IJHS | VOL 54.4 | DECEMBER 2019 ARTICLES On the Computation of Daily-motion in Ancient Indian Astronomy Anil Narayanan∗ Former Scientist, Indian Space Research Organization, 9106 Colgrove Ct, Lorton, VA 22079 (Received 21 November 2018; revised 29 August 2019) Abstract The computation of the true daily-motion of a heavenly body is an important precursory step in several calculations in Indian astronomy, like those for eclipses, planetary conjunctions, longitude corrections, etc. In this article we examine all aspects of the daily-motion calculation as given in the Sūryasiddhānta, an ancient Indian text of astronomy. It is observed that these algorithms are based on the standard Indian planetary model. In the standard model, the true longitude of a planet is determined by the combined effects of the manda and śīghra epicycles. In an analogous manner, the daily-motion of a planet is seen to be the sum of its manda and śīghra components. Computed results based on the daily-motion algorithms, when compared with actuality, were found to be considerably accurate. Key words: Daily-motion, Epicycle, Heliocentric, Manda, Śīghra, Sūryasiddhānta. 1 Introduction motions, which form the very core of any astronomical system. Yet another heliocentric feature can be found The true daily-motion of a planet is employed in several in the 4-step procedure for calculation of planetary lon- computations in Indian astronomy, and its calculation il- gitudes. The computation of the third or penultimate lustrates, among other things, the advanced nature of the step results in the heliocentric longitude of the planet, Indian science. As far as we can tell, no other ancient civ- with the fourth and final step merely converting the he- ilization, including the Greek and the Arab, has shown liocentric longitude to geocentric. These examples serve interest in computing the daily-motion of a planet as a to illustrate the curious undercurrent of heliocentricity stand-alone calculation. Thus, the conception and com- in the avowedly geocentric Indian planetary model. In putation of the daily-motion as a distinct entity appears to this present article, we will see that the daily-motion algo- be unique to Indian Astronomy. Uniqueness aside, three rithm adds yet another heliocentric feather in the cap for other features of this Indian computation are apt to draw Indian astronomy. our attention — first, the sheer brevity or conciseness of There are two definitions of daily-motion for a heavenly the calculation; second, the considerable accuracy of the body: (1) the mean daily-motion, and, (2) the true daily- computed result; third, and most interesting, the curious motion. The mean daily-motion is simply the average presence of heliocentric features in the computation. daily-motion of the planet taken over a sufficiently long Though the Indian planetary model is essentially geo- period of time, which, for all practical purposes, is a con- centric, some of its core features have intimations of he- stant. The true daily-motion, on the other hand, varies liocentricity. For example, consider the planetary mean- considerably from day to day. It is the arc traversed by the DOI: 10.16943/ijhs/2019/v54i4/49768 planet in the heavens on any particular day, as seen from *Email: [email protected] the earth. The magnitude of this true daily-motion oscil- ARTICLES IJHS | VOL 54.4 | DECEMBER 2019 lates about the mean daily-motion, sometimes being less the epicycle in one of two ways —CW (clock-wise) for the that it and at other times being greater. manda model and CCW for the śīghra model (Figure 1 de- Besides being an interesting parameter in itself, the picts the manda model). The angular rate of motion of true daily-motion is also an essential ingredient in sev- the actual planet on the epicycle equals the relative rate eral other computations in Indian astronomy, like those of motion between the mean planet and the apsis (apogee for eclipses, planetary conjunctions, rising and setting of for the manda and conjunction for the śīghra models re- planets, time of meridian crossing and determination of a spectively). correction factor for the true longitude. While the daily-motion formulae have been mentioned in several books and articles, there appear to have been few attempts at a rigorous analysis of the complete al- gorithm. The Indian text we will mainly refer is the Sūryasiddhānta (Burgess 1858), the oldest and most revered of all Indian works on astronomy. Like most In- dian texts, the Sūryasiddhānta is at places somewhat terse and succinct. Data and computation techniques are pre- sented in as compact a form as possible, with no explana- tory notes whatsoever. Rather than being a conventional textbook, it is more a concise aid to instruction for the ex- perienced teacher. The original text is estimated to be older than 3000 bce (Brennand 1896; Narayanan 2010, 2011). In this article, we will first analyze the complete set of daily-motion algorithms, and follow it up with some sam- ple computations based on these algorithms and compare the results with actuality. We begin by taking a brief look at the Indian planetary model. Figure 1 The Indian epicycle model. The manda epicycle scheme, as shown in Figure 1, 2 Overview of the Indian planetary starts off at time 푇0, at bottom right in the Figure with the model actual planet (푃) being farthest away from the earth. The planet at this point is said to be at its apogee (퐴). As seen The Indian planetary model employs the epicycle for its from the earth, the direction of the mean-planet (퐶) here basis. In this respect, it is similar to the Greek, Islamic coincides with the actual planet (푃). Sometime later, at and early European models. The Indian epicycle, how- time 푇1, the epicycle has moved CCW on the deferent by ever, differs from the rest in a curious way. Whilethe an angle 휃 (the anomaly), and its center is now at 퐶1. In other epicycles are of constant size, the radius of the In- that same time interval, the actual planet has moved CW dian epicycle changes with motion. That is, the epicycle on the epicycle, by the same angle, to 푃1. At this new loca- expands and contracts as it moves on the deferent circle. tion, the actual planet, as seen from the earth, no longer Thus, it is sometimes called the pulsating Indian epicycle. coincides with the mean-planet. The longitude of the ac- Figure 1 shows a schematic of the basic Indian epicy- tual planet is now the mean-planet’s longitude minus the cle model. The earth (퐸) is located at the center of the correction angle (훼), which can be found by simple geom- larger circle, the deferent. The smaller circle, called the etry. In this manner, the true manda-corrected longitude epicycle, moves on the deferent in a CCW (counter-clock- of the planet may be determined for any time. wise) manner, at an angular rate that equals the planet’s Note that there is an added complication in the Indian mean motion. The mean planet is situated at the center model which we have omitted in the above description. of the epicycle at 퐶. The actual planet (푃) revolves around As mentioned earlier, the radius of the Indian epicycle 406 IJHS | VOL 54.4 | DECEMBER 2019 ARTICLES is not constant but varies as a function of 휃. The inter- One other item to consider is the effect of epicycle pul- ested reader can look up Narayanan (2011) for details on sation — the phenomenon which increases and decreases how this additional feature is handled. This completes the the epicycle radius. To what degree are the manda and basic Indian epicycle model. For the Sun and the Moon, śīghra epicycle dimensions altered by pulsation? Table 1 whose orbits are centered on the earth, the manda epicy- shows the pulsation parameters of the manda and śīghra cle model yields longitudes with remarkable accuracy, far epicycles for each planet. In Indian astronomy, the size of more accurate than the Greek, Islamic or European mod- the epicycle is usually specified in terms of its circumfer- els (Narayanan 2011, 2013). However, for the five visible ence (퐶) rather than the radius (푟). The circumference of planets, whose motion is centered on the Sun, the manda the deferent is assumed to be 360 degrees and the epicycle model by itself would not have obviously provided accu- circumference is specified as a fraction of that in degrees. rate results. Both the Indians and the Greeks took diver- It can be seen from the max. and min. columns of Ta- gent paths to resolve the issue. While the Greeks resorted ble 1 that the magnitude of pulsation for both manda and to various mechanical devices like cranks and off-center śīghra epicycles is quite small, very fine in fact. Doubtless, designs, the Indians employed the simpler idea of a sec- the ancient Indians had good reasons for crafting such ond epicycle. fine control of the epicycle size. Today, with our as yetin- It appears that the ancient Indians perceived two other complete analysis of Indian astronomy, we can only mar- factors, apart from the mean motion, that influences the vel and conjecture about their reasons for doing so. progress of a planet in its orbit: (1) a decelerating entity, In modern astronomical terms, the manda correction called the manda, and, (2) an accelerating entity, called is related to the correction for the equation-of-center of a the śīghra. In Sanskrit, the words manda and śīghra mean planet’s orbit. Thus, the manda location for all the plan- slow and fast respectively.