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. 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.

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.

Synopsis of the book

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

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

Further references