by Vihari-Lala Mitra | 1891 | 1,121,132 words | ISBN-10: 8171101519
The English translation of the Yoga-vasistha: a Hindu philosophical and spiritual text written by sage Valmiki from an Advaita-vedanta perspective. The book contains epic narratives similar to puranas and chronologically precedes the Ramayana. The Yoga-vasistha is believed by some Hindus to answer all the questions that arise in the human mind, an...
Correctness of the Diagrams.
We have seen from the diagrams given in the following section, that the Tantrika formulists have spared no pains to divide the great circle of the Universe, filled by the omnipresence of Brahma and represented by the figure om, into several parts for the purpose of meditating His different hypostases, and contemplation of the various orders of creation. We are now to inquire as to whether these several divisions of a mathematical circle of 360 degrees are geometrically correct, or mere arbitrary partitions made by ignorant priests for their own amusement and deception of their proselytes.
The Heptagon and Nonagon.
Now for instance, the problem of inscribing a heptagon or a nonagon in a circle will at once startle a student of Euclid as altogether impossible, and identical with that which was celebrated among Greek geometricians as the problem of the trisection of the angle. If treated algebraically, it leads to a cubic equation with three real roots, the arithmetical value of which can be found only approximately.
The Lilavati's solution.
The author of the Lilavati has solved the problems, but given no account of the way in which he got the numbers stated by him; if they had been obtained by solution of the above mentioned equation, they would probably have been more accurate than they are. He only lays down an arbitrary rule, that the side of the heptagon is 52055/120000 of the diameter, and that of the nonagon 41081/120000 of the same. Neither of these is very far from the truth. The accurate value of the side of the heptagon lies between 82/182 and 105/242. The side of the nonagon lies between 13/38 and 105/307.
Commentators on Lilavati.
Among the commentators on Lilavati, Ramakrishna, Gangadhara, and Ranganatha have not attempted any demonstration of the problems in question, and have contented themselves with merely repeating the figures contained in the text. Ganesa confesses that the proof of the sides of the regular pentagon, heptagon and nonagon cannot be shown in a manner similar to that of the triangle, square and octagon.
But this is untrue of the pentagon; its side can be geometrically found as shown in Euclid Book IV. Prop 11; and the admission of Ganesa serves only to prove, that he was unacquainted with the Sanskrit translation of Euclid which contains a solution of this problem. Ganesa cannot mean only that the side of the pentagon is incommensurable with the diameter; for that is equally true of the triangle, square and octagon, inscribed in a circle.