The ascensional difference or the cara is a fundamental astronomical concept that is crucial in determining the durations of day and night, which are a function of the observer's latitude and the time of the year. Due to its importance, almost all astronomical texts prescribe a certain procedure for the determination of this element. The text Lagnaprakaraṇa-a hitherto unpublished manuscript attributed to Mādhava, the founder of the Kerala school of astronomy and mathematics-however discusses not one, but a number of techniques for the determination of the cara that are both interesting and innovative. The present paper aims to discuss these techniques.

The determination of the ascendant (udayalagna) or the rising point of the ecliptic is an important problem in Indian astronomy, both for its astronomical as well as socio-religious applications. Thus, astronomical works such as the Sūryasiddhānta, the Brāhmasphuṭasiddhānta, the Śiṣyadhīvṛddhidatantra, etc., describe a standard procedure for determining this quantity, which involves a certain approximation. However, Mādhava (c. 14 th century) in his Lagnaprakaraṇa employs innovative analytic-geometric approaches to outline several procedures to precisely determine the ascendant. This paper discusses the first method described by Mādhava in the Lagnaprakaraṇa.

The concept of the kālalagna is an important and innovative contribution of the Kerala school of astronomy, and is employed for a variety of astronomical computations in texts such as the Tantrasaṅgraha, the Candracchāyāgaṇita, the Karaṇapaddhati, and the Gaṇita-yukti-bhāṣā. This concept appears to have been first introduced by Mādhava (c. 14 th century), the pioneer of the Kerala school, in his Lagnaprakaraṇa. In this text, Mādhava makes innovative use of the kālalagna to determine the exact value of the udayalagna, or the ascendant, for the first time in the annals of Indian astronomy. This paper discusses the various techniques of determining the kālalagna described in the Lagnaprakaraṇa. 1 For instance, Amarasiṃha in the Digvargaprakaraṇa of the Amarakoṣa (1.3.230) defines the lagna as: राशीनामु दयो ल ं ते तु मे षवृ षादय: । (The rising [time] of the rāśis is lagna. They are meṣa, vṛṣa etc.).2 This standard procedure is perhaps best described in the Tripraśnādhikāra of the Śiṣyadhīvṛddhidatantra. For a detailed discussion, see Lalla (Śiṣyadhīvṛddhidatantra, pp. 61-69).3 Given the rising time of a rāśi, the rule of three is employed to determine the portion of a rāśi which rises in a desired amount of time. This introduces an error as the rāśi does not rise linearly with time.

The prāṇakalāntara, or the difference between the longitude and the corresponding right ascension, is an important astronomical parameter used in determining the ascendant (lagna), as well as the equation of time in Indian astronomy. This paper explores the different algorithms described to calculate the prāṇakalāntara in the Lagnaprakaraṇa, a hitherto unpublished manuscript attributed to Mādhava, the founder of the Kerala school of astronomy and mathematics. We also point out the interpretation of some of the algorithms in terms of epicyclic models.

Authored by the celebrated mathematician-astronomer Mādhava, the Lagnaprakaraṇa is an important astronomical text dedicated to the determination of the udayalagna or the ascendant, and is notable for its technical brilliancy and multiple approaches to a given problem. In continuation of our previous papers on this text, here we discuss some more methods for precisely determining the udayalagna as described in the second chapter of the Lagnaprakaraṇa.