95-482: Swathi Sangeethotsavam (Swathi Music Festival) is a ten-day festival of music celebrating the compositions of Maharaja Swathi Thirunal . The festival is held from 4 to 13 January every year at Kuthira Malika , Trivandrum , where the Maharaja is believed to have composed many of his works. The festival is a tribute to Swathi Tirunal and is exclusively dedicated to his compositions. The concerts are not ticketed. The festival
190-520: 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
285-418: 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
380-425: 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
475-525: A change in Indian customs that might lead to the happiest results. He was informed, on good authority, that there was not a child who had reached eight years of age not capable of reading and writing; but this distinguished prince, not satisfied with advancing the interests of elementary education, had established an observatory, and placed in it an English gentleman, a member of the Royal Society of London, and who
570-405: 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 ,
665-523: 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
760-499: A high place among the most enlightened of European Sovereigns had his destiny been so cast. You will be grieved to learn about the demise of His Highness the Rajah of Travancore. Among the native princes of India, he was distinguished for his superior intelligence and extensive acquirements in oriental literature. He is not unknown to fame in the European world, for most of you must be aware that
855-538: A monarch, Swathi Thirunal was incredibly hardworking and supremely committed to his kingdom and people. The appointment of General Cullen as the Resident of Travancore, was the beginning of the end for the Maharajah. Historian P. Shungunny Menon wrote: Resident Jerond Cullen assumed almost sovereign authority. Such was his oppressive intrusion in the administration. The king was made totally powerless. Compounding this atrocity
950-440: 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,
1045-428: 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
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#17328685684581140-552: A national calamity. The Journal the Royal Asiatic Society of Great Britain and Ireland ran an obituary in 1847 which mourned that, The early death of this enlightened and princely patron of true science, is a subject of just regret. Prince Rama Varma , renowned South Indian Classical musician and descendant of Swathi Thirunal, organizes the Swathi Sangeethotsavam , a 10-day music festival featuring exclusively
1235-451: 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
1330-444: 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
1425-518: 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
1520-598: 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
1615-455: A result of the revolt, the government changed the laws and allowed dalit men & women to wear upper clothes. Several taxes on the lower classes were also repealed by the Maharajah after protests. Another area where Swathi Thirunal took interest was in astronomy . He wished to compare Western findings with Indian knowledge. He had knowledge of observatories in Madras and others. Finding that there
1710-642: A similar set-up. The current observatory site was chosen on top of a laterite mount near the Kanakakunnu hill, which was observed as having the best western sky views in Eastern hemisphere, being near the equator and the Arabian sea. He was instrumental in buying telescopes and tools to Thiruvananthapuram (via ship route through the Middle east) from England. It became a part of the erstwhile Travancore University, but for some time
1805-427: 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
1900-717: A social reformer and iconoclast Ayya Vaikundar, an incarnation of Hindu Deity Vishnu ) severely criticized Swathi Thirunal for the then prevalent caste discrimination against the members of the lower classes in Travancore. He referred to the King as Ananthapuri Neechan (vile man of Ananthapuri); referred to the Brahmins and the British as Karineechanmar (vile Black cheaters) and Venneechanmar (vile White cheaters), respectively. The upper caste Hindus then complained to Swathi Thirunal that Vaikundar
1995-477: A son, Thiruvattar Chithira Nal Ananthapadmanabhan Chempakaraman Thampi . A few months later, for the care of the baby, the Maharajah married another lady called Neelamma Pillai Ammachi by adopting her into the Thiruvattar Ammaveedu. He later married Sundara Lakshmi in 1843, a Saiva Mudaliar dancer, after adopting her into Vadasseri Ammaveedu . The story of the dancer Sugandhavalli who didn't get along with
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#17328685684582090-600: 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
2185-518: 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
2280-494: 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
2375-411: 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
2470-540: 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
2565-400: 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
2660-413: 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
2755-465: Is conducted by Rama Varma Maharaja of Travancore Trust under the helm of Prince Rama Varma . The Government of Kerala used to conduct the festival in Kuthiramalika in memory of Swathi Thirunal . In the late 90s, after they decided to hold it in different places all over Kerala and discontinued the festival at Kuthiramalika, Rama Varma Maharaja took the initiative to continue the annual festival under
2850-409: 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
2945-437: 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
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3040-415: 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
3135-482: 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,
3230-726: 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
3325-593: The East India Company , and his officials and declared in the Durbar that she was entrusting the East India Company with the care of her child and expected the company to co-operate with him in future. In 1829 Swathi Thirunal reached majority at 16 and assumed full powers of ruler and reigned as the Maharajah of Travancore until his death in 1846. He had an elder sister, Gowri Rukmini Bayi , whose children ascended
3420-667: 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
3515-518: 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
3610-506: 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
3705-509: 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
3800-399: 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,
3895-621: 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
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3990-491: 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
4085-474: 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
4180-630: The Ammachi approached Irayimman Thampi for a solution. According to researchers he then created the famous Malayalam Padam (song) Prananaathan Enikku Nalkiya and told the Ammachi to sing it loudly in the King's presence; after hearing it the King was pleased and they reconciled immediately. This particular work of Thampi is considered by experts as one of the most beautiful Shringara (erotic) Padams available in Malayalam . Together they had three children but in 1839 Narayani Pillai Ammachi died, leaving behind
4275-602: The Diwan to resign when he heard that the Diwan had acted to favour a particular party in a land dispute. He started an English school in Thiruvananthapuram in 1834, which came to be called the Maharajah's Government Free School and later became Maharajah's High School and then Maharajah's College . It is now the University College . Later, similar schools were started at many other places. He also implemented reforms in
4370-461: The Earliest Times, 1878) records an incident between young Swathi Thirunal and Col. Welsh, a visiting British officer, that the word geometry and words like hexagon , heptagon and so on were derived from Sanskrit . Colonel Welsh summed up the boy King's genius as follows: Swati Tirunal, now thirteen... took up a book of mathematics and selecting the forty-seventh proposition of Euclid sketched
4465-514: The King's first wife, Narayani Pillai Thankachi, has been disproved by R.P. Raja as nothing but fiction in his research treatise 'New Light on Swathi Thirunal'. In 1845 the King constructed the Thanjavur Ammaveedu for his third consort. Sundara Lakshmi, a great devotee of Lord Ganapati and Kanjirottu Yakshi Amma, resided there until her death in 1856. Swathi Thirunal took over the reins of Travancore from his aunt, Gowri Parvati Bayi (she
4560-558: The Musnud (throne) the moment he had attained his 16th year. In 1829, at the age of sixteen, Maharajah Swathi Thirunal married Thiruvattar Ammachi Panapillai Amma Srimathi Ayikutty Narayani Pillai Thankachi, a famed beauty of the Thiruvattar Ammaveedu family, was an expert Carnatic singer and Veena player. Once there was a minor quarrel between Narayani Pillai Ammachi and her husband, the King. The quarrel continued for some days;
4655-614: The Oriental Manuscript Library were started by Swathi Thirunal, the Museum and Zoo in Thiruvananthapuram as well. The Maharajah was also an honorary member of the Royal Asiatic Society from 1843. Maharajah Swathi Thirunal also put an end to the barbaric punishment called the 'SUCHINDRAM KAIMUKKU' According to which the accused was forced to prove his innocence by dipping his hand in boiled ghee at Suchindram temple, and he
4750-405: The Travancore musnud consecutively. Her only daughter was the mother of Moolam Thirunal . He had a younger brother, Uthram Thirunal Marthanda Varma , who succeeded him in 1846 and ruled Travancore until his demise in 1860. Irayimman Thampi , the famous poet-composer wrote perhaps the most famous Malayalam lullaby Omanathinkal Kidavo ( ഓമനത്തിങ്കള്ക്കിടാവോ ), about Swathi Thirunal when he
4845-690: The Travancore Trust. The festival is organized by Prince Rama Varma , Carnatic musician and direct descendant of Swathi Tirunal. Swathi Thirunal Sri Swathi Thirunal Rama Varma III (16 April 1813 – 26 December 1846) was the Maharaja of the Kingdom of Travancore . He was a great musician and composer who has to his credit over 400 classical compositions in both Carnatic and Hindustani style. A code of laws, courts of justice, introduction of English education, construction of an observatory, installation of
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#17328685684584940-456: The age of 33, Maharajah Swathi Thirunal died on 26 December 1846. The demise of Maharajah Swathi Thirunal attracted the attention of even the foreign press. Allen's Indian Mail and Register of Intelligence of British &Foreign India, China, & All Parts of the East wrote: Both intellectually and morally, he was indeed far beyond his country and equals in rank; in both respects he might have taken
5035-584: 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
5130-439: The census, the population of Travancore was 128,068. Swathi Thirunal was also instrumental in bringing modern medicine to the state. He appointed a European as the palace physician. He was also given the responsibility of providing medical assistance to local people, for which hospitals were started. It is this post that was known as Surgeon General till the formation of Kerala State. He also started an engineering department, which
5225-451: The compositions of Maharaja Swathi Thirunal. Eminent Carnatic and Hindustani musicians participate in this unique musical event, which is conducted every year from 4 to 13 January at Kuthira Malika , Trivandrum and attracts music aficionados from across the globe. The award Swathi Sangeetha Puraskaram is instituted in the name of Maharajah Swathi Thirunal of Travancore to honour those musicians who have made valuable contributions to
5320-412: 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
5415-504: 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
5510-535: The deceased Rajah maintained an observatory at considerable expense, and that MR Caldecott was for a length of time, his highness's astronomer. The ephemeris emanating from the Travancore observatory was a valuable contribution to astronomical science ... The Rajah also supported an English school on a scale of liberality that perhaps has few precedents in other native states. He was a steady and staunch advocate of education, friend and patron of men of letters ... his loss will doubtless be greatly deplored by Travancoreans as
5605-478: The elder son. While in the womb itself, he was proclaimed King and thus was referred to as Garbha Sreemaan . He was born in Svati star, and this is the reason why he was named Swathi Thirunal. He reigned under the regency of his mother from 1813 to 1815 and then under the regency of his maternal aunt Gowri Parvati Bayi until 1829. When he was just four months old, his mother invited Colonel John Munro , representative of
5700-428: 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
5795-519: The field of music. It is also the highest honour for musicians by the Government of Kerala , India . In 1987, a Malayalam film titled Swathi Thirunal based on his life was released. It was directed by Lenin Rajendran . It stars Anant Nag in the title role, and Srividya , Ambika , Nedumudi Venu and Murali in other important roles. Sree Swathi Thirunal Maharaja , a 1967 documentary film about
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#17328685684585890-553: The figure on a country slate but what astonished me most was his telling us in English that Geometry was derived from the Sanskrit, which as Jaw metor (Jyamiti) to measure the earth and that many of our mathematical terms were also derived from the same source such as hexagon, heptagon, octagon... This promising boy is now, I conclude, sovereign of the finest country in India for he was to succeed to
5985-593: The first Government printing press, establishment of the first manuscripts library were amongst the many initiatives taken by Swathi Thirunal, as a King, to modernize Travancore. Swathi Thirunal was born into the Venad dynasty of the Matrilineal royal family of Travancore, which is now a part of Kerala , on 16 April 1813. He was the second child of Queen Gowri Lakshmi Bayi who ruled Travancore from 1810 to 1815, and Raja Raja Varma Koil Thampuran of Changanasseri Palace, and
6080-512: 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
6175-523: 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
6270-588: 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
6365-533: 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
6460-523: The king, directed by K. T. John, was produced by the Government of India 's Films Division . [1] : Articles and compilations by Dr Achuthsankar S Nair in Sruthi Magazine, June 2013 and Journal of Madras Music Academy, 2009 Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría ) 'land measurement'; from γῆ ( gê ) 'earth, land' and μέτρον ( métron ) 'a measure')
6555-546: 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
6650-405: The legal sector, starting Munsif , District and Appellate Courts and modernizing laws. He identified one Kandan Menon from Malabar and appointed him as Huzoor Diwan Peshkar to bring about legal reforms. Another of his achievements was to settle many land disputes by carrying out a resurvey of the land, in which also Menon helped him. He also conducted the first census of the state in 1836. As per
6745-411: 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
6840-429: 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,
6935-441: 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
7030-510: 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
7125-407: 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
7220-478: 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
7315-554: 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
7410-589: 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
7505-583: The then English scholar, Thanjavur Subba Rao as well. He continued to learn music by listening to accomplished musicians and practising himself. He encouraged both broad systems of Indian music , Hindustani and Carnatic music , though he was essentially a connoisseur of the Carnatic music tradition. He is credited with composing over 400 compositions in Carnatic and Hindustani music. Some of his favourite compositions were Padmanabha Pahi , Deva Deva , Devanke , Sarasijanabha and Sree Ramana Vibho . Swathi Thirunal
7600-409: 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
7695-830: Was administered as an independent government institution. It is now the oldest institution under the Kerala University. Started in 1837, some of the equipment is still to be seen at the Thiruvananthapuram observatory (now under the Department of Physics, University of Kerala ). In the early sixties, in relation to Indian Space Research Organisation, ISRO, the founder Dr. Vikram Sarabhai selected this astronomy observatory to study equatorial night skies. He assigned his doctoral students, notably Dr. A.P.J Kalam and Dr. K. Narayanan Nair, to collect data on cosmic rays and ionosphere. Trivandrum Public Library (now State Central Library ) and
7790-760: Was already an established astronomer from India and a member of the British and Canadian Astronomical Societies. The observatory benefited from the expertise of local English men, Colonel Fraser and Mr. Caldecott. A cotton mill expert John Caldecott, FRS was interested in astronomy but was self-taught, and later became one of its directors. As an industrial representative in Alapuzha, he used to make instruments for astronomical observations and initially mounted viewing instruments on top of mango tree in Residency of Kollam, Kochi and his Alappuzha homes. Raja Swathi Tirunal saw his collection and asked him to come to Thiruvananthapuram to start
7885-670: Was born. Both his aunt/foster mother, who was well-versed in music, and his father, a Sanskrit scholar, took special care about his education. Col. Munro also is said to have taken a keen interest in his education. He started learning Malayalam and Sanskrit at the age of six and English at the age of seven. The young Prince studied several languages, including Malayalam , Kannada , Tamil , Hindustani , Telugu , Marathi , Sanskrit , English and Persian . He impressed all his teachers, and even guests from abroad, with his keen understanding of not only languages but also other subjects like geometry. P. Sankunni Menon (A History of Travancore from
7980-549: Was cheating people by claiming to be God and as a result of the complaint, Swathi Thirunal ordered to arrest him but Vaikundar was not the cheater and later released by the government. In the then society, men and women belonging to the lower classes weren't allowed to cover their upper body as it was considered a privilege reserved for caste Hindus & people of other religions. Vaikundar exhorted dalits to fight against this and organized revolts to get their due rights which came to be known as Melmundu Samaram (Upper Cloth Revolt). As
8075-423: Was deeply interested in music right from childhood. Besides being an able ruler, he was a patron of music and was a musician himself. Researchers say that Swathi Thirunal affixed his compositions with the mudra Padmanabha , sarasijanaabha, etc. and its synonyms. His education in music started with the first lessons from Karamana Subrahmania Bhagavathar and Karamana Padmanabha Bhagavathar. Later, he studied music from
8170-477: Was educated by his prime minister— a rare tutor for a sovereign. The Rajah had established schools within his dominions—he had established a mathematical school under English superintendence; but he had done more—he had done what, he was sorry to say, had neither been done in England, Scotland, nor Ireland—be had established a school in every village of his dominions— and be gave education to every child, male and female –
8265-442: Was fluent in a number of languages including Malayalam , Sanskrit , Marathi , Telugu , Kannada , Hindustani , Bengali , Tamil , Oriya and English. This was a period when music and art were thriving in many parts of south India. The triumvirate of Carnatic music, Tyagaraja (1767–1847), Syama Sastri (1762–1827) and Muthuswami Dikshitar (1775–1835), lived and enriched music during this period. Swathi Thirunal's palace also
8360-503: Was home to many musicians and artistes of the period, including the famous Thanjavur Quartet brothers, Tyagaraja 's disciple Kannayya Bhagavathar, Ananthapadmanabha Goswami (a Maharashtrian singer known as Kokilakanthameru swami), Shadkala Govinda Marar , and many others. The literary works of Maharajah Swathi Thirunal include Bhakti Manjari', Syanandurapuravarnana Prabandham, Padmanabhasatakam, Muhanaprasa Antyaprasa Vyavastha, Ajamila, Kuchela Upakhyanas and Utsava Varnana Prabandha. As
8455-588: Was in that room – he meant Mr. Caldecott. In this observatory, observations were carried on with the same success as under British interests. The Rajah had also established a magnetical and meteorological observatory, having been led to do so by becoming acquainted with a report on Meteorology, published by the British Association. And the observations taken there were found to be as accurate as those taken in Edinburgh, Philadelphia, and other places. Swathi Thirunal
8550-488: Was placed under the command of one Lieutenant Horsley. The Karamana bridge was built at that time. Despite the progress achieved in varied fields under Swathi Thirunal's reign, the Kingdom of Travancore, like the rest of British India, was in the grip of extreme caste discrimination against Hindu lower classes ( dalits ). According to the followers of the movement called Ayya Vazhi (the path of Ayya Vaikundar) and historians,
8645-768: Was punished if the hand gets burnt. He is also credited with starting the first government press (the only press at that time was CMS Press in Kottayam ). A report on the English schools in Travancore appeared in The Gardner's Magazine of 1841, wrote about the administrative reforms brought in by Maharajah Swathi Thirunal: Rajah of Travancore, the great promoter of science in the East, was only twenty-eight years of age, and had not reigned more than ten years, yet, during that short period, he had caused himself to be distinguished by his accomplishments as well as by his' liberality. They would, no doubt, be interested in learning that this prince
8740-459: Was so much in common between western astronomy and Indian (eastern) astrological understanding of planets, stars and the known universe; Swathi Thirunal set the initiative to start an Astronomical Observatory. One of its directors would be his cousin, Raja Rama Varma Rohani Thirunal, who was the contemporary Raja of Mavelikara Palace, an important branch of the ruling Travancore Royal family related to Raja Swathi Thirunal Raja Rohani Thirunal (Rohini)
8835-520: Was the Regent for Swathi Thirunal in his boyhood) at the age of sixteen. He appointed his tutor, Sri Subba Rao, as the Prime Minister (Diwan). One of his first moves was to shift the government secretariat from Kollam (about seventy-five kilometers away) to Thiruvananthapuram . This enabled him to give personal attention to government affairs. He took steps to curb corruption in the government and told even
8930-596: 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
9025-431: Was the machinations of his aide Krishna Rao, who schemed with Cullen for his own personal gain. What ever the reason, the Resident's intrusion in the administration was unbearable for the young King. To compound his problems, the deaths of his elder sister, father, wife Narayani and all three children (Narayani's) made the Maharajah distraught. He increasingly sought silence and solitude, weakening his mind and body. Thus, at
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