Bhāskara I on the Construction of the Armillary Sphere

The armillary sphere is said to have been invented by the Greek astronomer Eratosthenes (276–194 BCE). Ptolemy’s Almagest (2nd century CE) contains a detailed description of the armillary sphere. However the armillary sphere described in Sanskrit texts on astronomy, from the seventh century onwards, is substantially different from the Greek armillary sphere.


Bhāskara I on the Construction of the Armillary Sphere
Peng Lu 1 . I N T ROD UCT ION T he armillary sphere is said to have been invented by the Greek astronomer Eratosthenes (276-194 bc). 1 Ptolemy's Almagest (second century ad) contains a detailed description of the armillary sphere. However the armillary sphere described in Sanskrit texts on astronomy, from the seventh century onwards, is substantially different from the Greek armillary sphere. 2 In Sanskrit, the armillary sphere is known as gola or gola-yantra. The word gola denotes primarily the "sphere." Āryabhaṭa and other astronomers use this term in the sense of "spherics" as opposed to gaṇita "mathematical in general." Thus Āryabhaṭa's Āryabhaṭīya (early sixth century ce, henceforth abh) has four chapters named Daśagītikā (Ten Gīti Stanzas Giving Parameters), Gaṇita (Mathematics), Kālakriyā (Time-reckoning), and Gola (Spherics). Here the chapter on Gola, expounds Āryabhaṭa's cosmology and geography, explains the apparent motions of the heavenly bodies, and gives solutions for problems in spherical trigonometry and rules for computing eclipses. 3 In addition, the term gola also denotes the armillary sphere consisting of several rings to represent the orbits of the celestial bodies in the heavens. 4 I wish to offer my special thanks to Professor S. R. Sarma for encouraging me to write this paper and for his constructive comments on the earlier version of this paper. I would also like to express my gratitude to my teacher Professor Michio Yano for his constant support. Finally my thanks are due to Wei Liu of Shanghai Conservatory of Music, China, for producing the 3dcg figures for my paper. 1 Bryant 1907: 18. 2 Ôhashi 1994: 272. 3 Pingree 1981 Āryabhaṭa uses the word gola once (abh 4.22) in the sense of the celestial globe also, but such usage is rare. All the other uses of the word gola in abh (1. 1, 2.7, 4.5-7, 4.15, 4.16, 4.23, 4.39) refer to spherics.
Brahmagupta discusses the armillary sphere in his Brāhmasphuṭasiddhānta (628 ad, henceforth bss), 5 but there he deals more with spherics than with the actual instrument. 6 About the same time (629 ad), Bhāskara I also describes the armillary sphere in his commentary on the Āryabhaṭīya (henceforth bab).
Subsequently Lalla (eighth century) in his Śiṣyadhīvṛddhidatantra, Śrīpati in his Siddhāntaśekhara (1039), Bhāskara II in his Siddhāntaśiromaṇi (1150) described the construction of the armillary sphere. Interestingly enough Lalla and others do not discuss the armillary sphere in the chapter Yantrādhyāya along with other astronomical instruments, but in a separate chapter called "tying or assembling the [various rings of the armillary] sphere" (gola-bandha).
Among the descriptions of the armillary sphere to be found in Sanskrit texts on astronomy, Bhāskara I's description is unique because it explains in detail how to construct the armillary sphere step by step: how to prepare a wooden ring by joining two or three strips of wood after making cuts or indents in the thickness at the ends, how to assemble two or more rings with appropriate indents at the points of intersection and so on. In this connection, he mentions three varieties of cuts (cheda) to be made for assembling the rings. Such practical details are not given in other Sanskrit texts. At the end of the passage, Bhāskara mentions that the armillary sphere he envisages is a free-standing one, but some others prefer to place the lower half of the armillary sphere in a pit dug in the level ground so that the horizon ring is level with the surface of the ground (see paragraph 20 of Bhāskara I's text and Figure 20 below).

B H Ā S K A R A'S D E S C R IPT ION OF THE GOL A
Before commenting on the Gola chapter of the Āryabhaṭīya, Bhāskara describes the armillary sphere (gola) first, because this instrument will be useful in explaining the statements on the spherics (gola) in the Āryabhaṭīya. In this paper, we follow the text of Bhāskara's commentary as given in the critical edition of 1976 by K. S.  and translate the passage on the armillary sphere and add explanatory notes. Bhāskara's description is so detailed that it is possible for us to reconstruct a gola-yantra with wooden rings. The successive stages of the construction are shown by different figures. We have added serial numbers to the paragraphs, and corrected the irregular sandhis that occur in Shukla's text. , and so on. In the same manner, here also, astronomers (sāṃvatsara) comprehend the real celestial sphere (gola) through certain [concepts] of geometry (kṣetragaṇita-viśeṣa) like the ring (vṛtta), the rod (śalākā), the thread (sūtra), the plumb line (avalambaka), and so on. 9 Therefore, [here] the demonstration of only the part will be undertaken, for it is impossible to demonstrate everything. While painting, who can paint the closing and opening in the flutter of the eyes?" 7 abh 1.1cd: आय भटीिण गदित गिणतं कालिबयां गोलम ् ॥    "Then he should insert a very smooth and straight iron rod (ayaḥśalākā) which is capable of bearing the weight of the shell of rings (pañjara), whose [two ends are] oval like the cow's tail (gopucchāyatavṛttā) 21 and which pierces through the intersections at the south and the north in such a way that its extremities (agra) project out on both sides. In the middle of it (i.e., the rod) make a spherical ball (samavṛttā) of clay or of any other [material to represent] as the earth. Thus, only one shell of rings is [enough] for all the planets. Although the planets are situated in different orbits, they are indeed seen as moving on a single orbit. Therefore only one shell of rings is [enough] here. Or, shells have to be shown separately (pṛthak pṛthak) for [each of] all the planets according to the size (pramāṇa) of their own different orbits (paricchinna-svakakṣyā)."  "Or one should firmly attach a three or four aṅgula's long smooth stalk of reed (śara-daṇḍikā) to the iron rod [projecting] out of the two intersections of the shell of rings. 22 Then taking [a ring] of an appropriate dimension representing the orbit of the sky (kha-kakṣyā), a ring in which [two] holes are made at distances equal to half the circumference (cakrārdhāntara) on both sides, one should place it in the south-north direction (dakṣiṇottarāvagāhin, lit., passing through the south and north [points]), and set the shell inside it by inserting the iron rod into the holes on both sides, so that that stalk of reed is embedded (avagāhin) in the boundary (i.e., interstice) of the two shells [along the iron rod]."  "One should affix as before another ring having the same size, which is embedded in the east and west directions, and in which intersections are made above and below. This is the prime vertical or east-west ring (sama-maṇḍala, literally "equal[izing] circle"). Again, one should affix another ring of the same size like a girdle [around the two rings] so that four intersections are produced in each cardinal direction, make holes at the points of the intersection in the south and north, insert the iron rod into them and then make [the third ring] firm. This is the horizon ring (kṣitija-maṇḍala). In this way, this armillary sphere (gola)

CO NC LUS I ON
In the preceding section we have followed Bhāskara's own words. I would like to add a few words on his method of constructing armillary sphere. The armillary sphere consists of two concentric shells with a clay terrestrial globe at the centre; the iron axis passes through the centres of all the three. The later writers call the history of science in south asia 3 (2015)  inner shell "sphere of stars" (bhagola or nakṣatragola) and the outer shell "sphere of the sky" (khagola). As mentioned in ¶ 2, Bhāskara regards the armillary sphere as a fictitious construction to understand the real celestial sphere. Already at the beginning of the commentary he states: "the armillary sphere (gola) is a means (upāya) to demonstrate the revolution of the planets and the situation of the earth." 23 Somewhat later, while commenting on Āryabhaṭīya 1.10, he asserts that, the armillary sphere (gola) is a shell of stars (bhapañjara), in which the motion of the planets is understood because the armillary sphere (gola) is a tool for the perception (pratipatti-hetu) of the true motion of the planets. 24 But besides measuring the chords of declination of the day circles by threads, Bhāskara does not mention other uses of the armillary sphere, whereas Lalla 23 BAB 1. history of science in south asia 3 (2015) 1-19 Adding the orbit of Mercury to the ecliptic. teaches how to determine time and the lagna (the point of sunrise on the east horizon) with the armillary sphere. 25 Moreover, Bhāskara does not explain clearly how to add the "vimaṇḍala" or the orbit rings of Mercury and Venus according to their śīghrocca scales. Since these two are inferior planets which always appear near the sun, the rings of their orbits are not like those of the moon or of the other planets, which are akin to the ecliptic ring. The rings of the orbits of Mercury and Venus ought to be smaller and added to the armillary sphere as shown in the Figure above. But if they are so added, they would hinder the rotation of the rings in the inner shell of the armillary sphere.