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

IV. 54 IV. THREE PROBLEMS 131 I: The circle on which the tip of the shadow moves on a day when the Sun is in the southern hemisphere. II: The circle on which the tip of the shadow moves on a day when the Sun is on the equator. III: The circle on which the tip of the shadow moves on a day when the Sun is in the northern hemisphere. A, B = rising and setting points of the Sun, on the day related to I. C, D = rising and setting points of the Sun on the day related to II, and E, F, related to III. AB, CD, and EF are the lines joining the respective rising and setting points and are parallel to one another. With reference to I, (i.e. for a day when the Sun is in the southern hemisphere), GA = HJ = Agrā (directed northward), JK = Sūryāgrā (directed northward) and HK = Agrā + Sūryāgrā, from which it is obvious that the Perpendicular is also directed northward. With reference to II, (i.e. for a day when the Sun is on the equator), the Sun rises at C itself and sets at D itself, and therefore the Agrā is zero. LM is the Sūryāgrā (directed northward) and the ‘Per- pendicular’ = Sūryāgrā ∓ Agrā, is also LM. With reference to III, (i.e. for a day when the Sun is in the northern hemisphere), Agrā = QP = NO (directed southward) and PR or OS is the Sūryāgrā (directed north). At a time sufficiently near sunrise or sunset, for which OS is the Sūryāgrā, the Perpendicular is NS (directed southward). This is the case where Agrā is deductive but numerically greater than the Sūryāgrā. At a time sufficiently near noon, for which PR is the Sūryāgrā, the Perpendicular is QR got by PR – PQ, QP being numerically less than PR. (iii) Agrā: This is the amplitude, and forms the distance between the parallel lines constituting the prime vertical and the line joining the rising and setting points. This is also the sine of the angles of the rising and setting points made from the East or West points, respectively. The author has given the formula for this in V. 39, without mentioning its name Agrā, as also here without men- tioning its name. The derivation of the formula has been given by us there. When the Sun is in the northern hemisphere, this is north, and when in the southern, it is south. But here, as we are dealing with the shadow, we have reversed the directions. One thing must be mentioned in this connection: TS and NP interpret the word Sūryāgrā as Agrā or ‘Sine of the amplitude of the Sun’, evidently assuming the derivation sūryasya agrā = Sūryāgrā, i.e. Agrā itself, because the context is the Sun here. As for Sūryāgrā itself, they simply call it ‘a sine’. They have failed to notice that if taken thus, the formula for getting them would become wrong. Even if somehow, by changing the order of words in the sentence, we make the formulae agree in this work, in the next work of getting the sun from the direction of the shadow, it would be impossible to secure agreement between the words there. But we must mention here that in the Mahābhāskarīya, Agrā is termed ‘Arkāgrā’, evidently by the derivation, arkasya agrā arkāgrā. Sūryāgrā is there called Śaṅkvagra, as we have already said. (Vide Mahābhāskarīya, III. 53-54). But here we have no choice except to go by the text. (iv) Perpendicular: From what we have already said, and from the Fig. 18, it can readily be seen that (Sūryāgrā ∓ Agrā) is the distance between the Prime vertical and the tip of the Great shadow. This is called ‘Bhuja’ by other authors. The Bhuja corresponding to the shadow is got from this by the proportion,