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100-490: Sankhyayana Brahmana is the second available and preserved Brahmana text of the Rigaveda . The text is divided into 30 chapters and 226 Khandas . It is said that Kaushitaki was the teacher of Sankhyayana. He imparted the knowledge of the text to his disciple Sankhyayana. Then Sankhyayana nominated the name of the text as Kaushitaki Brahmana after his teacher name. But later the text was called as Sankhyayana Brahmana. The text

200-520: 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

300-418: 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

400-425: 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

500-689: A Brahmana-proper, although it has been published as one. Linked with the Krishna (Black) Yajurveda, it is 'actually part of the Vadhula Shrauta Sutra'. S. Sharva states that in 'the brahmana literature this word ['brahmana'] has been commonly used as detailing the ritualism related to the different sacrifices or yajnas ... The known recensions [i.e. schools or Shakhas ] of the Vedas , all had separate brahmanas. Most of these brahmanas are not extant .... [ Panini ] differentiates between

600-602: A ceremony by which people of non-Aryan stock could be admitted into the Aryan family'. The Sadvimsa Brahmana is also of the Kauthuma Shakha, and consists of 5 adhyayas (lessons or chapters). Caland states it is 'a kind of appendix to the [Panchavimsha Brahmana], reckoned as its 26th book [or chapter]... The text clearly intends to supplement the Pancavimsabrahmana, hence its desultory character. It treats of

700-909: A commentary on the Vedas, so that even common people would be able to understand the meaning of the Vedic Mantras. Madhavacharya told him that his younger brother Sayana was a learned person and hence he should be entrusted with the task'. Modak also lists the Brahmanas commented upon by Sayana (with the exception of the Gopatha): For ease of reference, academics often use common abbreviations to refer to particular Brahmanas and other Vedic, post-Vedic (e.g. Puranas ), and Sanskrit literature. Additionally, particular Brahmanas linked to particular Vedas are also linked to (i.e. recorded by) particular Shakhas or schools of those Vedas as well. Based on

800-405: 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 ,

900-523: 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

1000-435: A mantra infallible, while one mistake made it powerless. Scholars suggest that this orthological perfection preserved Vedas in an age when writing technology was not in vogue, and the voluminous collection of Vedic knowledge were taught to and memorized by dedicated students through Svādhyāya , then remembered and verbally transmitted from one generation to the next. It seems breaking silence too early in at least one ritual

1100-440: 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,

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1200-428: 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

1300-451: 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

1400-444: 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

1500-528: 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

1600-939: A portion, could be reckoned as part of the Brâhmana literature of the Rig-veda (see Aitareya-âranyaka, Introduction, p. xcii), and that hence the Upanishad might be called the Upanishad of the Brâhmana of the Kaushîtakins'. W. Caland states that of the Samaveda , three Shakhas (schools or branches) 'are to be distinguished; that of the Kauthumas, that of the Ranayaniyas, and that of the Jaiminiyas'. Visnu

1700-602: 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

1800-427: 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

1900-600: 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

2000-518: 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

2100-494: 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

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2200-411: 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

2300-540: 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

2400-535: Is a divine purification. Whatever here is unappeased of the sacrifice and whatever is impure, for all that, water forms the means of appeasing. So by water they appease it. It seems that this Brahmana has not been fully translated to date, or at least a full translation has not been made available. Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría )  'land measurement'; from γῆ ( gê )  'earth, land' and μέτρον ( métron )  'a measure')

2500-400: 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

2600-413: 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

2700-468: Is also linked with the Ashvalayana Shakha. The text itself consists of eight pañcikā s (books), each containing five adhyaya s (chapters), totaling forty in all. C. Majumdar states that 'it deals principally with the great Soma sacrifices and the different ceremonies of royal inauguration'. Haug states that the legend about this Brahmana, as told by Sayana , is that the 'name "Aitareya"

2800-620: Is also mentioned in the commentary of Brahman Sutra by Shankracharya . Similarly the Vedic sage and Sanskrit grammarian Panini in his text Astadhyayai mentioned about the text. It is said that the text Sankhyayana Brahmana is generally not mentioned in Puranas but in Agni Purana its mention is found at the second shloka of chapter 271. "bhedaḥ sāṅkhyāyanaścaika āśvalāyano dvitīyakaḥ, śatāni daśa mantrāṇāṃ brāhmaṇā dvisahasrakaṃ" In

2900-620: Is below and night to what is on the other side. In fact, the sun never sets. Nor does it set for him who has such a knowledge. Such a one becomes united with the sun, assumes its form, and enters its place. As detailed in the main article, the Aitareya Brahmana (AB) is ascribed to the sage Mahidasa Aitareya of the Shakala Shakha (Shakala school) of the Rigveda , and is estimated to have been recorded around 600-400 BCE . It

3000-447: Is by Indian tradition traced to Itara ... An ancient Risi had among his many wives one who was called Itara . She had a son Mahidasa by name [i.e. Mahidasa Aitareya]... The Risi preferred the sons of his other wives to Mahidasa, and went even so far as to insult him once by placing all his other children in his lap to his exclusion. His mother, grieved at this ill-treatment of her son, prayed to her family deity ( Kuladevata ), [and]

3100-826: Is considered to be an appendix to the Panchavismsha / Tandya Brahmana. The Adbhuta Brahmana is from the last part of the Sadvimsa Brahmana and deals with 'omens and supernatural things'. Attributed by Caland to the Kuthuma-Ranayaniya Shakha, but by Macdonell to the Tandin Shakha . Also called the Devatadhyaya Brahmana. The Mantra Brahmana (also called the Samaveda-Mantrabrahmana, SMB)

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3200-409: 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

3300-595: Is divided into thirty chapters [adhyayas] and 226 Khanda[s]. The first six chapters dealing with food sacrifice and the remaining to Soma sacrifice. This work is ascribed to Sankhyayana or Kaushitaki'. S. Shrava disagrees, stating that it 'was once considered that [the] Kaushitaki or Samkhayana was the name of the same brahmana... [but the Samkhayana] differs, though slightly, from the Kaushitaki Brahmana'. C. Majumdar states that it 'deals not only with

3400-939: Is from the first two chapters of the Chandogya Brahmana (also called the Chandogyaopanishad and the Upanishad Brahmana); the remaining chapters of the Chandogya Brahmana form the Chandogya Upanishad . Also called the Catapatha Brahmana (CB; this abbreviation also denotes the Mâdhyandina recension ) Part of the Taittiriya Aranyaka ; explains the Pravargya rite. Generally not considered

3500-437: 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

3600-415: 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

3700-544: Is permissible in the Satapatha (1.1.4.9), where 'in that case mutter some Rik [ Rigveda ] or Yagus-text [ Yajurveda ] addressed to Vishnu ; for Vishnu is the sacrifice, so that he thereby regains obtains a hold on the sacrifice , and penance is there by done by him'. Recorded by the grammarian Yaska , the Nirukta , one of the six Vedangas or 'limbs of the Vedas' concerned with correct etymology and interpretation of

3800-482: 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,

3900-586: Is the sacrifice ; what here (on this day) is not brought about, that he brings about through Vishnu (who is) the sacrifice. Caland states that the Panchavimsha / Tandya Brahmana of the Kauthuma Shakha consists of 25 prapathakas (books or chapters). C. Majumdar states that it 'is one of the oldest and most important of Brahmanas. It contains many old legends, and includes the Vratyastoma ,

4000-753: 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

4100-844: The Brahmin priest. There are fifteen verses in this chapter which discusses about the activities of Prajapati , part of the Brahmin priests and Haviryajna . The chapters onwards to the sixth chapter discuss about the Yajnas related to the Soma . These Yajnas are called as Somayajna . This Hinduism-related article is a stub . You can help Misplaced Pages by expanding it . Brahmana Divisions Sama vedic Yajur vedic Atharva vedic Vaishnava puranas Shaiva puranas Shakta puranas The Brahmanas ( / ˈ b r ɑː m ə n ə z / ; Sanskrit : ब्राह्मणम् , IAST : Brāhmaṇam ) are Vedic śruti works attached to

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4200-690: 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

4300-523: 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

4400-567: The Monier-Williams Sanskrit dictionary, 'Brahmana' means: M. Haug states that etymologically , 'the word ['Brahmana' or 'Brahmanam'] is derived from brahman which properly signifies the Brahma priest who must know all Vedas , and understand the whole course and meaning of the sacrifice ... the dictum of such a Brahma priest who passed as a great authority, was called a Brahmanam'. S. Shrava states that synonyms of

4500-518: 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

4600-509: The Puranas (e.g. Bhagavata Purana , Canto 4, Chapter 8-12). The gods and the Asuras were in conflict over these worlds. From them Agni departed, and entered the seasons. The gods, having been victorious and having slain the Asuras, sought for him; Yama and Varuna discerned him. Him (the gods) invited, him they instructed, to him they offered a boon. He chose this as a boon, '(Give) me

4700-509: 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

4800-613: The Rigveda . A.B. Keith , a translator of the Aitareya and Kausitaki Brahmanas, states that it is 'almost certainly the case that these two [Kausitaki and Samkhyana] Brahmanas represent for us the development of a single tradition, and that there must have been a time when there existed a single... text [from which they were developed and diverged]'. Although S. Shrava considers the Kausitaki and Samkhyana Brahmanas to be separate although very similar works, M. Haug considers them to be

4900-520: The Samhitas (hymns and mantras) of the Rig , Sama , Yajur , and Atharva Vedas. They are a secondary layer or classification of Sanskrit texts embedded within each Veda, which explain and instruct on the performance of Vedic rituals (in which the related Samhitas are recited). In addition to explaining the symbolism and meaning of the Samhitas , Brahmana literature also expounds scientific knowledge of

5000-662: The Soma , but also other sacrifices'. Keith estimates that the Kaushîtaki-brâhmana was recorded around 600–400 BCE, adding that it is more 'scientific' and 'logical' than the Aitareya Brahmana, although much 'of the material of the Kausitaki, and especially the legends, has been taken over by the Brahmana from a source common to it and the Aitareya, but the whole has been worked up into a harmonious unity which presents no such irregularities as are found in

5100-754: The Subrahmanya formula, of the one-day-rites that are destined to injure ( abhicara ) and other matters. This brahmana, at least partly, is presupposed by the Arseyakalpa and the Sutrakaras'. Caland states that the Adbhuta Brahmana, also of the Kauthuma Shakha, is the 'latest part [i.e. 5th adhyaya of the Sadvimsa Brahmana], that which treats of Omina and Portenta [ Omens and Divination ]'. Majumdar agrees. Caland states that

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5200-412: The Vedas , references several Brahmanas to do so. These are (grouped by Veda): Both apply to the Śukla (White) Yajurveda. The 14th Century Sanskrit scholar Sayana composed numerous commentaries on Vedic literature, including the Samhitas , Brahmanas, Aranyakas , and Upanishads . B.R. Modak states that 'king Bukka [1356–1377 CE] requested his preceptor and minister Madhavacharya to write

5300-545: The Vedic Period , including observational astronomy and, particularly in relation to altar construction, geometry . Divergent in nature, some Brahmanas also contain mystical and philosophical material that constitutes Aranyakas and Upanishads . Each Veda has one or more of its own Brahmanas, and each Brahmana is generally associated with a particular Shakha or Vedic school. Less than twenty Brahmanas are currently extant, as most have been lost or destroyed. Dating of

5400-399: 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,

5500-631: 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

5600-496: 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

5700-474: 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

5800-569: The Aitareya. It is clearly a redaction of the tradition of the school made deliberately after the redaction of the Aitareya'. Max Müller states that the Kaushitaki Upanishad – also called the Kaushitaki Brahmana Upanishad (KBU) – 'does not form part of the Kaushîtaki-brâhmana in 30 adhyâyas which we possess, and we must therefore account for its name by admitting that the Âranyaka , of which it formed

5900-472: The Brahmans, consists, according to the opinion of the most eminent divines of Hindustan , of two principal parts, viz. Mantra [ Samhita ] and Brahmanam... Each of the four Vedas ( Rik , Yajus , Saman , and Atharvan ) has a Mantra, as well as a Brahmana portion. The difference between both may be briefly stated as follows: That part which contains the sacred prayers, the invocations of the different deities,

6000-558: The Brâhma n as are thus our oldest sources from which a comprehensive view of the sacrificial ceremonial can be obtained, they also throw a great deal of light on the earliest metaphysical and linguistic speculations of the Hindus . Another, even more interesting feature of these works, consists in the numerous legends scattered through them. From the archaic style in which these mythological tales are generally composed, as well as from

6100-540: The Earth ( Bhumi ), who appeared in her celestial form in the midst of the assembly, placed him on a throne ( simhasana ), and gave him as a token of honour for his surpassing all other children in learning a boon (vara) which had the appearance of a Brahmana [i.e. the Aitareya]'. P. Deussen agrees, relating the same story. Notably, The story itself is remarkably similar to the legend of a Vaishnava boy called Dhruva in

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6200-503: The Kauthuma Shakha is 'in 3 prapathakas [books or chapters]... It deals with the deities to which the samans are addressed'. Dalal adds that the 'first part of the Devatadhyaya is the most important as it provides rules to determine the deities to whom the samans are dedicated. Another section ascribes colours to different verses, probably as aids to memory or for meditation... [It] includes some very late passages such as references to

6300-744: The Kauthumas, i.e. the Gramegeya-gana / Veya-gana and the Aramyegeya-gana / Aranya-gana]'. The nature of the ganas noted are discussed in the same text. As illustrated below, this Brahmana is virtually identical to the Jaiminiya Arsheya Brahmana of the Jaiminiya Shakha . Caland states that the Vamsha Brahmana of the Kauthuma Shakha is 'in 3 khandas [books]... it contains the lists of teachers of

6400-461: The Samaveda'. Notably, Dalal adds that of the 53 teachers listed, the 'earliest teacher, Kashyapa , is said to have received the teaching from the god, Agni '. He should proceed thus: Having taken a water-pot or a water-jar he should go pouring it out from the garhapatya to the ahavaniya with the verse: "Here Visnu strode". The rc [RigVeda verse, e.g. 1.22.17] is a divine purification, water

6500-609: The Samavidhana Brahmana of the Kauthuma Shakha is 'in 3 prapathakas [books or chapters]... its aim is to explain how by chanting various samans [hymns of the Samaveda ] some end may be attained. It is probably older than one of the oldest dharmasastras, that of Gautama'. M. S. Bhat states that it is not properly a Brāhmaṇa text, but belongs to the Vidhāna literature. Caland states that the Daivata Brahmana of

6600-416: The Sankhyayana Brahmana, the rules and instructions of human conduct have been described. Apart from the human conduct, it also describes some astronomical events in its passages. The first six chapters of the text describe about the Yajna of food and the remaining chapters describe about the Yajna of Soma. The Soma Yajna in the Vedic tradition is considered as the major subject of the text. The chapters of

6700-433: The abbreviations and Shakhas provided by works cited in this article (and other texts by Bloomfield , Keith , W. D, Whitney , and H.W. Tull), extant Brahmanas have been listed below, grouped by Veda and Shakha . Note that: The Kausitaki and Samkhyana are generally considered to be the same Brahmana. Also called the Cankhayana Brahmana. The Panchavismsha and Tandya are the same Brahmana. The Sadvimsa Brahmana

6800-531: 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

6900-412: 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

7000-407: 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

7100-418: The fact that not a few of them are found in Brâhma n as of different schools and Vedas , though often with considerable variations, it is pretty evident that the ground-work of many of them goes back to times preceding the composition of the Brâhma n as'. The Indira Gandhi National Centre for the Arts (IGNCA) states that while 'the Upanishads speculate on the nature of the universe, and the relationship of

7200-428: 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

7300-501: The final codification of the Brahmanas and associated Vedic texts is controversial, as they were likely recorded after several centuries of oral transmission. The oldest Brahmana is dated to about 900 BCE , while the most recent are dated to around 700 BCE. Brahmana (or Brāhmaṇam , Sanskrit : ब्राह्मणम्) can be loosely translated as ' explanations of sacred knowledge or doctrine ' or ' Brahmanical explanation'. According to

7400-579: 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

7500-568: The fore-offering and the after-offerings for my own, and the ghee of the waters and make of plants.' Therefore they say 'Agni's are the fore-offerings and the after-offerings; Agni's is the butter.' Then indeed did the gods prosper, the Asuras were defeated. He prospers himself, his foe is defeated, who knows thus. The Indira Gandhi National Centre for the Arts (IGNCA) states that the 'Kaushitaki Brahmana [is] associated with Baskala Shakha of [the] Rigveda and [is] also called Sankhyayana Brahmana. It

7600-526: 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

7700-481: The four yugas or ages'. Caland states that the Samhitopanishad Brahmana of the Kauthuma Shakha is 'in 5 khandas [books]... It treats of the effects of recitation, the relation of the saman [hymns of the Samaveda ] and the words on which it is chanted, the daksinas to be given to the religious teacher'. Dalal agrees, stating that it 'describes the nature of the chants and their effects, and how

7800-598: 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

7900-537: 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

8000-552: 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

8100-411: 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

8200-429: 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,

8300-426: The old and the new brahmanas... [he asked] Was it when Krishna Dvaipayana Vyasa had propounded the Vedic recensions? The brahmanas which had been propounded prior to the exposition of recensions by [ Vyasa ] were called as old brahmanas and those which had been expounded by his disciples were known as new brahmanas'. The Aitareya , Kausitaki, and Samkhyana Brahmanas are the two (or three) known extant Brahmanas of

8400-438: The one and the many, the immanent and transcendental, the Brahmanas make concrete the world-view and the concepts through a highly developed system of ritual-yajna. This functions as a strategy for a continuous reminder of the inter-relatedness of man and nature, the five elements and the sources of energy'. The Brahmanas are particularly noted for their instructions on the proper performance of rituals, as well as explanations on

8500-441: 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

8600-468: The performance of Vedic sacrifices , and Arthavada praises the rituals, the glory of the Devas and so on. The belief in reincarnation and transmigration of soul started with [the] Brahmanas... [The] Brahmana period ends around 500 BC[E] with the emergence of Buddhism and it overlaps the period of Aranyakas , Sutras , Smritis and the first Upanishads '. M. Haug states that the 'Veda, or scripture of

8700-514: 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

8800-407: 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

8900-482: 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

9000-529: The riks or Rig Vedic verses were converted into samans. Thus it reveals some of the hidden aspects of the Sama Veda '. Caland states that the Arsheya Brahmana of the Kauthuma Shakha is ''in 3 prapathakas [books or chapters]... This quasi-brahmana is, on the whole, nothing more than an anukramanika, a mere list of the names of the samans [hymns of the Samaveda ] occurring in the first two ganas [of

9100-472: The sacred verses for chanting at the sacrifices , the sacrificial formulas [is] called Mantra ... The Brahmanam [part] always presupposes the Mantra; for without the latter it would have no meaning... [they contain] speculations on the meaning of the mantras, gives precepts for their application, relates stories of their origin... and explains the secret meaning of the latter'. J. Eggeling states that 'While

9200-554: 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

9300-402: The same work referred to by different names. The sun does never set nor rise. When people think the sun is setting (it is not so). For, after having arrived at the end of the day, it makes itself produce two opposite effects, making night to what is below and day to what is on the other side...Having reached the end of the night, it makes itself produce two opposite effects, making day to what

9400-589: 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

9500-486: The symbolic importance of sacred words and ritual actions. Academics such as P. Alper, K. Klostermaier and F.M, Muller state that these instructions insist on exact pronunciation (accent), chhandas (छन्दः, meters), precise pitch, with coordinated movement of hand and fingers – that is, perfect delivery. Klostermaier adds that the Satapatha Brahamana , for example, states that verbal perfection made

9600-574: The text have nine verses. The chapter discusses about the Agnihotra . The third chapter discusses about the offerings to the new and the full moon . It contains nine verses. The fourth chapter of the text discusses about some special Yajnas attributed to sages. There fourteen verses in the chapter. The fifth chapter discusses about the four monthly Yajnas. In this chapter there are ten verses attributed to Viswadevas , Varunapraghasas , Sakamedhas and father, etc. The sixth chapter discusses about

9700-539: The text is called Adhyaya . There are 30 Adhyayas in the text. The first Adhyaya having five verses discusses about the establishment of Agni . The verses are attributed to the sacred offerings to different forms of Agni, the attainment of pre and post offerings by Agni, the time of re-establishment of Agni, pre and post offerings and the portion of butters. The last verse is attributed to the Vibhaktis and offerings to Goddess Aditi . The second Adhyaya or chapter of

9800-409: 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

9900-439: The word 'Brahmana' include: R. Dalal states that the 'Brahmanas are texts attached to the Samhitas [hymns] – Rig , Sama , Yajur and Atharva Vedas – and provide explanations of these and guidance for the priests in sacrificial rituals'. S. Shri elaborates, stating 'Brahmanas explain the hymns of the Samhitas and are in both prose and verse form... The Brahmanas are divided into Vidhi and Arthavada. Vidhi are commands in

10000-596: 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

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