The author of the Lílávatí 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. [Sidenote: Commentators on Lílávatí.] Among the commentators on Lílávatí, Rámakrishna, Gangádhara, and Ranganátha 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. [Sidenote: The Pentagon.] 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. THE FIGURES. OF OM (ON OR EN) OF HINDU ONTOLOGY. I. Mudrá, Madawar, Sphere or Sphaira. A Symbol of the Universe and Universalia. A System of the Universal Religion. ॐकार मुद्रा । अखण्डमण्डलाकारं व्याप्तयेनचराचर स्वरूपं । II. The circle O, An Emblem of infinity and Eternity. A Type of the Catholic Theism of Hindus. ॐकार वृत्तं । अनाद्यनन्तसनातन व्राह्मधर्म्म​स्वरूपञ्च । III. The convexity of O. A Type of the Extramundane, Unknowable and Absolute Supreme Brahma. Significant of Agnoism and Agnosticism. ॐवृत्तपरिधेःपरं । अपरिमेयाज्ञेयतुरीयपरव्र​ह्म स्वरूपं । IV. The concavity of O. Emblem of Intramundane Immensity of knowable Nature and its God Brahma . And Indicative of Gnosticism and Pantheism. ॐकारबृत्तमध्यं । प्रज्ञेय निर्गुणब्रह्मतथाज्ञेयब्रह्माण्डस्वरूपं । V. The circle with the Central point or Monad. A Symbol of the Definite and known world and its God. And signifying the Monotheism of all nations. ॐबृत्तं सविन्दुमध्यं । सगुणब्रह्मविन्दुब्रह्माण्डवीजमपर तैजस ब्रह्मस्वरूपं । 1. The circle with the central A, अ Alif or Unit. Emblematical of the unity of a Personal God. And the Primary unity of all things in Nature. And significant of unitarianity or _Advaita matam_. ॐकार बृत्त केन्द्रस्थ​अकार सहितं । सोपाधिक एकमेवाद्वितीयमतरूपं ।