1. The property that the second order differences of sines are proportional to the sines themselves was known in early Indian mathematics, dating back to Aryabhata I in the 5th century AD.
2. The paper describes various formulations of this proportionality found in important Indian astronomical works from the 5th to 15th centuries. It also outlines Nilakantha Somayaji's ingenious geometric proof of the property from his commentary on Aryabhatiya in the early 16th century.
3. The Indian mathematical method of computing sine tables using this difference property, which involves an implied differential process, was considered novel by Western scholars despite being used in India since ancient times
Bhaskara II, an Indian mathematician from the 12th century, derived the formula for the surface area of a sphere. He used a crude form of integration by dividing the surface into thin circular sections and approximating their areas. This led to the formula of circumference x diameter. Earlier, Aryabhata II had provided the same formula, though Bhaskara criticized another mathematician, Lalla, for providing an incorrect formula. Bhaskara's derivation showed originality compared to Archimedes' method, though it lacked the same level of ingenuity.
This document discusses ancient Indian values of Pi that were used in mathematics. It provides 5 ancient approximations of Pi that were used in India and other parts of the world:
1. The simplest approximation of Pi = 3, which was used by the ancient Babylonians, in the Bible, and in ancient Chinese works. Similar approximations are found in ancient Indian texts.
2. The "Jaina value" of Pi = 22/7, which was frequently used in Jain texts and first appeared in India around the 1st century AD.
3. The Archimedean value of Pi = 22/7, which was recognized as a satisfactory approximation after Archimedes showed Pi is between 3 1/
This document discusses fractional parts of Aryabhata's sine differences that are found in Govindasvami's commentary on Mahabhashya. Specifically:
1. Govindasvami's commentary provides the sexagesimal fractional parts (seconds and thirds) of Aryabhata's 24 tabular sine differences, allowing for a more accurate sine table with values given to the second order of sexagesimal fractions.
2. These fractional parts found in Govindasvami's commentary are subtracted from or added to Aryabhata's sine differences to improve the accuracy of the table.
3. In addition to improving Aryabhata's sine table, the
1) The ancient Indian mathematician Bhaskara II first described the addition and subtraction theorems for sines in his work Jyotpatti from 1150 AD. These theorems state that the sine of the sum of two arcs is equal to the sum of the sines of individual arcs multiplied by their cosines, and the sine of the difference of two arcs is equal to the difference of the sines.
2) Bhaskara II provided two formulas based on these theorems to tabulate sines at intervals of 225 minutes and 60 minutes. These formulas allow constructing tables of 24 sines and 90 sines respectively using recurrence relations.
3) The article discusses Bhaskara II's contribution to trigon
The document summarizes rules for calculating the perimeter and area of an ellipse from an ancient Indian mathematics text from the 9th century AD. It provides the ancient text's approximate formulas for perimeter and area, and explains how they were likely empirically derived by comparing an ellipse to simpler shapes like a semicircle or circular segment. It also gives the correct elliptic integral formulas, and shows how the ancient approximations compare to the precise solutions for a numerical example.
The document presents a new interpretation of a rule in the Ganita-siira-sa4graha (GSS), an ancient Indian mathematics text, for calculating the surface area of a spherical segment.
Previous interpretations by Rangacharya and Jain assumed the rule treated the spherical segment as a flat circular base. The document suggests an alternate interpretation where the Sanskrit term "viskambha" refers to the curvilinear breadth of the segment, not its diameter. This new interpretation matches the true mathematical formula and has less error than the previous views.
1. Madhava of Sangamagrama formulated a rule equivalent to the Gregory series for calculating the inverse tangent function.
2. The rule expresses the inverse tangent as a sum of terms involving sines and cosines, equivalent to the Gregory series.
3. Ancient Indian mathematicians provided proofs of the rule, equivalent to modern proofs through term-by-term integration, showing they understood the concept of infinite series well before Western mathematicians formulated them explicitly.
Bhaskara II, an Indian mathematician from the 12th century, derived the formula for the surface area of a sphere. He used a crude form of integration by dividing the surface into thin circular sections and approximating their areas. This led to the formula of circumference x diameter. Earlier, Aryabhata II had provided the same formula, though Bhaskara criticized another mathematician, Lalla, for providing an incorrect formula. Bhaskara's derivation showed originality compared to Archimedes' method, though it lacked the same level of ingenuity.
This document discusses ancient Indian values of Pi that were used in mathematics. It provides 5 ancient approximations of Pi that were used in India and other parts of the world:
1. The simplest approximation of Pi = 3, which was used by the ancient Babylonians, in the Bible, and in ancient Chinese works. Similar approximations are found in ancient Indian texts.
2. The "Jaina value" of Pi = 22/7, which was frequently used in Jain texts and first appeared in India around the 1st century AD.
3. The Archimedean value of Pi = 22/7, which was recognized as a satisfactory approximation after Archimedes showed Pi is between 3 1/
This document discusses fractional parts of Aryabhata's sine differences that are found in Govindasvami's commentary on Mahabhashya. Specifically:
1. Govindasvami's commentary provides the sexagesimal fractional parts (seconds and thirds) of Aryabhata's 24 tabular sine differences, allowing for a more accurate sine table with values given to the second order of sexagesimal fractions.
2. These fractional parts found in Govindasvami's commentary are subtracted from or added to Aryabhata's sine differences to improve the accuracy of the table.
3. In addition to improving Aryabhata's sine table, the
1) The ancient Indian mathematician Bhaskara II first described the addition and subtraction theorems for sines in his work Jyotpatti from 1150 AD. These theorems state that the sine of the sum of two arcs is equal to the sum of the sines of individual arcs multiplied by their cosines, and the sine of the difference of two arcs is equal to the difference of the sines.
2) Bhaskara II provided two formulas based on these theorems to tabulate sines at intervals of 225 minutes and 60 minutes. These formulas allow constructing tables of 24 sines and 90 sines respectively using recurrence relations.
3) The article discusses Bhaskara II's contribution to trigon
The document summarizes rules for calculating the perimeter and area of an ellipse from an ancient Indian mathematics text from the 9th century AD. It provides the ancient text's approximate formulas for perimeter and area, and explains how they were likely empirically derived by comparing an ellipse to simpler shapes like a semicircle or circular segment. It also gives the correct elliptic integral formulas, and shows how the ancient approximations compare to the precise solutions for a numerical example.
The document presents a new interpretation of a rule in the Ganita-siira-sa4graha (GSS), an ancient Indian mathematics text, for calculating the surface area of a spherical segment.
Previous interpretations by Rangacharya and Jain assumed the rule treated the spherical segment as a flat circular base. The document suggests an alternate interpretation where the Sanskrit term "viskambha" refers to the curvilinear breadth of the segment, not its diameter. This new interpretation matches the true mathematical formula and has less error than the previous views.
1. Madhava of Sangamagrama formulated a rule equivalent to the Gregory series for calculating the inverse tangent function.
2. The rule expresses the inverse tangent as a sum of terms involving sines and cosines, equivalent to the Gregory series.
3. Ancient Indian mathematicians provided proofs of the rule, equivalent to modern proofs through term-by-term integration, showing they understood the concept of infinite series well before Western mathematicians formulated them explicitly.
1. According to Jaina cosmography, the Jambu Island is circular with a diameter of 100,000 yojanas.
2. Ancient Jaina texts provided approximations for the circumference of the Jambu Island using simple formulas like C=3D. More accurate values were also calculated using the formula C=√(10D2) which is better than C=3D.
3. The article examines various circumference values found in ancient Jaina texts like the Tiloya-Pannatti and expresses them using the detailed unit system from the text to compare to the precise modern calculation of the circumference.
1) The ancient Indian text Baudhayana's Sulba Sutra from around 800-400 BC contains a rule for approximating the square root of two. The rule increases the side of a square by one-third and one-fourth parts to get an approximation of 1.41421.
2) This approximation method of linear interpolation was popular in ancient India. The rule can be explained through a two-term interpolation formula.
3) Repeated use of this interpolation process yields the four-term approximation found in the Sulba Sutra, equivalent to methods used by later mathematicians like Neugebauer. While an improvement, the Indian approximation was still less accurate than the sexagesimal
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help boost feelings of calmness, happiness and focus.
1. Aryabhata, an Indian astronomer and mathematician from the 5th century AD, approximated pi (π) to 3.1416 in his famous work the Aryabhatiya. This approximation correct to four decimal places was one of the most accurate approximations of pi used anywhere in the ancient world.
2. Aryabhata expressed pi as a fraction 62832/20000, which can be expressed in continued fractions as 3 + (4/16). This value of pi was later used and referenced by many Indian and foreign mathematicians and astronomers over subsequent centuries.
3. Scholars debate whether Aryabhata's value of pi was influenced by the Greeks
Narayana Pandita, an important ancient Indian mathematician who lived in the 14th century, developed a method for approximating quadratic surds (square roots of non-perfect squares). His method involved finding integer solutions to the indeterminate equation x2 - Ny2 = 1 and taking the ratio of the solutions as the approximation. The article describes Narayana's method using the example of approximating √10 and relates his method to other ancient approaches like the binomial approximation and Newton's method. It highlights the significance of Narayana's works in the development of ancient Indian mathematics.
1) Ancient Indian mathematicians developed rational approximations to trigonometric functions like sine, cosine, and versine that were found in Sanskrit works from the 7th to 17th centuries AD.
2) These approximations took the form of algebraic formulas relating the functions to angles in degrees. For example, one approximation for sine was sin(A) ≈ 16A(180-A)/(32400-A^2).
3) These formulas, which were popular in India, can be found in many original Indian mathematical works and show that such approximations were well-established by the 7th century.
The document discusses addition and subtraction theorems for the sine and cosine functions in medieval Indian mathematics. It provides statements of the theorems as found in several important Indian works from the 12th to 17th centuries. These theorems are equivalent to the modern mathematical formulas for trigonometric addition and subtraction. The document also outlines various proofs and derivations of the theorems found in Indian works, which indicate how Indians understood the rationales behind the theorems.
1) Brahmagupta, an ancient Indian mathematician from the 7th century AD, developed formulas for calculating the area and diagonals of a cyclic quadrilateral using only the lengths of its four sides.
2) His formula for accurate area involved taking the square root of the product of terms involving the differences between half sums of opposite sides.
3) His expressions for the diagonals involved dividing and multiplying terms involving products of the sides, and took the square root of the results. These formulas are considered some of the most remarkable in Hindu geometry.
Bhaskara I, an Indian mathematician from around 600 AD, developed an elegant algebraic formula for approximating trigonometric sine values. This formula, described in his text Mahabhaskariya, expresses sine as a function of the angular arc in degrees. Several subsequent Indian texts, including works by Brahmagupta, Varahamihira, and Bhaskara II, reference or derive equivalent forms of Bhaskara I's approximation formula. The document discusses the accuracy of the formula and compares it to actual sine values, providing historical context on the development and transmission of trigonometric concepts in early Indian and Western mathematics.
This document summarizes a rule from the ancient Indian mathematics text Lilavati for computing the sides of regular polygons inscribed in a circle. The rule provides coefficients that when multiplied by the circle's diameter and divided by 12,000 yield the side lengths of triangles, squares, pentagons, etc. up to nonagons. While the rationales for some coefficients are clear, others are less accurate, possibly due to limitations in available sine tables or knowledge of exact solutions. The document discusses methods that may have been used to derive the coefficients and compares the original values to more accurate modern calculations.
1) The document discusses an ancient Indian mathematics problem involving using gnomons (vertical rods) to measure the height of a lamp post and the distance between shadow tips.
2) It presents the original rule from the Aryabhatiya, and provides examples and explanations of how to use the rule to calculate heights and distances.
3) The rule involves using ratios between shadow lengths, distance between shadow tips, and gnomon length to determine the "upright" distance and lamp post height.
Brahmagupta was an ancient Indian mathematician from the 7th century AD who made important contributions to mathematics. This document discusses one of Brahmagupta's formulas for calculating the volume of frustum-like solids (solids with parallel ends of different shapes and sizes). The formula provides a method to calculate both the practical volume and gross volume, and uses the difference between the two to find the accurate volume. The formula is shown to apply to specific cases like truncated wedges and cones. The formula demonstrates Brahmagupta's sophisticated mathematical abilities and had influence in both India and other parts of the ancient world.
1) Neelakantha Somaydji was an important mathematician from medieval India who lived from 1443-1543 AD and wrote several astronomical works.
2) In one of his works called Golasara, Neelakantha provides a formula for computing the length of a small circular arc given the Indian Sine and Versed Sine.
3) The formula, which is equivalent to the modern formula, enables the approximate computation of an angle when its sine or cosine is given.
This document summarizes the key points from a journal article about Ramanujan's formula for an infinite series. It discusses 12 classes of related infinite series and evaluates them in closed form for certain positive integer values. Several theorems are proved relating the different series and evaluating specific cases. The document aims to provide corrected versions and new proofs of some of Ramanujan's formulas for infinite series.
This document contains questions from an Advanced Mathematics exam for a fourth semester Bachelor's degree. It includes questions on topics such as vectors, lines and planes, motion, vector calculus, and Laplace transforms. Students were instructed to answer 15 full questions choosing from the total of 18 questions provided.
1982 a simple molecular statistical treatment for cholestericspmloscholte
1. The document presents a statistical mechanical treatment of a simplified molecular model for describing the cholesteric phase.
2. The model combines the Maier-Saupe interaction, which describes the nematic phase, with a twist interaction that induces helical twisting of the nematic director.
3. The model is solved using a mean field approximation, allowing analytical expressions to be obtained for relevant quantities like order parameters and free energy as a function of temperature.
Some aspects of the oldest nearby moving cluster (Ruprecht 147)Premier Publishers
Based on the membership data retrieved from the Two Micron All Sky Survey (2MASS), we have computed some parameters of the moving open cluster Ruprecht 147, like, vertex, velocity, distance, distance modulus, and center of the cluster. All of these aspects were carried out using an algorithm due to Sharaf et al. (2000), into which error estimates of these parameters will be established in closed analytical forms (e.g. standard and probable errors of the vertex coordinates, angular distance, velocity of the cluster, parallaxes of member stars, and distance of the cluster).
Finally, we compared our results with other published values, which is in good agreement.
This document contains questions from a Material Science and Metallurgy exam. It covers various topics:
- Crystal structures of BCC, FCC and HCP lattices and their properties. Diffusion of iron atoms in BCC lattice.
- Mechanical properties in the plastic region from stress-strain diagrams. True and conventional strain expressions. Twinning mechanism of plastic deformation.
- Fracture mechanisms based on Griffith's theory of brittle fracture. Factors affecting creep. Fatigue testing and S-N curves for materials.
- Solidification process and expression for critical nucleus radius. Cast metal structures. Solid solutions and Hume-Rothery rules. Phase diagrams and Gibbs phase rule.
Stability criterion of periodic oscillations in a (13)Alexander Decker
This document discusses domination problems on isosceles triangular chessboards using different chess pieces. It examines placing a minimum number of pieces such that all unoccupied positions are attacked (the domination number). For a single piece type, it determines the domination number and possible solutions for rooks, bishops, and kings on isosceles triangular boards. It also considers domination numbers when using two piece types together, such as kings and rooks or kings and bishops. Key results include formulas for the domination number and total solutions in terms of the board size for each piece type.
1. According to Jaina cosmography, the Jambu Island is circular with a diameter of 100,000 yojanas.
2. Ancient Jaina texts provided approximations for the circumference of the Jambu Island using simple formulas like C=3D. More accurate values were also calculated using the formula C=√(10D2) which is better than C=3D.
3. The article examines various circumference values found in ancient Jaina texts like the Tiloya-Pannatti and expresses them using the detailed unit system from the text to compare to the precise modern calculation of the circumference.
1) The ancient Indian text Baudhayana's Sulba Sutra from around 800-400 BC contains a rule for approximating the square root of two. The rule increases the side of a square by one-third and one-fourth parts to get an approximation of 1.41421.
2) This approximation method of linear interpolation was popular in ancient India. The rule can be explained through a two-term interpolation formula.
3) Repeated use of this interpolation process yields the four-term approximation found in the Sulba Sutra, equivalent to methods used by later mathematicians like Neugebauer. While an improvement, the Indian approximation was still less accurate than the sexagesimal
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help boost feelings of calmness, happiness and focus.
1. Aryabhata, an Indian astronomer and mathematician from the 5th century AD, approximated pi (π) to 3.1416 in his famous work the Aryabhatiya. This approximation correct to four decimal places was one of the most accurate approximations of pi used anywhere in the ancient world.
2. Aryabhata expressed pi as a fraction 62832/20000, which can be expressed in continued fractions as 3 + (4/16). This value of pi was later used and referenced by many Indian and foreign mathematicians and astronomers over subsequent centuries.
3. Scholars debate whether Aryabhata's value of pi was influenced by the Greeks
Narayana Pandita, an important ancient Indian mathematician who lived in the 14th century, developed a method for approximating quadratic surds (square roots of non-perfect squares). His method involved finding integer solutions to the indeterminate equation x2 - Ny2 = 1 and taking the ratio of the solutions as the approximation. The article describes Narayana's method using the example of approximating √10 and relates his method to other ancient approaches like the binomial approximation and Newton's method. It highlights the significance of Narayana's works in the development of ancient Indian mathematics.
1) Ancient Indian mathematicians developed rational approximations to trigonometric functions like sine, cosine, and versine that were found in Sanskrit works from the 7th to 17th centuries AD.
2) These approximations took the form of algebraic formulas relating the functions to angles in degrees. For example, one approximation for sine was sin(A) ≈ 16A(180-A)/(32400-A^2).
3) These formulas, which were popular in India, can be found in many original Indian mathematical works and show that such approximations were well-established by the 7th century.
The document discusses addition and subtraction theorems for the sine and cosine functions in medieval Indian mathematics. It provides statements of the theorems as found in several important Indian works from the 12th to 17th centuries. These theorems are equivalent to the modern mathematical formulas for trigonometric addition and subtraction. The document also outlines various proofs and derivations of the theorems found in Indian works, which indicate how Indians understood the rationales behind the theorems.
1) Brahmagupta, an ancient Indian mathematician from the 7th century AD, developed formulas for calculating the area and diagonals of a cyclic quadrilateral using only the lengths of its four sides.
2) His formula for accurate area involved taking the square root of the product of terms involving the differences between half sums of opposite sides.
3) His expressions for the diagonals involved dividing and multiplying terms involving products of the sides, and took the square root of the results. These formulas are considered some of the most remarkable in Hindu geometry.
Bhaskara I, an Indian mathematician from around 600 AD, developed an elegant algebraic formula for approximating trigonometric sine values. This formula, described in his text Mahabhaskariya, expresses sine as a function of the angular arc in degrees. Several subsequent Indian texts, including works by Brahmagupta, Varahamihira, and Bhaskara II, reference or derive equivalent forms of Bhaskara I's approximation formula. The document discusses the accuracy of the formula and compares it to actual sine values, providing historical context on the development and transmission of trigonometric concepts in early Indian and Western mathematics.
This document summarizes a rule from the ancient Indian mathematics text Lilavati for computing the sides of regular polygons inscribed in a circle. The rule provides coefficients that when multiplied by the circle's diameter and divided by 12,000 yield the side lengths of triangles, squares, pentagons, etc. up to nonagons. While the rationales for some coefficients are clear, others are less accurate, possibly due to limitations in available sine tables or knowledge of exact solutions. The document discusses methods that may have been used to derive the coefficients and compares the original values to more accurate modern calculations.
1) The document discusses an ancient Indian mathematics problem involving using gnomons (vertical rods) to measure the height of a lamp post and the distance between shadow tips.
2) It presents the original rule from the Aryabhatiya, and provides examples and explanations of how to use the rule to calculate heights and distances.
3) The rule involves using ratios between shadow lengths, distance between shadow tips, and gnomon length to determine the "upright" distance and lamp post height.
Brahmagupta was an ancient Indian mathematician from the 7th century AD who made important contributions to mathematics. This document discusses one of Brahmagupta's formulas for calculating the volume of frustum-like solids (solids with parallel ends of different shapes and sizes). The formula provides a method to calculate both the practical volume and gross volume, and uses the difference between the two to find the accurate volume. The formula is shown to apply to specific cases like truncated wedges and cones. The formula demonstrates Brahmagupta's sophisticated mathematical abilities and had influence in both India and other parts of the ancient world.
1) Neelakantha Somaydji was an important mathematician from medieval India who lived from 1443-1543 AD and wrote several astronomical works.
2) In one of his works called Golasara, Neelakantha provides a formula for computing the length of a small circular arc given the Indian Sine and Versed Sine.
3) The formula, which is equivalent to the modern formula, enables the approximate computation of an angle when its sine or cosine is given.
This document summarizes the key points from a journal article about Ramanujan's formula for an infinite series. It discusses 12 classes of related infinite series and evaluates them in closed form for certain positive integer values. Several theorems are proved relating the different series and evaluating specific cases. The document aims to provide corrected versions and new proofs of some of Ramanujan's formulas for infinite series.
This document contains questions from an Advanced Mathematics exam for a fourth semester Bachelor's degree. It includes questions on topics such as vectors, lines and planes, motion, vector calculus, and Laplace transforms. Students were instructed to answer 15 full questions choosing from the total of 18 questions provided.
1982 a simple molecular statistical treatment for cholestericspmloscholte
1. The document presents a statistical mechanical treatment of a simplified molecular model for describing the cholesteric phase.
2. The model combines the Maier-Saupe interaction, which describes the nematic phase, with a twist interaction that induces helical twisting of the nematic director.
3. The model is solved using a mean field approximation, allowing analytical expressions to be obtained for relevant quantities like order parameters and free energy as a function of temperature.
Some aspects of the oldest nearby moving cluster (Ruprecht 147)Premier Publishers
Based on the membership data retrieved from the Two Micron All Sky Survey (2MASS), we have computed some parameters of the moving open cluster Ruprecht 147, like, vertex, velocity, distance, distance modulus, and center of the cluster. All of these aspects were carried out using an algorithm due to Sharaf et al. (2000), into which error estimates of these parameters will be established in closed analytical forms (e.g. standard and probable errors of the vertex coordinates, angular distance, velocity of the cluster, parallaxes of member stars, and distance of the cluster).
Finally, we compared our results with other published values, which is in good agreement.
This document contains questions from a Material Science and Metallurgy exam. It covers various topics:
- Crystal structures of BCC, FCC and HCP lattices and their properties. Diffusion of iron atoms in BCC lattice.
- Mechanical properties in the plastic region from stress-strain diagrams. True and conventional strain expressions. Twinning mechanism of plastic deformation.
- Fracture mechanisms based on Griffith's theory of brittle fracture. Factors affecting creep. Fatigue testing and S-N curves for materials.
- Solidification process and expression for critical nucleus radius. Cast metal structures. Solid solutions and Hume-Rothery rules. Phase diagrams and Gibbs phase rule.
Stability criterion of periodic oscillations in a (13)Alexander Decker
This document discusses domination problems on isosceles triangular chessboards using different chess pieces. It examines placing a minimum number of pieces such that all unoccupied positions are attacked (the domination number). For a single piece type, it determines the domination number and possible solutions for rooks, bishops, and kings on isosceles triangular boards. It also considers domination numbers when using two piece types together, such as kings and rooks or kings and bishops. Key results include formulas for the domination number and total solutions in terms of the board size for each piece type.
This document appears to contain questions from an engineering mathematics exam. It includes questions on several topics:
1. Differential equations, evaluating integrals using Cauchy's integral formula, Bessel functions, and Legendre polynomials.
2. Vector calculus topics like divergence and curl of vector fields, and finding equations of planes and lines.
3. Probability and statistics problems involving binomial, normal and Poisson distributions.
4. Graph theory questions about planar graphs, chromatic polynomials, and finding minimum spanning trees.
5. Combinatorics problems involving counting arrangements and distributions with restrictions.
The document discusses scattering cross sections and scattering amplitudes. It defines differential and total scattering cross sections, relating them to the scattering amplitude. The scattering amplitude f(θ,φ) is derived using Born approximation by solving the Schrodinger equation for scattering. The scattering amplitude is expressed as an integral involving the scattering potential V(r) and is related to the experimentally observable differential scattering cross section.
This summary provides the key details from the document in 3 sentences:
The document investigates the structure of unital 3-fields, which are fields where addition requires 3 summands rather than the usual 2. It is shown that unital 3-fields are isomorphic to the set of invertible elements in a local ring R with Z2Z as the residual field. Pairs of elements in the 3-field are used to define binary operations that allow reducing the arity and connecting the 3-field to binary algebra. The structure of finite 3-fields is examined, proving properties like the number of elements being a power of 2.
The document discusses the four quantum numbers that describe electrons in atoms: principal quantum number (n), azimuthal/angular momentum quantum number (l), magnetic quantum number (m), and spin quantum number (s). N determines the shell and average distance from the nucleus. L determines the subshell and shape. M gives orbital orientation. S indicates electron spin direction as clockwise or counterclockwise. Each number has specific allowed values and properties that provide the electron's precise location.
The document summarizes a two-day international seminar on the history of mathematics held in New Delhi in 2012. Over 150 mathematicians from 16 countries participated in the event, which celebrated the 125th birth anniversary of Srinivasa Ramanujan and covered various aspects of mathematics history, especially ancient Indian history. There were 11 sessions over the two days featuring talks and papers from scholars. The seminar concluded with positive feedback and was considered a great success in arranging such a large international event.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
Best practices for project execution and deliveryCLIVE MINCHIN
A select set of project management best practices to keep your project on-track, on-cost and aligned to scope. Many firms have don't have the necessary skills, diligence, methods and oversight of their projects; this leads to slippage, higher costs and longer timeframes. Often firms have a history of projects that simply failed to move the needle. These best practices will help your firm avoid these pitfalls but they require fortitude to apply.
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At Techbox Square, in Singapore, we're not just creative web designers and developers, we're the driving force behind your brand identity. Contact us today.
Digital Marketing with a Focus on Sustainabilitysssourabhsharma
Digital Marketing best practices including influencer marketing, content creators, and omnichannel marketing for Sustainable Brands at the Sustainable Cosmetics Summit 2024 in New York
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https://www.productmanagementtoday.com/frs/26903918/understanding-user-needs-and-satisfying-them
We know we want to create products which our customers find to be valuable. Whether we label it as customer-centric or product-led depends on how long we've been doing product management. There are three challenges we face when doing this. The obvious challenge is figuring out what our users need; the non-obvious challenges are in creating a shared understanding of those needs and in sensing if what we're doing is meeting those needs.
In this webinar, we won't focus on the research methods for discovering user-needs. We will focus on synthesis of the needs we discover, communication and alignment tools, and how we operationalize addressing those needs.
Industry expert Scott Sehlhorst will:
• Introduce a taxonomy for user goals with real world examples
• Present the Onion Diagram, a tool for contextualizing task-level goals
• Illustrate how customer journey maps capture activity-level and task-level goals
• Demonstrate the best approach to selection and prioritization of user-goals to address
• Highlight the crucial benchmarks, observable changes, in ensuring fulfillment of customer needs
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Discover timeless style with the 2022 Vintage Roman Numerals Men's Ring. Crafted from premium stainless steel, this 6mm wide ring embodies elegance and durability. Perfect as a gift, it seamlessly blends classic Roman numeral detailing with modern sophistication, making it an ideal accessory for any occasion.
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Storytelling is an incredibly valuable tool to share data and information. To get the most impact from stories there are a number of key ingredients. These are based on science and human nature. Using these elements in a story you can deliver information impactfully, ensure action and drive change.
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How to Implement a Real Estate CRM SoftwareSalesTown
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Starting a business is like embarking on an unpredictable adventure. It’s a journey filled with highs and lows, victories and defeats. But what if I told you that those setbacks and failures could be the very stepping stones that lead you to fortune? Let’s explore how resilience, adaptability, and strategic thinking can transform adversity into opportunity.
1. 1 5 hs 7, tl't 2, tl^&l
EARLY INDIANS ON SECOND OR,DER,
SINE DIFFERENCES
R. C. Guptl
Assistant Professor of llathemat'ics, Birla Institute of Technology,
P.O. Mesra, Ranchi, (Bihar)
(Receiuecl, JuIy 1972)
31
The well known property that the socond ordor diffsrences of sines aro pro-
portional to tho sines themselves was knorvn evon to iryabhata I (born A,
D. 476) whoso Aryabh.tliya is tho earliest extant historicsl work (of the dated
type) containing a sine tablo, The paper describes the various forms of tho
propottionality factor involvod in the mathemabical formula expressing
tho above property,
Relovant references and rules aro givon from the
Indian astronomical works guch as Argabhatiya, Surya-Siddhdnla, Golo,sdra
and Tantra-Saqgraha (A.D. f 500).
The commoncry of Nilakantha Somaydji(born A. D. 1443) on the Aryabhatrya
discusses the property in details and contains an ingenious geometrical
proof of it. The paper gives a brief description of this proof which is merely
bosed on tho similarity of triangles.
Tho Indian mabhomatical method based on the implied differontial process
is founcl, in the words of Delambrc, "neither amongst the Greeks nor
omongst tho Arabs."
1. IxrnonucrroN
Lei (n being a positiveinteger)
So : 'B si:nnh ,.. (t)
Dr:Sr
Dn+ t : Sr * r - Sr . (9
It is easily seen tha,t
D ,-D n * , - F .S , ... (3)
where the propoftionality factor -F (indepenclentof ra) is given by
I :2 (I-c o g h ). ... (4)
Relation (3) represents the fact that in a, set of equidistant tabulated fndian.Sines
defined by (1), the differences of the first Sine-differences(Bn+l-/S,), that is, the
second Sine-differences (Dn-Dn*r) are proportional to the sines B, themselves.
This fact seems to be recognised in India almost since the very begiiLning of fndian
Trigonometry. In Section 2 below we sha,ll describe some of the forms of the
rule (3) alongwith va,riousforms of the factor -F as iound in important Indian lvorks.
In Section3 we shall outline an'Indian proof of the rule'as found in Nilaka4lha
Somayd,ji's AryabhaQiyo-Bhd1ya (: NAB) which was written in the early part of
the sixteenth century of our era,.
YOL .7 , N o. 2.
2. 8i R. C. GUPTA
2. Fonus or rnp Rur,n
It is oasy to see thir,t /
trt : (Dr_D2)lDr ...(5)
tr4ren the norru (r'aclius or ,Sanustotus) R is equal to 3-138minntes and the uniform
tabular interval /r, is equal bo 225 minutes (as is the case with the usual fndian
Sine Tables), we have
Dr : 3438 sin 225' : 224'86 nearl.v'
Dr : 3438 sin 450'-3'138 sin 225' : 213'89nearly,
Dr-D-r: 0'97 : I aPproximatel.v'
Using this value antl (5), rve can put (3) as
D n t.t : D n-S nl D t " ' (6)
A rule rvhich is equivalent to (6) is founcll in tlte Aryabha{iya II,12 of Aryabhala
I (born 476 A.D.) which is the earliest extant historical rvork of the datecl type
containing a Sine table. The rule founde in the Surya-Sidclhd,nta, 15-16 is also
Il,
equivalent to (6) according to thc interpretations of the commcntators Mallikdrjuna
(1I78 A.D.) anci Rimakrsna (I.172 A.D.). The -l/rl.B also acceptsthat t'he Surya-
Sitld,hdntarule is s&me as above and further gives au exact form of the rule (3)
lvhich can be expressedin our notat'ion as follorvs3
D n+t : D n- S o'(DL- Dr)I D t
or,
D n + r: D D r+ ... + D ,).( D r- D 2)D L.
|
" -(D r+
Te Gola.sdro III, l3-1{ gives a rule equir-alent to{
S u -r : S ,-[(2 /n ).{ " t? s in 90" --B si n (90" -D )}' S n* D n+ r]
which implies (3) rvit'h
F :2 @-n cosh)/" rR .
The NAB (part I, p. 53) quotes the Golasdra-rule and further aclds tirat rve
equivalent'ly have
F :2(R versh)/-R .
The actual value of .F (independent of r?) is given by
F : (2 sin 112'5')2: l/233'53 very nearly.
The Tantrasar.ngraha(:78) II,4 givessthe value of the reciprocal of -F as 233.5
and the commenbator therebf even gives it as
233+32160
which is almost equal to the true value.
A rule equivalent to (3) occurs in the ?S II, S-9 (p. 18), v'hich was m'itten in
A.D. 1500, as follorvs :
oFi*qr?iqr<(' taai gq) qt({d ((: r
srr?rsqr4rR?nfr (q'r( (cv€-fcFil(g',rkd: IIq ||
ilr*qr'g gsr{TCrEqT'
fafrcrkfr ficr( r
s{+{c<Eq-Esrr}(t: frrsgqt{il: | |( | |
3. EARLY INDIANS ON SECOND ORDER O'""''ENCES 83
"*U
'Trvice the difference between the last and the last-but-one (Sines)is the multi.
plier; the semi-diameter is the divisor. The first Sine then (that is, when operated
by the multiplier and divisor defined above) becomes the difference of the initial
Sine-differences. lVith those very multiplier ancl divisor (operated upon) the
tabular Sines starting from the second, (rve get) the successivedifferences of
Sine-differencesrespectively.' That is,
2[.Rsin 90"-.R sin (90'-h)] : Multiplier, .L1;
Semidiametet ot radius .B : Divisor D.
Then
(tll lD)S, : Dt-Dz
(MlD)Sn: Dn-Dn+t, n:2,3, ,,.,
So that we have
D,-Dn+r: 2(l-cos D)S,
whichr is equivalent to (3).
Finally, we also ha,Ye
I : (c rd h )2 1 B 2 ... (A )
where crd b denotes the full chord of the arc h in a circle of radius ,8. lviih (A)
as the yalue of the proportionality factor, the N;lB (part I, p. 52) gives the verbal
statement of-the rule (3) as follorvs (NAB rvas composed after ?B)
ilqgfrq.rilgqrqr: gcwsq-rqli girqK!,
ffif g1lrq.K:I s'af (Eq-siqr+Tqq
I
'X'or the Sine at any arc-junction (that is, at any point rvhere two adjacent
elemental arcs meet,) the square of tho full chord is the multiplier; the square of
the radius is the divisor. The result (of operating the Sine by multiplier and
divisor) is the difference of the (trvo adjacent) Sine-differences.'
That is,
D ,- D n * , :' S' ' (c rd h )2R 2 .
| ...(7)
From this, the .lf.llB rightly concludes that
'Uilsrrgtrlf$iq eqrcqq-dni1l€3 1'
'The (numerical) increases of the Sine.differences is proportional to the very
Sines.'
3. Pnoon oF TEE RrrLE
An Indian proof of the rule (7) as found in the NAB (pafi I, pp. 48-52) may
be briefly outlined in the modern language as follows :
Make tho reference circle on a lovel gtound and draw the reference lines XOX'
and YOY' (seethe accompanfng figure where only a quadrant is shown). Mark
the parts of the arc on the circumference (by points, such as L, LItNt which are
at tho &roual interval l,).
4. 84 R. C. GUFIA
!.
I
t
r
t.'
l
I
o X
Take a rcd OQ equal in length to the radius -& and fix firmly and crossly (ancl
symmetrically) another rod jllrY whose length is equal to tho full chorcl of the
(elenrental) arc h at the point P rvhich is at a distance equal to the Versed Sine of
half the elemental arc D from the end Q of the first rod.
The sides of the similar triangles NKII and OAQ are proportional. There-
fore. bv the Rule of Three we have
NK : OA.LINIOQ
x IK :
QA .MN IOQ
fn other worcls rve have*
Lemma, I : The difference of Sines, corresponding to the end-points of an5r
elemental arc, is proportional to the Cosine at the middle of the arc;
Lemma II '. The difference of Cosines, corresponding to the enrl-points of
any element arc, is proportional to the Sine at the middle of the arc;
the proporbionality factor in both casesbeing
: (chord of the arc)/Radius : (crd ft.)/.B
* The Sanslsit text (q-+qTT€qRGTr . qr-e-slizfil as quoted in the lLlB,
Cs:),
statos the Lemmas as two Rules of Three. ,Seb Gupta, R.C., Some fmportant IndianMathematical
llethods as Conceived in Sanskrit Lenguage, peper presonted at the International Sanskrit
Conferenco, New Delhi, llarch 1972, p. 3. For a nice stetemont of tho Lemmas, oee Gupta,
R,. C., Second Order Interpolatioo in Indisn lVletherqatics etc., .f, J.E.5., Vol. 4 (196g),
p. 95, verses 7-8.
5. EARLY INDIAS ON SECOD ORDEIi, SI)iE DIFFEREiiCES do
Thus, in our s1'rnbolsrve ]tave (rvhen arc 'lIX: nh)
D ,+ r: @ rc lh ).OA l R
and, similarly
D,, : (crd h).OBiR.
Therefore,
D, - Dn * , - (c rc lh .(OB-OA )l n . ...(8)
Norv tlre seconilhaf (T,V) of the first. (los'er) arc LTM ancl the first haff QIQ)
of the second (upper) arc illQli together fonn the arc T,IIQ *'hose longth is equal
to that of an elemental arc 1.. Thus rve can place the above frame of tu'o rocls
such that the raclial rod coincicles rvith O-il1 ancl the cross radial rod (therefore)
coinciclesrvit'h the full chorcl of the arc ?Q. ancl consiclerthe proportionality of sides
as before.
frr ot'her rvorcls rve use Lentnta /1 for thc arc 7Q. This will mean that the
difference of the Cosines, OB ancl O-{, corresl>ondingto the encl-points T and.Q,
rvill be proportional to the Sine, ,]1C, at thr. micldle point M of the arc TQ. That is,
rve have
OB-OA : (crd h).:llClR
: (crcli,).(-Esin nh)lR.
Hence by (8)
D n -D ,,., : (c rd tr):.(J s i rt n h ) l R 2
?
rvhich is equivalenb to (7).
4. CoNcr,uorNc Rnrnnrs
An Indian mcthoclof conrputingtabular Sinesbv using a process given basicallS'
by the tule expressedby (7) has been regarclecl curioug b;' Delambre whom Dat'ta6
quotes as remarliing thus :
"This clifferential procoss has not upto norv beeu enrployed except b5r Briggs
(c. f6l5 A.D.) rvho lrirnself dicl not larorv t.hat the const'antfactor rvas the squarc
of the chorcl or the intcrvrr,l (taking unit ratlius). ancl rvho coulcl not obtain it, except
by comparing the second differences obtainecl in a different rnanner. The fndians
also hacl probably clone tlr.e same; they obtair-rthe methocl of differences only from
a table calculated previousl.v bv a geonretrical process. Here then is a methorl
which the fndians possessecl and which is founcl neither arnongst the Greeks nor
amongst the Arabs".
Like Delambre, BurgessTalso thinks that the property, that the second differ-
ences of Sines are proportional to Sines themselvcs, 'rvas knour to the Hindus
only by observation. Had their trigonometry sufficed to demonstra.te it, they
might easily have constructecl much more complete and accurate table of Sines'.
Datta (op. cil.), borvever, sees no reason to suspect that fndians obtained the
above formula (6) by inspection after having calculatecl t'he table by a different
method; "there is no doubt that the early Hindus lvert in possession necessary of
resourcesto deriye the formula". he adds.
6. 86 GIII{TA : EAII,LY I){DIAS ON SECOIfD ON,DER,SINE DIFFXR,ENCES
Finall5' it nr.ay be statecl that various geometrical proofs of the rule have been
givens by moclern schola'rs lilie Nervton, I(rishnas'rvami Ayvanger, Naraharawa
and Srini'r'asiengar. Hori'over, it nray be pointecl out that the rule given by (T)
is exact, ancl not approximato as assumecl b_v some of the above scholars. The
exposition ancl the limiting forms of th.e nrles and results from the NAB and
Yu,kti-Bhdqd. (lTth century A.D.) as given b1' Sarast'atie shoulcl also be noted.
Many other moclern proofs hilvrr been givcn,l0
R,nlonpxcns aND Xott,
I The Aryabhaliyo (rvith the commentary of Paramedvara) edited by H.Kern, Leiclen 1874; p.30.
For a fi esh moclern esposition of tho rulc see Sen, S. N. : Aryabhato's llathem atics, Bulletht
National Institute oJ Scicnces of Itvlia No. 21 (196:)), p. 213.
2 Tho Silrya Siddhdnta (rrith the comment&ry of Paramesivara) editecl by K. S. Shukla, Lucknorv
1957:'p.27. For references to the commentators }lallikdrjuna ancl Ramakrsr.ra see Lucknorv
University transcripts No. 45747 ancl No. 457{9 respectively.
3 The Aryabhaliyct with Lhe Bhagya (gloss) of l{ilakar.rtha Part I (Ganita) edited by S. Sambasiva
Sastri, Trivanclrum, 1930; p. i16.
a Golasdra of Nilakaltha SorntryEji editcd by K. V. Sarma, Hoshiarpur 1970; p. 19.
6 "fhe Tantrasamlyaha of Niltlktrnthtr, Somasutvan (rvith commentary of Sankara, Variar) eclitecl
b y S. K. Pilla i, T r iva n cln r m l9 5E ; p. 17.
6 Drrtta, B. B. : Hinclrr Contribution to llathematics. Bulletht Allahabad Uniu. Math. Assoc..
Vo ls. I & 2 ( 1 9 2 7 - 2 9 ) ; p . 6 :1 .
7 Burgess, E. (translator) Silryo, Siddhdnta. Calcutta reprint lgl)5; p. 62.
6 (i) Bu r g e ss, E., o p . cit., p . s:ji r vhe.rc I{. A . N ervton's prcof i s quoted.
( i i) Ayya n g a r ' , A. A. K.: l' h c Hinr-l u S i ne.Tabl es. ,J. Irtrl i an fuIath. S oc., V ol . 15 (1921),
fir ' st p a r t, p . l:.1 3 .
( i i i ) Na r a h a r a yya , S. N.: No te s o n the H i ncl u Tabl es of S i nes, J.Incl i a,n Math. S oa., V ol . l 5
( 1 9 2 - { ) , No te s a n d Qu e stio n s, p p. l 0S -110.
(iv) Srinivasiengar, C. N.. ?/rc History oJ Ancient Indian Mathematics, Celcutta, lg67; p. 52.
e Sarusn'athi, T. A., The Devclopmr.nt of llathcmatical Series in Inclia after Bhaskara II.
Bulletin oJ the National Irt.st. of Scierrcesof Inrlia No. 2l (106:]), pp. 335-339.
ro See Bina Chatterjee (editor ancl tlanslator): The KhantlakhidyaLa of Brahmagupta. New
De lh i a n d Ca lcu tta , 1 0 7 0 . Vo l. I, pp. f98-205.