Karanapaddhati is an astronomical treatise in

Sanskrit Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...

attributed to

Puthumana Somayaji Puthumana Somayaji (c.1660–1740) was a 17th-century astronomer-mathematician from Kerala, India. He was born into the Puthumana or Puthuvana (in Sanskrit, Nutanagriha or Nuthanvipina) family of Sivapuram (identified as present day Thrissur). The ...

, an

astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

of the

Kerala school of astronomy and mathematics The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Tirur, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta S ...

. The period of composition of the work is uncertain. C.M. Whish, a civil servant of the

East India Company The East India Company (EIC) was an English, and later British, joint-stock company founded in 1600 and dissolved in 1874. It was formed to trade in the Indian Ocean region, initially with the East Indies (the Indian subcontinent and South ...

, brought this work to the attention of

Europe Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a subcontinent of Eurasia and it is located entirel ...

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

. The sixth chapter contains

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...

expansions for the value of the mathematical constant π, and expansions for the trigonometric sine, cosine and

inverse tangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Sp ...

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

Thrissur Thrissur (), formerly Trichur, also known by its historical name Thrissivaperur, is a city and the headquarters of the Thrissur district in Kerala, India. It is the third largest urban agglomeration in Kerala after Kochi and Kozhikode, and t ...

Kerala Kerala ( ; ) is a state on the Malabar Coast of India. It was formed on 1 November 1956, following the passage of the States Reorganisation Act, by combining Malayalam-speaking regions of the erstwhile regions of Cochin, Malabar, South ...

India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...

. 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 system the number of days in the '' Kali Yuga'', 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 Tantrasamgraha, or Tantrasangraha, (literally, ''A Compilation of the System'') is an important astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics. The ...

Nilakantha Somayaji Keļallur Nilakantha Somayaji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehens ...

(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'' : Rotation and revolutions of the

planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...

s in one ''

mahayuga A ''Yuga'' Cycle ( ''chatur yuga'', ''maha yuga'', etc.) is a cyclic age (epoch) in Hindu cosmology. Each cycle lasts for 4,320,000 years (12,000 divine years) and repeats four ''yugas'' (world ages): '' Krita (Satya) Yuga'', ''Treta Yuga'', ''D ...

''; the number of civil days in a ''mahayuga''; the solar months, lunar months, intercalary months; ''

kalpa Kalevan Pallo (KalPa) is a professional ice hockey team which competes in the Finnish Liiga. They play in Kuopio, Finland at the Olvi Areena. Team history Established in 1929 as ''Sortavalan Palloseura'' in Sortavala, the club relocated to Kuop ...

'' and the four ''

yuga A ''yuga'', in Hinduism, is generally used to indicate an age of time. In the ''Rigveda'', a ''yuga'' refers to generations, a long period, a very brief period, or a yoke (joining of two things). In the ''Mahabharata'', the words ''yuga'' and ...

s'' and their durations, the details of '' Kali Yuga'', calculation of the ''Kali'' era from the Malayalam Era, 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'' :

Parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

s connected with Kali era, the positions of the planets, their angular motions, various parameters connected with

Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...

. :''Chapter 3'' : Mean center of Moon and various parameters of Moon based on the

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

and

longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...

of the same, the constants connected with

. :''Chapter 4'' :

Perigee An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any ell ...

and

apogee An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any ell ...

of the

Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin at ...

, corrections to be given at different occasions for the

, constants for

, Mercury,

Jupiter Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousandth t ...

Venus Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never f ...

, Saturn in the respective order, the

perigee An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any ell ...

and

of all these planets, their

conjunction Conjunction may refer to: * Conjunction (grammar), a part of speech * Logical conjunction, a mathematical operator ** Conjunction introduction, a rule of inference of propositional logic * Conjunction (astronomy), in which two astronomical bodies ...

, their conjunctions possibilities. :''Chapter 5'' : Division of the

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

manvantara A ''manvantara'', in Hindu cosmology, is a cyclic period of time identifying the duration, reign, or age of a Manu, the progenitor of mankind. In each ''manvantara'', seven Rishis, certain deities, an Indra, a Manu, and kings (sons of Manu) ar ...

s for this kalpa, further details on the four yugas. :''Chapter 6'' : Calculation of the

circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...

of a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

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 chord, the arrow, the

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

s, their relations among a variety of parameters; methods to memorize all these factors using the katapayadi system. :''Chapter 7'' :

Epicycle In the Hipparchian, Ptolemaic, and Copernican systems of astronomy, the epicycle (, meaning "circle moving on another circle") was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, S ...

s of the Moon and the Sun, the apogee and perigee of the planets; sign calculation based on the

zodiac The zodiac is a belt-shaped region of the sky that extends approximately 8° north or south (as measured in celestial latitude) of the ecliptic, the apparent path of the Sun across the celestial sphere over the course of the year. The pat ...

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

epicycle In the Hipparchian, Ptolemaic, and Copernican systems of astronomy, the epicycle (, meaning "circle moving on another circle") was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, S ...

s and apogee for all the planets, their hypotenuse. :''Chapter 8'' : Methods for the determination of the

and

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 In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...

expressions for the constant π and infinite series expansions for the

trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

. These series also appear in

and their proofs are found in Yuktibhāṣā.

Series expressions for π

Series 1 The first series is specified in the verse which translates into the formula π/4 = 1 - 1/3 + 1/5 - 1/7 + ... Series 2 A second series is specified in the verse and this can be put in the form π = 3 + 4 Series 3 A third series for π is contained in which is π = 3 + 6

Series expansions of trigonometric functions

The following verse describes the infinite series expansions of the sine and cosine functions. 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

function. The specified expansion is tan⁻¹ x = x - x³ / 3 + x⁵ / 5 - ...

References

Venketeswara Pai R, K Ramasubramanian, M S Sriram and M D Srinivas, Karanapaddhati of Putumana Somayaji, Translation with detailed Mathematical notes, Jointly Published by HBA (2017) and Springer (2018).

Further references

*Open Library reference to Karana-paddhati with two commentarie

* * * *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 40007

(Retrieved on 13 January 2010) {{Scientific Research in Kerala , state=collapsed Kerala school of astronomy and mathematics Hindu astronomy Astronomy books Indian mathematics Indian astronomy texts