Karanapaddhati
Encyclopedia
Karanapaddhati is an astronomical treatise in Sanskrit
attributed to Puthumana Somayaji
, an astronomer
-mathematician
of the Kerala school of astronomy and mathematics. The period of composition of the work is uncertain. C.M. Whish, a civil servant of the East India Company
, brought this work to the attention of Europe
an scholars for the first time in a paper published in 1834. The book is divided into ten chapters and is in the form of verses in Sanskrit
. The sixth chapter contains series expansions for the value of the mathematical constant π
, and expansions for the trigonometric sine
, cosine and inverse tangent functions.
in Kerala
, India
.
The period in which Somayaji lived is also uncertain. There are several theories in this regard.
and infinite series expansions for the trigonometric functions. These series also appear in Tantrasangraha and their proofs are found in Yuktibhāṣā
.
The first series is specified in the verse
vyāsāccaturghnād bahuśaḥ pr̥thaksthāt tripañcasaptādyayugāhr̥ tāni
vyāse caturghne kramaśastvr̥ṇam svaṁ kurjāt tadā syāt paridhiḥ susuksmaḥ
which translates into the formula
π/4 = 1 - 1/3 + 1/5 - 1/7 + ...
Series 2
A second series is specified in the verse
vyāsād vanasamguṇitāt pr̥thagāptaṁ tryādyayug-vimulaghanaiḥ
triguṇavyāse svamr̥naṁ kramasah kr̥tvāpi paridhirāneyaḥ
and this can be put in the form
π = 3 + 4 { 1 / ( 33 - 3 ) + 1 / ( 53 - 5 ) + 1 / ( 73 - 7 ) + ... }
Series 3
A third series for π is contained in
vargairyujāṃ vā dviguṇairnirekairvargīkṛtair-varjitayugmavargaiḥ
vyāsaṃ ca ṣaḍghanaṃ vibhajet phalaṃ svaṃ vyāse trinīghne paridhistadā syāt
which is
π = 3 + 6 { 1 / ( (2 × 22 - 1 )2 - 22 ) + 1 / ( (2 × 42 - 1 )2 - 42 ) + 1 / ( (2 × 62 - 1 )2 - 62 ) + ... }
The following verse describes the infinite series expansions of the sine
and cosine functions.
cāpācca tattat phalato'pi tadvat cāpāhatāddvayādihatat trimaurvyā
labdhāni yugmāni phalānyadhodhaḥ cāpādayugmāni ca vistarārdhāt
vinyasya coparyupari tyajet tat śeṣau bhūjākoṭiguṇau bhavetāṃ
These expressions are
sin x = x - x3 / 3! + x5 / 5! - ...
cos x = 1 - x2 / 2! + x4 / 4! - ...
Finally the following verse gives the expansion for the inverse tangent function.
vyāsārdhena hatādabhiṣṭaguṇataḥ koṭyāptamaādyaṃ phalaṃ
jyāvargeṇa vinighnamādimaphalaṃ tattatphalaṃ cāharet |
kṛtyā koṭiguṇāsya tatra tu phaleṣvekatripañcādibhir-
bhakteṣvojayutaistajet samajutiṃ jīvādhanuśiśaṣate ||
The specified expansion is
tan−1 x = x - x3 / 3 + x5 / 5 - ...
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
attributed to Puthumana Somayaji
Puthumana Somayaji
Puthumana Somayaji was a 15th century mathematician from Kerala, India. He was born into the Puthumana Illam, and is believed to have been a contemporary of Vatasseri Namboodiri....
, an astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
-mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
of the Kerala school of astronomy and mathematics. The period of composition of the work is uncertain. C.M. Whish, a civil servant of the East India Company
East India Company
The East India Company was an early English joint-stock company that was formed initially for pursuing trade with the East Indies, but that ended up trading mainly with the Indian subcontinent and China...
, brought this work to the attention of Europe
Europe
Europe is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...
an scholars for the first time in a paper published in 1834. The book is divided into ten chapters and is in the form of verses in Sanskrit
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
. The sixth chapter contains series expansions for the value of the mathematical constant π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
, and expansions for the trigonometric sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
, cosine and inverse tangent functions.
Author and date of Karanapaddhati
Nothing definite is known about the author of Karanapaddhati. The last verse of the tenth chapter of Karanapaddhati describes the author as a Brahamin residing in a village named Sivapura. Sivapura is an area surrounding the present day ThrissurThrissur
This article is about the city in India. For the district, see Thrissur district. For the urban agglomeration area of Thrissur see Thrissur Metropolitan Area...
in Kerala
Kerala
or Keralam is an Indian state located on the Malabar coast of south-west India. It was created on 1 November 1956 by the States Reorganisation Act by combining various Malayalam speaking regions....
, India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
.
The period in which Somayaji lived is also uncertain. There are several theories in this regard.
- C.M. Whish, the first westerner to write about Karanapaddhati, based on his interpretation that certain words appearing in the final verse of Karanapaddhati denote in katapayadi systemKatapayadi systemKaṭapayādi system of numerical notation is an ancient Indian system to depict letters to numerals for easy remembrance of numbers as words or verses...
the number of days in the Kaliyuga, concluded that the book was completed in 1733 CE. Whish had also claimed that the grandson of the author of the Karanapaddhati was alive and was in his seventieth year at the time of writing his paper.
- Based on reference to Puthumana Somayaji in a verse in Ganita Sucika Grantha by Govindabhatta, Raja Raja Varma placed the author of Karanapaddhati between 1375 and 1475 CE.
- An internal study of Karanapaddhati suggests that the work is contemporaneous with or even antedates the Tantrasangraha of Nilakantha SomayajiNilakantha SomayajiKelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...
(1465–1545 CE).
Synopsis of the book
A brief account of the contents of the various chapters of the book is presented below.- Chapter 1 : RotationRotationA rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
and revolutions of the planetPlanetA planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
s in one mahayuga; the number of civil days in a mahayuga; the solar months, lunar months, intercalary months; kalpa and the four yugaYugaYuga in Hindu philosophy is the name of an 'epoch' or 'era' within a cycle of four ages. These are the Satya Yuga, the Treta Yuga, the Dvapara Yuga, and finally the Kali Yuga. According to Hindu cosmology, life in the universe is created, destroyed once every 4.1 to 8.2 billion years, which is...
s and their durations, the details of kaliyuga, calculation of the Kali era from the Malayalam EraMalayalam calendarMalayalam calendar is a solar and sidereal Hindu calendar used in Kerala, India. The origin of the calendar has been dated as 825 CE....
, calculation of Kali days; the true and mean position of planets; simple methods for numerical calculations; computation of the true and mean positions of planets; the details of the orbits of planets; constants to be used for the calculation of various parameters of the different planets.
- Chapter 2 : ParameterParameterParameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
s connected with Kali era,the positions of the planets, their angular motions, various parameters connected with MoonMoonThe Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
.
- Chapter 3 : Mean center of Moon and various parameters of Moon based on the latitudeLatitudeIn geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
and longitudeLongitudeLongitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
of the same, the constants connected with MoonMoonThe Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
.
- Chapter 4 : PerigeePerigeePerigee is the point at which an object makes its closest approach to the Earth.. Often the term is used in a broader sense to define the point in an orbit where the orbiting body is closest to the body it orbits. The opposite is the apogee, the farthest or highest point.The Greek prefix "peri"...
and apogee of the MarsMarsMars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
, corrections to be given at different occasions for the MarsMarsMars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
, constants for MarsMarsMars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
, MercuryMercury (planet)Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
, JupiterJupiterJupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...
, VenusVenusVenus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...
, SaturnSaturnSaturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...
in the respective order, the perigeePerigeePerigee is the point at which an object makes its closest approach to the Earth.. Often the term is used in a broader sense to define the point in an orbit where the orbiting body is closest to the body it orbits. The opposite is the apogee, the farthest or highest point.The Greek prefix "peri"...
and apogee of all these planets, their conjunction, their conjunctions possibilities.
- Chapter 5 : Division of the kalpa based on the revolution of the planets, the number of revolutions during the course of this kalpa, the number of civil and solar days of earth since the beginning of this kalpa, the number and other details of the manvantaraManvantaraManvantara or Manuvantara , or age of a Manu , the Hindu progenitor of mankind, is an astronomical period of time measurement. Manvantara is a Sanskrit sandhi, a combination of words manu and antara, manu-antara or manvantara, literally meaning the duration of a Manu, or his life span .Each...
s for this kalpa, further details on the four yugas.
- Chapter 6 : Calculation of the circumferenceCircumferenceThe circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....
of a circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
using variety of methods; the division of the circumference and diameters; calculation of various parameters of a circle and their relations; a circle, the arc, the chordChord (geometry)A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...
, the arrow, the angleAngleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
s, their relations among a variety of parameters; methods to memorize all these factors using the katapayadi systemKatapayadi systemKaṭapayādi system of numerical notation is an ancient Indian system to depict letters to numerals for easy remembrance of numbers as words or verses...
.
- Chapter 7 : Epicycles of the Moon and the Sun, the apogee and perigee of the planets; sign calculation based on the zodiacZodiacIn astronomy, the zodiac is a circle of twelve 30° divisions of celestial longitude which are centred upon the ecliptic: the apparent path of the Sun across the celestial sphere over the course of the year...
al sign in which the planets are present; the chord connected with rising, setting, the apogee and the perigee; the method for determining the end-time of a month; the chords of the epicycles and apogee for all the planets, their hypotenuse.
- Chapter 8 : Methods for the determination of the latitudeLatitudeIn geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
and longitudeLongitudeLongitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
for various places on the earth; the R-sine and R-cosine of the latitude and longitude, their arc, chord and variety of constants.
- Chapter 9 : Details of the Alpha aeries sign; calculation of the positions of the planets in correct angular values;; calculation of the position of the stars, the parallax connected with latitude and longitude for various planets, Sun, Moon and others stars.
- Chapter 10 : Shadows of the planets and calculation of various parameters connected with the shadows; calculation of the precision of the planetary positions.
Infinite series expressions
The sixth chapter of Karanapaddhati is mathematically very interesting. It contains infinite series expressions for the constant πPi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
and infinite series expansions for the trigonometric functions. These series also appear in Tantrasangraha and their proofs are found in Yuktibhāṣā
Yuktibhasa
Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...
.
Series expressions for π
Series 1The first series is specified in the verse
vyāsāccaturghnād bahuśaḥ pr̥thaksthāt tripañcasaptādyayugāhr̥ tāni
vyāse caturghne kramaśastvr̥ṇam svaṁ kurjāt tadā syāt paridhiḥ susuksmaḥ
which translates into the formula
π/4 = 1 - 1/3 + 1/5 - 1/7 + ...
Series 2
A second series is specified in the verse
vyāsād vanasamguṇitāt pr̥thagāptaṁ tryādyayug-vimulaghanaiḥ
triguṇavyāse svamr̥naṁ kramasah kr̥tvāpi paridhirāneyaḥ
and this can be put in the form
π = 3 + 4 { 1 / ( 33 - 3 ) + 1 / ( 53 - 5 ) + 1 / ( 73 - 7 ) + ... }
Series 3
A third series for π is contained in
vargairyujāṃ vā dviguṇairnirekairvargīkṛtair-varjitayugmavargaiḥ
vyāsaṃ ca ṣaḍghanaṃ vibhajet phalaṃ svaṃ vyāse trinīghne paridhistadā syāt
which is
π = 3 + 6 { 1 / ( (2 × 22 - 1 )2 - 22 ) + 1 / ( (2 × 42 - 1 )2 - 42 ) + 1 / ( (2 × 62 - 1 )2 - 62 ) + ... }
Series expansions of trigonometric functions
The following verse describes the infinite series expansions of the sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
and cosine functions.
cāpācca tattat phalato'pi tadvat cāpāhatāddvayādihatat trimaurvyā
labdhāni yugmāni phalānyadhodhaḥ cāpādayugmāni ca vistarārdhāt
vinyasya coparyupari tyajet tat śeṣau bhūjākoṭiguṇau bhavetāṃ
These expressions are
sin x = x - x3 / 3! + x5 / 5! - ...
cos x = 1 - x2 / 2! + x4 / 4! - ...
Finally the following verse gives the expansion for the inverse tangent function.
vyāsārdhena hatādabhiṣṭaguṇataḥ koṭyāptamaādyaṃ phalaṃ
jyāvargeṇa vinighnamādimaphalaṃ tattatphalaṃ cāharet |
kṛtyā koṭiguṇāsya tatra tu phaleṣvekatripañcādibhir-
bhakteṣvojayutaistajet samajutiṃ jīvādhanuśiśaṣate ||
The specified expansion is
tan−1 x = x - x3 / 3 + x5 / 5 - ...
Further references
- Open Library reference to Karana-paddhati with two commentaries.http://openlibrary.org/b/OL19358577M/Karana-paddhati_with_two_commentaries.
- Indian National Science Academy has started a project in 2007–08 titled "A Critical Study of Karana-paddhati of Putumana Somayaji and Preparation of English Translation with Mathematical Notes" by Dr. K Ramasubramanian, Assistant Professor, Dept. of History, Indian Institute of Technology,, Powai, Mumbai 400076.http://insaindia.org/compro05.htm (Retrieved on 13 January 2010)