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
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
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
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. 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
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
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
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. 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.
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.
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.
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.
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/
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.
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.
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 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.
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. 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.
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
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) 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 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.
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.
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.
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/
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.
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.
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 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.
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. 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.
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
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) 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.