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 dayThrissur in Kerala, India. 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 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 Somayaji.
Synopsis of the book
A brief account of the contents of the various chapters of the book is presented below.
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
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
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
Open Library reference to 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.
