पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

Chapter Four THREE PROBLEMS — TIME, PLACE AND DIRECTION ४. चतुर्थोऽध्यायः त्रिप्रश्नाधिकारः Introductory Problems on Time, Place and Direction, involving spherical trigonometry, are dealt with in this chapter. The first fifteen verses are devoted to the construction of a table of sines. As this kind of matter does not involve constants specific to any siddhānta and is commonly found in all siddhāntas, we cannot say which Siddhānta this belongs to, Pauliśa or Saura, the only two siddhāntas meant to be expounded in detail by the author. Probably it is the author’s own, meant for both, or taken from both. Two things point to this conclusion: In the part of the work dealing with the Saura, viz., chs. IX, X, XI, XIII, XIV, XV, XVI, and XVII, no space is given to the sine tables, though required, and to the problems dealt with here, and therefore if these are not meant for Saura, it would be imperfect though almost full. On the other hand, certain redundant and crude rules point to this chapter’s connection with the Pauliśa, as also its position in the chapter distribution in the PS text. [ज्यानयनम्] षष्टिशतत्रयपरिधेर्वर्गदशांशात् पदं स विष्कम्भः | तदिहां(शच)तुष्कं संप्रकल्प्य रा(श्य)ष्टभागज्या || १ || Table of R Sines

1. Take the circumference as measured in 360 units, square it, take the tenth part of the square, and find its square root. The result is the diameter of the circle in the units taken. We assume the diameter to be 4°, (i.e., 240′) and hereunder give the tabular sines of angles for 3° 45′ interval. The rule is: diameter = √circumference²/10. It comes to this: d = c/√10. The formula, d = c/π is well known, and the author has taken √10 as an approximation for π which is incommensurable and usually represented by the approximate values, 22/7, 355/133, 3.1416 etc. The Sūrya Siddhānta too gives √10 as the value of π in its instruction to find the circumference of the earth from its diameter (I.59.): “The earth’s diameter is 1600 yojanas. Square this, multiply by 10, and find the square root. This is the earth’s circumference.” By thus taking √10 for π, an error of about 0.0067% results, and for a circumference of 21,600′, we get the radius 3415′, instead of the well-known 3438′. But it must be mentioned here that this error does not affect the computation of the sines 1a-b. A.B. परिधे वर्ग c. A.B. तदिहांशाश्चतुष्कं (B. ॰ष्क) b. A. विष्कुम्भः d. A. संप्रकल्प्य; B. प्रकल्प्य. A.B. राश्याष्ट०