Karanapaddhati

Karanapaddhati
Author Puthumana Somayaji
Country India
Language Sanskrit
Subject Astronomy/Mathematics
Publication date
 1733 CE (?)

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 European scholars for the first time in a paper published in 1834.[1] 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.[2]

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 in Kerala, India.

The period in which Somayaji lived is also uncertain. There are several theories in this regard.[3]

Synopsis of the book

A brief account of the contents of the various chapters of the book is presented below.[5]

Chapter 1 : Rotation and revolutions of the planets in one mahayuga; the number of civil days in a mahayuga; the solar months, lunar months, intercalary months; kalpa and the four yugas and their durations, the details of kaliyuga, 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 : Parameters connected with Kali era,the positions of the planets, their angular motions, various parameters connected with Moon.
Chapter 3 : Mean center of Moon and various parameters of Moon based on the latitude and longitude of the same, the constants connected with Moon.
Chapter 4 : Perigee and apogee of the Mars, corrections to be given at different occasions for the Mars, constants for Mars, Mercury, Jupiter, Venus, Saturn in the respective order, the perigee 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 manvantaras for this kalpa, further details on the four yugas.
Chapter 6 : Calculation of the circumference of a circle 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 angles, their relations among a variety of parameters; methods to memorize all these factors using the katapayadi system.
Chapter 7 : Epicycles of the Moon and the Sun, the apogee and perigee of the planets; sign calculation based on the zodiacal 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 latitude and longitude 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 π and infinite series expansions for the trigonometric functions. These series also appear in Tantrasangraha and their proofs are found in Yuktibhāṣā.

Series expressions for π

Series 1

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 ) + ... }

Series expansions of trigonometric functions

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 - ...

References

  1. 1 2 Charles Whish (1834), "On the Hindu Quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Sastras, the Tantra Sahgraham, Yucti Bhasha, Carana Padhati and Sadratnamala", Transactions of the Royal Asiatic Society of Great Britain and Ireland (Royal Asiatic Society of Great Britain and Ireland) 3 (3): 509–523, doi:10.1017/S0950473700001221, JSTOR 25581775
  2. ↑ Datta, Bibhutibhushan; A.N. Singh (1993). "Uses of series in India". Indian Journal of History of Science 28 (3): 103–129.
  3. 1 2 3 Bag, Amulya Kumar (1966). "Trigonometrical series in the Karanapaddhati and the probable date of the text" (PDF). Indian Journal of History of Science (Indian National Science Academy) 1 (2): 98–106.
  4. ↑ Rajaraja Varma Vadakkumkuur. History of Sanskrit Literature in Kerala (1–6 Volumes) 1. p. 529.
  5. ↑ N. Gopalakrishnan (2004). Baharatheeya Vijnana / Saastra Dhaara ( Handbbok of Ancient Indian Scientific Books) (PDF). Heritage Publication Series 78. Thiruvanannthapuram, India: Indian Institute of Scientific Heritage. pp. 18–20. Retrieved 12 January 2010.

Further references

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