in congregational prayers. Sheikh

5828-753: The Sulba Sutras . According to ( Hayashi 2005 , p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . In the Bakhshali manuscript , there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs

5952-690: The Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ( c.  1890 BC ), and the Babylonian clay tablets , such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum . Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated

6076-523: The Lambert quadrilateral and Saccheri quadrilateral , were part of a line of research on the parallel postulate continued by later European geometers, including Vitello ( c.  1230  – c.  1314 ), Gersonides (1288–1344), Alfonso, John Wallis , and Giovanni Girolamo Saccheri , that by the 19th century led to the discovery of hyperbolic geometry . In the early 17th century, there were two important developments in geometry. The first

6200-518: The Oxford Calculators , including the mean speed theorem , by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with

6324-550: The Presanctified Liturgy ), but only one person may be ordained to each order at any given service, that is, at most one bishop, one presbyter, and one deacon may be ordained at the same liturgy. In most Protestant churches, ordination is the rite by which their various churches: For the sake of authorization and church order, and not for reason of 'powers' or 'ability', individuals in most mainline Protestant churches must be ordained in order to preside at

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6448-509: The Riemann surface , and Henri Poincaré , the founder of algebraic topology and the geometric theory of dynamical systems . As a consequence of these major changes in the conception of geometry, the concept of " space " became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics . The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements , one of

6572-726: The Unitarian Universalist Association , candidates for "ministerial fellowship" with the denomination (usually third-year divinity school students) are reviewed, interviewed, and approved (or rejected) by the UUA Ministerial Fellowship Committee (MFC). However, given the fundamental principle of congregational polity , individual UU congregations make their own determination on ordination of ministers, and congregations may sometimes even hire or ordain persons who have not received UUA ministerial fellowship, and may or may not serve

6696-664: The Universal Algebra (1898), included the first systematic exposition in English of the theory of extension and the exterior algebra . With the rise of differential geometry the exterior algebra was applied to differential forms . In 1995 Lloyd C. Kannenberg published an English translation of The Ausdehnungslehre and Other works. For an introduction to the role of Grassmann's work in contemporary mathematical physics see The Road to Reality by Roger Penrose . Grassmann's mathematical ideas began to spread only towards

6820-702: The University of Tübingen . Note: Extensive online bibliography , revealing substantial contemporary interest in Grassmann's life and work. References each chapter in Schubring. Ordination Ordination is the process by which individuals are consecrated , that is, set apart and elevated from the laity class to the clergy , who are thus then authorized (usually by the denominational hierarchy composed of other clergy) to perform various religious rites and ceremonies. The process and ceremonies of ordination vary by religion and denomination. One who

6944-399: The complex plane using techniques of complex analysis ; and so on. A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry,

7068-590: The exterior product , also called "combinatorial product" (in German: kombinatorisches Produkt or äußeres Produkt “outer product”), the key operation of an algebra now called exterior algebra . (One should keep in mind that in Grassmann's day, the only axiomatic theory was Euclidean geometry , and the general notion of an abstract algebra had yet to be defined.) In 1878, William Kingdon Clifford joined this exterior algebra to William Rowan Hamilton 's quaternions by replacing Grassmann's rule e p e p = 0 by

7192-514: The 1860s and 1870s various mathematicians came to ideas similar to that of Grassmann's, but Grassmann himself was not interested in mathematics anymore. Adhémar Jean Claude Barré de Saint-Venant developed a vector calculus similar to that of Grassmann, which he published in 1845. He then entered into a dispute with Grassmann about which of the two had thought of the ideas first. Grassmann had published his results in 1844, but Saint-Venant claimed that he had first developed these ideas in 1832. One of

7316-631: The 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis , and introducing

7440-496: The 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss 's Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space . This implies that surfaces can be studied intrinsically , that is, as stand-alone spaces, and has been expanded into

7564-474: The 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry. Points are generally considered fundamental objects for building geometry. They may be defined by

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7688-463: The Book of Judges, there is a repetitive cycle of sin and deliverance. There is also a proposition regarding the cyclical offenses: "In those days Israel had no king; all the people did whatever seemed right in their own eyes" (Jdg. 21:25). Based partially upon the leadership of the prophetess, Deborah, some Protestant and non-denominational organizations grant ordination to women. Other denominations refute

7812-804: The Catholic Church. Some have even begun protest churches. Policy regarding the ordination of women differs among the different denominations of Judaism . Most Orthodox congregations do not allow female rabbis, while more liberal congregations began allowing female rabbis by the middle of the twentieth century. Most Abrahamic religions condemn the practice of homosexuality and the Bible has been interpreted that in Romans 1 that homosexuals are "worthy of death". Interpretation of this passage, as with others potentially condemning homosexuality varies greatly between and within different denominations. Beginning in

7936-561: The Catholic and Anglican churches recognize Orthodox ordinations. In the Catholic and Anglican churches, ordinations have traditionally been held on Ember Days , though there is no limit to the number of clergy who may be ordained at the same service. In the Eastern Orthodox Church, ordinations may be performed any day of the year on which the Divine Liturgy may be celebrated (and deacons may also be ordained at

8060-462: The Eucharist. In common use , however, the term priest , when unqualified, refers to the order of presbyter, whereas presbyter is mainly used in rites of ordination and other places where a technical and precise term is required. Ordination of a bishop is performed by several bishops; ordination of a priest or deacon is performed by a single bishop. The ordination of a new bishop is also called

8184-575: The Gewerbeschule in Berlin. A year later, he returned to Stettin to teach mathematics, physics, German, Latin, and religious studies at a new school, the Otto Schule. Over the next four years, Grassmann passed examinations enabling him to teach mathematics, physics , chemistry , and mineralogy at all secondary school levels. In 1847, he was made an "Oberlehrer" or head teacher. In 1852, he

8308-569: The Melchizedek or Aaronic priesthood, perform confirmations , bless and anoint the sick with oil, bless and dedicate graves, and other such rites. There are five offices within the Melchizedek Priesthood to which one could potentially be ordained: "Ordination to an office in the Aaronic Priesthood is done by or under the direction of the bishop or branch president. Ordination to an office in the Melchizedek Priesthood

8432-775: The NKT-IKBU ordination consists of the Five Precepts of a lay person, plus five more precepts created by Kelsang Gyatso . He is said to view them as a "practical condensation" of the 253 Vinaya vows of fully ordained monks. There are also no formal instructions and guidelines for the behaviour of monks and nuns within the NKT. Because the behaviour of monks and nuns is not clearly defined "each Resident Teacher developed his or her own way of 'disciplining' monks and nuns at their centres ...". Kelsang Gyatso's ordination has been publicly criticised by Geshe Tashi Tsering as going against

8556-522: The Protestant Reformation and the loosening of authority structures within many denominations, most Protestant groups re-envisioned the role of the ordained priesthood. Many did away with it altogether. Others altered it in fundamental ways, often favoring a rabbinical-type married minister of teaching (word) and discarding any notion of a sacrificial priesthood. A common epithet used by Protestants (especially Anglicans) against Catholics

8680-554: The Sōtō school developed the practice of ordaining laypeople after death, thus allowing monastic funeral rites to be used for them as well. The Buddhist ordination tradition of the New Kadampa Tradition -International Kadampa Buddhist Union (NKT-IKBU) is not the traditional Buddhist ordination, but rather one newly created by Kelsang Gyatso . Although those ordained within this organisation are called 'monks' and 'nuns' within

8804-686: The Tibetan tradition, and donated €50,000 for further research. The "1st International Congress on Buddhist Women's Role in the Sangha: Bhikshuni Vinaya and Ordination Lineages" was held at the University of Hamburg from 18–20 July 2007, in cooperation with the University's Asia-Africa Institute. Although the general tenor was that full ordination was overdue, the Dalai Lama presented a pre-drafted statement saying that more time

8928-733: The United States of America ordains women as deacons, priests and bishops. The Lutheran Evangelical Protestant Church ordains women at all levels including deacon, priest and bishop. Other denominations leave the decision to ordain women to the regional governing body, or even to the congregation itself; these include the Christian Reformed Church in North America and the Evangelical Presbyterian Church . The ordination of women in

9052-590: The angles between plane curves or space curves or surfaces can be calculated using the derivative . Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem . Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in

9176-583: The claim of a precedent based on Deborah's example because she is not specifically described as ruling over Israel, rather giving judgments on contentious issues in private, not teaching publicly, neither did she lead the military. Her message to her fellow judge Barak in fact affirmed the male leadership of Israel. The United Church of Canada has ordained women since 1932. The Evangelical Lutheran Church in America ordains women as pastors, and women are eligible for election as bishops. The Episcopal Church in

9300-412: The concept of angle and distance, finite geometry that omits continuity , and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space , or simply a space is a mathematical structure on which some geometry

9424-464: The congregation as its principal minister/pastor. The ordination of women is often a controversial issue in religions where either the office of ordination, or the role that an ordained person fulfills, is traditionally restricted to men, for various theological reasons. The Christian priesthood has traditionally been reserved to men. Some claim that women were ordained deacons in the first millennium of Christianity, but their claims are disputed. After

9548-513: The contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes ( c.  287–212 BC ) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of

9672-600: The core teachings of Buddhism and against the teachings of Tsongkhapa, the founder of the Gelugpa school from which Kelsang Gyatso was expelled As Unitarian Universalism features very few doctrinal thresholds for prospective congregation members, ordinations of UU ministers are considerably less focused upon doctrinal adherence than upon factors such as possessing a Masters of Divinity degree from an accredited higher institution of education and an ability to articulate an understanding of ethics, spirituality and humanity. In

9796-520: The end of his life. Thirty years after the publication of A1 the publisher wrote to Grassmann: “Your book Die Ausdehnungslehre has been out of print for some time. Since your work hardly sold at all, roughly 600 copies were used in 1864 as waste paper and the remaining few odd copies have now been sold out, with the exception of the one copy in our library.” Disappointed by the reception of his work in mathematical circles, Grassmann lost his contacts with mathematicians as well as his interest in geometry. In

9920-428: The field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry (also known as combinatorial geometry ), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits

10044-408: The first mathematicians to appreciate Grassmann's ideas during his lifetime was Hermann Hankel , whose 1867 Theorie der complexen Zahlensysteme . […], he developed […] some of Hermann Grassmann's algebras and W.R. Hamilton's quaternions . Hankel was the first to recognise the significance of Grassmann's long-neglected writings and was strongly influenced by them. In 1872 Victor Schlegel published

10168-408: The first part of his System der Raumlehre , which used Grassmann's approach to derive ancient and modern results in plane geometry . Felix Klein wrote a negative review of Schlegel's book citing its incompleteness and lack of perspective on Grassmann. Schlegel followed in 1875 with a second part of his System according to Grassmann, this time developing higher-dimensional geometry. Meanwhile, Klein

10292-520: The first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem . Pythagoras established the Pythagorean School , which is credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history. Eudoxus (408– c.  355 BC ) developed the method of exhaustion , which allowed the calculation of areas and volumes of curvilinear figures, as well as

10416-526: The former in topology and geometric group theory , the latter in Lie theory and Riemannian geometry . A different type of symmetry is the principle of duality in projective geometry , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem. A similar and closely related form of duality exists between

10540-526: The growing call for the ordination of women, Pope John Paul II issued the statement Ordinatio sacerdotalis in 1995. In it, he gave reasons why women cannot be ordained, and defined that the Holy Spirit had not conferred the power to ordain women upon the Church. In the wake of this definitive statement, many theologians considered the issue settled, but many continue to push for the ordination of women in

10664-598: The idea of metrics . For instance, the Euclidean metric measures the distance between points in the Euclidean plane , while the hyperbolic metric measures the distance in the hyperbolic plane . Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity . In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning

10788-537: The idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin ) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry . Omar Khayyam (1048–1131) found geometric solutions to cubic equations . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals , including

10912-537: The last years of his life he turned to historical linguistics and the study of Sanskrit . He wrote books on German grammar , collected folk songs, and learned Sanskrit. He wrote a 2,000-page dictionary and a translation of the Rigveda (more than 1,000 pages). In modern studies of the Rigveda , Grassmann's work is often cited. In 1955 a third edition of his dictionary was issued. Grassmann also noticed and presented

11036-903: The late 20th century, and more so in the early 21st century, several mainline denominational sects of Christianity and Judaism in the US and Europe endorsed the ordination of openly LGBT persons. See LGBT clergy in Christianity . The United Church of Christ ordained openly gay Bill Johnson in 1972, and lesbian Anne Holmes in 1977. While Buddhist ordinations of openly LGBT monks have occurred, more notable ordinations of openly LGBT novitiates have taken place in Western Buddhism. Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría )  'land measurement'; from γῆ ( gê )  'earth, land' and μέτρον ( métron )  'a measure')

11160-496: The latter half of the 20th century was an important issue between Anglicans and Catholics since the Catholic Church viewed the ordination of women as a huge obstacle to possible rapprochement between the two churches. The Catholic Church has not changed its view or practice on the ordination or women, and neither have any of the Orthodox churches; these churches represent approximately 65% of all Christians worldwide. In response to

11284-552: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral . Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry . Al-Mahani (b. 853) conceived

11408-539: The lineage of fully ordained nuns was not brought to Tibet by the Indian Vinaya masters, hence there is no rite for the ordination of full nuns. However th 14th Dalai Lama has endeavored for many years to improve this situation. In 2005, he asked fully ordained nuns in the Dharmaguptaka lineage, especially Jampa Tsedroen , to form a committee to work for the acceptance of the bhiksuni lineage within

11532-730: The mandate (approval) of the Pope , as the guarantor of the Church's unity. Moreover, at least three bishops are to perform the consecration, although the Apostolic See may dispense from this requirement in extraordinary circumstances (for example, in missionary settings or times of persecution). In the Catholic Church, those deacons destined to be ordained priests are often termed transitional deacons ; those deacons who are married before being ordained, as well as any unmarried deacons who chose not to be ordained priests, are called permanent deacons . Those married deacons who become widowers have

11656-411: The most influential books ever written. Euclid introduced certain axioms , or postulates , expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of

11780-429: The multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. In differential geometry,

11904-510: The notion of a vector space . He went on to develop those methods in his Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik ( A1 ) and its later revision Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet ( A2 ). In 1844, Grassmann published his masterpiece ( A1 ) commonly referred to as the Ausdehnungslehre , which translates as "theory of extension" or "theory of extensive magnitudes". Since A1 proposed

12028-441: The only instruments used in most geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis , parabolas and other curves, or mechanical devices, were found. The geometrical concepts of rotation and orientation define part of

12152-461: The organisation, and wear the robes of traditional Tibetan monks and nuns, in terms of traditional Buddhism they are neither fully ordained monks and nuns (Skt.: bhikshu , bhikshuni ; Tib.: gelong, gelongma) nor are they novice monks and nuns (Skt.: sramanera, srameneri; Tib.: gestul, getsulma). Unlike most other Buddhist traditions, including all Tibetan Buddhist schools, which follow the Vinaya,

12276-514: The physical world, geometry has applications in almost all sciences, and also in art, architecture , and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem , a problem that was stated in terms of elementary arithmetic , and remained unsolved for several centuries. During

12400-407: The placement of objects embedded in the plane or in space. Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space ). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of

12524-691: The possibility of seeking ordination to the priesthood in exceptional cases. While some Eastern churches have in the past recognized Anglican ordinations as valid, the current Anglican practice, in many provinces, of ordaining women to the priesthood —and, in some cases, to the episcopate—has caused the Orthodox generally to question earlier declarations of validity and hopes for union. The Catholic Church has never recognized Anglican orders as valid. Anglicanism recognizes Catholic and Orthodox ordinations; hence, clergy converting to Anglicanism are not "re-ordained". With respect to Lutheranism , "the Catholic Church has never officially expressed its judgement on

12648-482: The properties that they must have, as in Euclid's definition as "that which has no part", or in synthetic geometry . In modern mathematics, they are generally defined as elements of a set called space , which is itself axiomatically defined. With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that

12772-530: The religion permits any baptized adult male in good standing to officiate at a baptism, wedding, or funeral. In the Church of Jesus Christ of Latter-day Saints , a rite of ordination is performed to bestow either the Aaronic or Melchizedek priesthood (Hebrews 5:4–6) upon a worthy male member. As in the Anglican, Catholic and Orthodox traditions, great care is taken to assure that the candidate for priesthood

12896-443: The rule e p e p = 1. (For quaternions , we have the rule i = j = k = −1.) For more details, see Exterior algebra . A1 was a revolutionary text, too far ahead of its time to be appreciated. When Grassmann submitted it to apply for a professorship in 1847, the ministry asked Ernst Kummer for a report. Kummer assured that there were good ideas in it, but found the exposition deficient and advised against giving Grassmann

13020-522: The sacraments ( Baptism and Holy Communion ), and to be installed as a called pastor of a congregation or parish. Some Protestant traditions have additional offices of ministry to which persons can be ordained. For instance: For most Protestant denominations that have an office of bishop , including certain Lutheran and many Methodist churches, this is not viewed as a separate ordination or order of ministry. Rather, bishops are ordained ministers of

13144-554: The same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves , which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , surfaces are described by two-dimensional 'patches' (or neighborhoods ) that are assembled by diffeomorphisms or homeomorphisms , respectively. In algebraic geometry, surfaces are described by polynomial equations . A solid

13268-550: The same order as other pastors, simply having been "consecrated" or installed into the "office" (that is, the role) of bishop. However, some Lutheran churches also claim valid apostolic succession . Some Protestant churches – especially Pentecostal ones – have an informal tier of ministers. Those who graduate from a bible college or take a year of prescribed courses are licensed ministers. Licensed ministers are addressed as "Minister" and ordained ministers as "Reverend." They, and also Evangelical pastors, are generally ordained at

13392-586: The saying "nine rabbis do not constitute a minyan, but ten cobblers can". ) Recently, in some denominations , semikhah , or semicha lehazzanut , may refer to the ordination of a hazzan (cantor); while others use the term "investiture" to describe the conferral of cantorial authority. The tradition of the ordained monastic community ( sangha ) began with the Buddha , who established orders of monks and later of nuns . The procedure of ordination in Buddhism

13516-552: The second canon thereof states, "Let a presbyter, deacon, and the rest of the clergy, be ordained by one bishop"; the latter canons, whatever their origin, were imposed on the universal church by the Seventh Ecumenical Council , the Second Council of Nicaea , in its first canon. Only a person ordained to the priesthood may administer certain sacraments (most especially, hear confessions , anointing

13640-471: The sense of "lean on", and hence "to be authorized". While the Hebrew word semikhah is rendered as "ordination" in English, a rabbi is not a priest per se , but primarily functions as a legal scholar and teacher of Torah ; and in fact, for many religious purposes the presence of a rabbi is not necessary. (For example, at prayer, a minyan (quorum) of ten lay-people is both necessary and sufficient; thus

13764-526: The sick – unction – or celebrating any Mass – the Eucharist). The Catholic Church teaches that one bishop is sufficient to consecrate a new bishop validly (that is, for an episcopal ordination actually to take place). In most Christian denominations that retain the practice of ordination, only an already ordained (consecrated) bishop or the equivalent may ordain bishops, priests, and deacons. However, Canon Law requires that bishops always be consecrated with

13888-589: The study of Euclidean concepts such as points , lines , planes , angles , triangles , congruence , similarity , solid figures , circles , and analytic geometry . Euclidean vectors are used for a myriad of applications in physics and engineering, such as position , displacement , deformation , velocity , acceleration , force , etc. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics , econometrics , and bioinformatics , among others. In particular, differential geometry

14012-452: The theory of linear independence in a way that is astonishingly similar to the presentation one finds in modern linear algebra texts. He defines the notions of subspace , linear independence , span , dimension , join and meet of subspaces, and projections of elements onto subspaces. [...] few have come closer than Hermann Grassmann to creating, single-handedly, a new subject. Following an idea of Grassmann's father, A1 also defined

14136-409: The theory of manifolds and Riemannian geometry . Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries ) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and

14260-474: The theory of the tides. In 1840, he did so, taking the basic theory from Laplace 's Traité de mécanique céleste and from Lagrange 's Mécanique analytique , but expositing this theory making use of the vector methods he had been mulling over since 1832. This essay, first published in the Collected Works of 1894–1911, contains the first known appearance of what is now called linear algebra and

14384-697: The validity of orders as they have been handed down by episcopal succession in these two national Lutheran churches" (the Evangelical Lutheran Church of Sweden and the Evangelical Lutheran Church of Finland ) though it does "question how the ecclesiastical break in the 16th century has affected the apostolicity of the churches of the Reformation and thus the apostolicity of their ministry". Some Eastern Orthodox churches recognize Catholic ordinations while others "re-ordain" Catholic clergy (as well as Anglicans) who convert. However, both

14508-408: Was advancing his Erlangen program , which also expanded the scope of geometry. Comprehension of Grassmann awaited the concept of vector spaces , which then could express the multilinear algebra of his extension theory. To establish the priority of Grassmann over Hamilton, Josiah Willard Gibbs urged Grassmann's heirs to have the 1840 essay on tides published. A. N. Whitehead 's first monograph,

14632-540: Was appointed to his late father's position at the Stettin Gymnasium, thereby acquiring the title of Professor. In 1847, he asked the Prussian Ministry of Education to be considered for a university position, whereupon that Ministry asked Ernst Kummer for his opinion of Grassmann. Kummer wrote back saying that Grassmann's 1846 prize essay (see below) contained "commendably good material expressed in

14756-838: Was finished in 827 CE at Enryaku-ji temple on Mount Hiei , and was the first in Japan. Prior to this, those wishing to become monks/nuns were ordained using the Hinayana precepts, whereas after the Mahayana ordination platform, people were ordained with the Bodhisattva precepts as listed in the Brahma Net Sutra . Pabbajja is an ordination procedure for novice Buddhist monks in the Theravada tradition. The legitimacy of fully ordained nuns ( bhikkhuni/bhiksuni ) has become

14880-480: Was indeed "older" (i.e., less synthetic) than Sanskrit. This meant that genealogical and typological classifications of languages were at last correctly separated in linguistics, allowing significant progress for later linguists. These philological accomplishments were honored during his lifetime. He was elected to the American Oriental Society and in 1876 he received an honorary doctorate from

15004-412: Was required to reach a decision, thus nullifying the intentions of the congress. In Medieval Sōtō Zen , a tradition of posthumous ordination was developed to give the laity access to Zen funeral rites . Chinese Ch’an monastic codes, from which Japanese Sōtō practices were derived, contain only monastic funeral rites; there were no provisions made for funerals for lay believers. To solve this problem,

15128-403: Was that Catholics were a 'priest-ridden' people. Hatred for priests was a common element of anti-Catholicism and pogroms against Catholics focused on expelling, killing, or forcefully 'laicizing' priests. Beginning in the twentieth century, many Protestant denominations began re-evaluating the roles of women in their churches. Many now ordain women. According to the biblical book of Judges ,

15252-596: Was the creation of analytic geometry, or geometry with coordinates and equations , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics . The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections , especially as they relate to artistic perspective . Two developments in geometry in

15376-697: Was the field that most interested him when he returned to Stettin in 1830 after completing his studies in Berlin. After a year of preparation, he sat the examinations needed to teach mathematics in a gymnasium, but achieved a result good enough to allow him to teach only at the lower levels. Around this time, he made his first significant mathematical discoveries, ones that led him to the important ideas he set out in his 1844 paper Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik , here referred to as A1 , later revised in 1862 as Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet , here referred to as A2 . In 1834 Grassmann began teaching mathematics at

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