Determination of Kalalagna in the Lagnaprakarana

Abstract

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.

Precise Determination of the Ascendant in Lagnaprakarana – II

“…Now, applying the cosine rule in the spherical triangle Γ𝐸𝑅, we have cos 𝛼 𝑒 = cos(𝜆 𝑚 + 90) cos 𝜇, 7 It may be noted that a similar usage has been encountered in verse 36 (see our previous paper), where the term koṭikrānti refers to the 'declination' derived from the cosine of the longitude of the madhyalagna.…”

“…See our discussion of verse 30 in[7].24 In Figure8a, the pole of the ecliptic is in the western hemisphere. Instead, when the pole of the ecliptic is in the eastern hemisphere, we…”

Precise Determination of the Ascendant in the Lagnaprakaraṇa – I

“…9 Except when the points are separated by 180 degrees, two points are sufficient to define a unique great circle on a sphere. 10 See verse 30 in [7]. to the meridian ecliptic point (madhyalagna).…”

“…which is the same as (7). Now, a brief note on the advantage of the choice of form of the rule prescribed by (7). Naively, it may appear that expanding sin( ± ) and determining the sum or difference of products of the sine and cosine functions could be more cumbersome than directly determining the sine of the total quantity.…”

“…1 In the same paper, we have briefly noted the state of udayalagna computations in Indian astronomy prior to Mādhava, and also remarked upon the approximations involved therein. 2 DOI: 10.16943/ijhs/2019/v54i3/49741 * Corresponding author: aditya.kolachana@gmail.com 1 See the introduction to [7]. 2 The standard procedure adopted by Indian astronomers prior to Mādhava to determine the udayalagna is perhaps best described in the In the first chapter of the Lagnaprakaraṇa, Mādhava discusses several procedures (many of them novel) to determine astronomical quantities such as the prāṇakalāntara (difference between the longitude and right ascension of a body), cara (ascensional difference of a body), and kālalagna (the time interval between the rise of the vernal equinox and a desired later instant).…”