ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

BRAHMAGUPTA AND ŚRĪṢEṆA 331 systems, and he very much resented such interferences in pure academic life of this country. He was opposed to Āryabhaṭa for a different reason. Āryabhaṭa was universally regarded as an authority in this country, and the conservatism was so deep that even where it could be shown by direct observation or on valid theoretical grounds, that a particular concept was erroneous or less accurate, people still chose to adhere to it, since they had the backing of Āryabhaṭa's authority. Brahma- gupta was against this nonscientific attitude. Needless to say, Brahmagupta was not always fair to Āryabhaṭa in his criticism ; he overdid in enumerating the shortcomings of Āryabhaṭa's system, as if he was personally jealous of his wide popularity. Brahmagupta's feelings against Lāṭadeva, Śrīṣeṇa, Viṣṇu- candra and others would be seen from the following passage in the Brāhmasphuṭasiddhānta : From the fact that Śrīṣeṇa, Viṣṇucandra, Pradyumna, Āryabhaṭa, Lāṭa, and Siṁha contradict one another regarding eclipses and similar topics, their ignorance is proved daily. The criticisms which I have passed on Āryabhaṭa are, with the requisite modifications, to be applied to the doctrines of each of these teachers as well. I will, however, make some further critical re- marks on Śrīṣeṇa and others. Śrīṣeṇa took from Lāṭa the rules concerning the mean motions of the Sun, and the Moon, the Moon's apogee and her node, and the mean motions of Mars, Mercu- ry's Śīghra, Jupiter, Venus's Śīghra, and Saturn ; he took elapsed years and the revolutions of yuga (yuga- yāta-varṣa-bhagaṇa) from Vasiṣṭha and the Padakaraṇa of Vijayānandi; further took from Āryabhaṭa the rules concerning the apogee, epicycles and nodes, and those referring to the true motions of the planets and thus the Romaka Siddhānta which was (or is) a heap of jewels (as it were) has, by Śrīṣeṇa, been made into a patched rag (as it were)¹.

1. श्रीषेणविष्णुचन्द्र प्रद्युम्नार्यभटलाटसिंहानाम् । ग्रहणादिविसंवादात् प्रतिदिवसं द्विगुणमज्ञत्वम् ॥ [Cont. on Page 330]

332 BRAHMAGUPTA A GREAT CRITIC Brahmagupta very emphatically says about his system that so long as people would be finding concordance between the observed and theoretical results (dṛggaṇitaikyam) in respect of solar and lunar eclipses, his Brāhma Siddhānta would be held in esteem¹. In other systems, whatever concordance appears to be bet- ween the observation and calculation, of eclipses etc., it is, Brahmagupta says, merely accidental or by chance, as the maxim of letters bored by an insect in wood or paper². युक्त्याऽऽर्यभटोक्तानि प्रत्येकं दूषणानि योज्यानि । [Cont. from Page 329] श्रीषेणप्रभृतीनां कानि चिदन्यानि वक्ष्यामि ॥ लाटात् सूर्यशशांकौ मध्याबिन्दूच्च चन्द्रपातौ च । कुजबुधशीघ्रबृहस्पति सितशीघ्र शनैश्चरान् मध्यान् ॥ युगयातवर्षभगणान् वासिष्ठाद्विजयनन्दि कृतपादात् । मन्दोच्च परिधिपातस्पष्टीकरणाद्यमार्यभटात् ॥ श्रीषेणेन गृहीत्वा रत्नोच्चयरोमकः कृतः कन्था । एतानेव गृहीत्वा वासिष्ठो विष्णुचन्द्रेण ॥ —BrSpSi. XII. 46-50 2. चन्द्ररवि ग्रहणेन्दुच्छायादिषु सर्वदा यतो ब्राह्मे । दृग्गणितैक्यं भवति स्फुटसिद्धान्तस्ततो ब्राह्मः ॥ —BrSpSi. XI. 61 3. अन्योर्न कदाचिदपि ग्रहणादिषु भवति दृष्टिगणितैक्यम् । यद्भवति तद् घुणाक्षरमतोऽस्फुटभ्यां किमेताभ्याम् ॥ —BrSpSi. XI, 51. —: 0 :— Reference Brahmagupta : Tantraparīkṣādhyāya in BrSpSi. K.S. Shukla : The Mahābhāskarīya and the Laghubhāskarīya. H.T. Colebrooke : Miscellaneous Essays, Vol. II., 1872. G. Thibaut and Sudhākara Dvivedi : The Pañcasiddhāntikā, Preface, 1889.

CHAPTER XIII Brahmagupta and Astronomical Instruments The Twenty-second Chapter of the Brāhmasphuṭasiddhānta is known as the Yantrādhyāya or a chapter on instruments. There is a description of seventeen types of time-reckoning instru- ments (Kāla-yantra)¹ :

1. Dhanuryantra—Bow instrument.
2. Turyagolaka yantra—Quadrant (one-fourth sphere)
3. Cakra yantra=wheel or circle.
4. Yaṣṭi yantra—a pole or staff instrument.
5. Śaṅku yantra—Gnomon.
6. Ghaṭikā yantra—a clock or pot instrument.
7. Kapāla yantra—Bowl or potsherd instrument.
8. Karttarī yantra—Scissor or knife ; cutter.
9. Pīṭha yantra=Pedastal or seat instrument.
10. Salila yantra—Water-leveller.
11. Brahma or Śāṇa yantra—For describing circles.
12. Avalamba Sūtra—Threads with plumbs (Plumb lines).
13. Karṇa or chāyā-karṇa—A set of squares for diagonals.
14. Chāyā or śaṅku-chāyā—Sundial. ──────────────────────────────────────────────────
15. सप्तदश कालयन्त्राण्यतो धनुस्तुर्यगोलकं चक्रम् । यष्टिः शंकुर्घटिका कपालकं कर्त्तरी पीठम् ॥ सलिलं भ्रमीऽवलम्बः कर्णश्छाया दिनार्थमर्कोच्चः । नतकालज्ञानार्थं तेषां संसाधनान्यष्टौ ॥ —BrSpSi. XXIII, 5-6

334 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS 15. Dinārdha yantra—Midday measure instrument. 16. Arka yantra—Sun-instrument. 17. Akṣa or Palāñśa yantra—Small degree measure arc instrument. Salila yantra is used for levelling; since a liquid such as water seeks its own level, it can be utilised to know whether a surface has been levelled or not.¹ Bhrama or Śāṇa is used for drawing circles. Avalambaka or plambline is used for adjusting vertical line. Karṇa is used in connection with angles and diagonals. From Salila (no. 10) to the last (no. 17); these eight are used for adjustments and are basically important. The dhanuryantra is used for nata and unnata kāla ghaṭikās. On the paridhi or the circumference of the cakra-yantra are indicated the twelve rāśis, ending up to Mīna (XXII. 18). Brahmagupta has described the yaṣṭi yantra and shown how it could be used to give time at different parts of the day, and from its shadow dṛgjyā and other characteristics can be calculat- ed. This instrument can also be used for ascertaining the solar- lunar differences, and for fixing up the directions. It can be used for determining various heights and altitudes. The karttarī yantra is of the shape of a pair of scissors with two semi-circular blades, fastended to a string at the centre; at the centre is fixed a pin or a pole which casts shadows. Setting up of the Gnomon Here it would be interesting to describe the setting of a gnomon, which K.S. Shukla has given in details while commenting on the Mahābhāskarīya (IV. 1) : After having tested the level of the ground by means of water, draw a neat circle with a pair of compasses (karkaṭa) (At the centre of that circle, set up a vertical gnomon). The gnomon should be large, cylindrical, massive, and tested for its perpendicularity by means of four threads with plumbs (avalmbaka) tied to them.

1. सलिलेन समं साध्यं भ्रमेण वृत्तमवलम्बकेनोर्ध्वम् । तिर्यक् कर्णेनान्यैः कथितैश्च नव प्रवक्ष्यामि ॥ —BrSpSi XXII. 7.

SETTING UP OF THE GNOMON 335 Bhāskara I in his commentary on the Āryabhaṭīya tells us that there was a difference of opinion amongst astronomers in his time regarding the shape and size of gnomon (also called style). Some astronomers prescribed a gnomon with its one-third in the bottom of the shape of a prism on a square base (caturasra), one- third in the middle of the shape of a cow's tail (go-pucchākāra). and one-third at the top of the shape of a spear-head (Śūlākāra) and some others prescribed a square prismoidal (samacaturasra). gnomon. The followers of Āryabhaṭa I, he informs us, prescrib- ed the use of a broad (pṛthu), massive (guru), and large (dīrgha cylindrical gnomon, made of excellent timber and free from any hole, a scar or knot on its body. In the above stanza, Bhāskara I prescribes this last kind of gnomon : the other two kinds he proves in the commentary to be defective and so he rejects them. For getting the shadow'end easily and correctly the cylindri- cal gnomon was surmounted by a fine cylindrical iron or wooden nail fixed vertically at the centre of the upper end. The nail was taken to be longer than the radius of the gnomon, so that its shadow was always seen on the ground. Certain writers, Bhāskara I tells us in the commentary, prescribed a gnomon of half a cubit (=12 aṅgulas) in length and having twelve divisions. But according to Bhāskara I (although it was the usual custom) there was no such hard and fast rule. The gnomon could be of any length and any number of divisions. The gnomon should, however, be large enough, so that the rings of graduation on the gnomon may be clearly seen on the shadow. A broad and massive gnomon was preferred because it was unaffected by the wind. Brahmagupta describes gnomon which at the bottom is two aṅgulas wide, pointed as a needle., 12 aṅgulas in length, and full of holes from the basic circular part to the pointed extremity. (BrSpSi. XXII. 39). As regards testing the level of the ground, Bhāskara I observes : When there is no wind, place a jar (full) of water upon a tripod on the ground which has been made plane by means of eye or thread, and bore a (fine) hole (at the bottom of the jar) so that the water may

336 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS have continuous flow. Where the water falling on the ground spreads in a circle, there the ground is in perfect level; where the water accumulates after departing from the circle of water, it is low; and where the water does not reach, there it is high. (Bhāskara's Commentary on the Āryabhaṭīya, II. 13). After the ground was levelled, a prominently distinct circle was drawn on the ground as stated in the text (MBh. III. 1). In the time of Śaṅkaranārāyaṇa (869 A.D.), there it seems that all lines were drawn on the ground with sandal paste (candana- kṣodārdra). The above circle having been thus drawn and coated with sandal paste, another small concentric circle was drawn with the radius of the gnomon. The gnomon was then placed vertically with the periphery of its base in coincidence with that circle. The gnomon was thus set up exactly in the middle of the bigger circle. The verticality of the gnomon was tested by means of four plumb lines hung on the four sides of the gnomon Gnomon Used for Finding the Directions The rule in this connection has been described by Brahmagupta in BrSpSi. III. 1. The same rule in other words has been described by Bhāskara I in MBh. III. 2. In the Vāsanā Bhāṣya, Pṛthūdaka Svāmī describes the details of determining the directions. The level of the ground is ascertained by means of water and a gnomon of 12 aṅgulas is set up. Find out two points where the shadow of the gnomon enters into and passes out of the circle. Bhāskara prescribes drawing out a fish figure with these points. The thread line which goes through the mouth and tail of the fish figure indicates the north and south directions with respect to the gnomon. Brahmagupta says that if the Sun is on the eastern side, then where the shadow-point enters circle (in the forenoon) that point would be the west, and the point where it emerges out (in the afternoon) is the east. As the Sun moves along the ecliptic, its declination changes. By the the time the shadow moves between the forenoon and afternoon points as given above, the Sun , traverses some distance of the ecliptic and, so, theoretically speaking, its

GNOMON USED FOR FINDING THE DIRECTIONS 357 declination gets changed. It follows. therefore, that the East- West line in the above determination is not the true position of the actual East-West line. Brahmagupta (628 A. D.) was the first Hindu astronomer who prescribed the determination of the East-West line with proper allowance for the change in the Sun's declination. (Shukla) The details of the method intended by him have been supplied by his commentator Pṛthūdaka Svāmī (860 A. D.). Bhāskara and Brahmagupta both give another method of determining directions : (BrSpSi. III. 2 ; MBh. III. 3) : With the three points (at the ends of the three shadows of the gnomon) corresponding to (any three) different times (in the day), draw two fish-figures (each with two of the three points) in accordance with the usual method. From the point of intersection of the lines passing through the mouth and tail (of the two fish-figures), determine the north and south directions. (MBh. III. 3). Brahmagupta in his rule is more precise : The point where the lines passing through the two fish-figures, which are drawn by means of three shadow ends (of the gnomon), intersect each other is for places in the northern hemisphere, the south direction, (if the midday shadow falls to the north of the foot of the gnomon). If the midday shadow falls towards the south of the foot of the gnomon, it is the north direction. (BrSpSi. III. 2). This rule is obviously based on the assumption that the the locus of the end of the shadow of the gnomon is a circle. In tact the locus for places whose latitude is less than (90°—the obliquity of the ecliptic), this locus is a hyperbola. Brahmagupta has made numerous uses of gnomon. He and Bhāskara, for example, both give the rules for finding the latitude and colatiude and the zenith distance and altitude of the Sun by finding out the length of the shadow and the length of the gnomon (BrSpSi. III. 10 ; MBh. III. 5) ; also rule for the determination of the latitude with the help of the Sun's meridian zenith distance and declination (BrSpSi. III. 13 ; MBh.

338 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS III. 17) ; also rule for finding the Sun's altitude (BrSpSi. III. 27 ; MBh. III. 24) (The Sun's altitude for the night has been called by Brahmagupta as pātāla-śaṅku, BrSpSi. XV. 9). Golayantra or Armillary Sphere The first mention of the Golayantra or the armillary sphere is in the Āryabhaṭīya (Golapāda. 22)¹ which was a uniformly round circle made of wood or of bamboo and which was of uniform weight or density alround. It was levelled with mercury, oil or water. A śalākā or pin (or rod) was fixed in it in the south-north direction. Its description from the com- mentary Bhaṭadīpikā of Paramādiśvara is given here : A sphere of wood. uniformly round on all sides and with uniform density, and also light is made to revolve round an iron axis fixed north-south without friction (oil may be introduced to avoid friction). To the backside of the sphere, is fixed a nālaka full of water which has the length equal to the circumference of the sphere: and which has a hole at the bottom. Now a thread, connected to the hook of the wooden ball (on the top side) passing over another small ball (in the same axis of the wooden ball) is attached to the mercury lobe by its other end. The mercury lobe is placed on the level of water and water is allo- wed to flow through the bottom hole and with water mercury lobe also goes down. The time in which the above hook of ball comes to bottom (180°) is noted. The experiment is repeated with oil. The use of this mechanism is to revolve the ball by water or oil¹

1. काष्ठमयं समवृत्तं समन्ततस्सम गुरु लघु गोलम् । पारतं तैलजलैस्तं भ्रमयेत्स्वधिया च कालसमम् ॥ Arya. IV. 22 काष्ठमयं वंशादि काष्ठेन निर्मितं समवृत्तं सर्वतोवृत्तं समन्ततस्सम गुरुं सर्वावयवेषु समं गुरुत्वं यथा भवति तथा कृतं । लघुमगुरुं एवं भूतं गोलं कृत्वा पारतादिभिस्तं स्वधिया च कालसमं भ्रमयेत् । अयमर्थः । भूमिष्ठ दक्षिणोत्तरस्तम्भयोरुपरि गोलप्रोतायश्शलाकाया अग्रे स्थापयेत् । गोलदक्षिणोत्तरा- च्छिद्रे च तैलेन सिञ्चेत् यथा निस्सङ्गो गोलो भ्रमति । गोलस्यापरतो गोलपरिधिसंमित दैर्ध्यं साधश्छिद्रं जलपूर्णं नलकं निदध्यात् ततो गोलस्यापरस्वस्तिक कीलकं विधाय तस्मिन्सूत्रस्यैक मग्रं बद्ध्वादौ विषुवन्मण्डलपृष्ठेन प्राङ्मुखं नीत्वा तदग्रबद्धं पारतपूर्णमलावु जलपूर्णे नलके निदध्यात् ततो नलकस्याध [Cont. on Page 337]

GOLAYANTRA OR ARMILLARY SPHERE 339 In the Arabic epitome of the Almagest entitled Tahriru'l mejesti. the armillary sphere, Za ul halk, is thus described : Two equal circles are placed at right angles, the one representing the ecliptic, the other the solstitial colure. Two pins pass throught the poles of the ecliptic and two other pins are placed on the poles of the equator. On the first two pins are suspended a couple of circles moving, the one within, the other without, the first mentioned circles, and representing two seconda- ries of the ecliptic. On the two other pins a circle is placed, which encompasses the whole instrument, and within which the different circles turn; it repre- sents the meridian. Within the inner secondary of the ecliptic, a circle is fitted to it, in the same plane and turning in it. This is adapted to measure latitudes. To this internal circle, two apertures or sights, opposite to each other, and without its plane are adapted like the sights of an instrument for altitudes. The armil- lary sphere is complete when consisting of these six circles. The ecliptic and secondaries are to be gra- duated as minutely as may be practicable. It is best to place both secondaries, as by some directed, within the ecliptic (instead of placing one of them without it), that the complete revolution of the outer secondary may not be obstructed by the pins at the poles of the equator. The meridian likewise should be doubled, or made to consist of two circles; the external one gra- duated and the internal one moving within it. Thus the pole may be adjusted at its proper elevation above the horizon of any place. The instrument so constructed consists of seven circles. It is remarked that when the circle representing the meridian, is placed in the plane of the true meridian, so छिद्रं विकृतं कुर्यात् तेन जलं निस्स्रवति । नलकाच्च जलमधो गच्छति । तद्वशाच्च तदस्थमलावु पारतपूर्त्या गुरुत्वाज्जलेन सहाधो गच्छद् गोलं प्रत्यङ्मुखमाकर्षति । एवं त्रिंशद् घटिकाभिर्व्वसंमितं यथा जलं भवति गोलस्य चार्धं भ्रमति तथा स्वबुद्ध्या जलनिस्स्रावो योज्यः । इति । अमृतस्रावयोगेन कालभ्रमण साधनम् । गुणबीजसमाकृष्टं गोलयंत्रं प्रकल्पयेत् ॥ SuSi. XIII, 16-17

340 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS that it cuts the plane of the horizon at right angles, and one of the poles of the equator is elevated above the horizon conformably with the latitude of the place; then the motions of all the circles round the poles represent the motions of the universe. After rectifying the meridian, if it be wished to observe the Sun and Moon together, the outer secon- dary of the ecliptic must be made to intersect the ecliptic at the Sun's place for that time: and the solstitial colure must be moved until the place of intersection be opposite to the Sun. Both circles are thus adjusted to their true places; or if any object but the Sun, be observed, the colure is turned until the object be seen in its proper place, on that secondary referred to the ecliptic: the circle representing the ecliptic being at the same time in the plane of the true ecliptic and in its proper situation. Afterwards, the inner secondary is turned towards the Moon (or to any star intended to be observed), and the smaller circle within it, bearing the two sights is turned, until the Moon, (or to any star intended to be observed), and the smaller circle within it, bearing the two sights, is turned, until the Moon be seen in the line of the apertures. The intersection of the secondary circle and ecliptic is the place of the Moon in longitude: and the arc of the secondary, between the aperture and the ecliptic, is the latitude of the Moon on either side (North or South). (From Colebrooke's Miscellaneous Essays). The same instrument, as described by Montucla from the text of Ptolemy (1. 3. c. 2) consists of six circles: first a large circle representing the meridian; next four circles united toge- ther, representing the equator, ecliptic and two colures, and turning within the first circle on the poles of the equator, lastly a circle turning on the poles of the ecliptic, furnished with sights and nearly touching, on its concave side, the circumference of the ecliptic. The armillary sphere described by the Arabian epitomiser, differs, therefore; from Ptolemy's in omitting the equator and

GOLAYANTRA OR ARMILLARY SPHERE 341 equinoctial colure, and adding an inner secondary of the ecliptic, which as well as the meridian, is doubled. According to Lalande, the astrolobe of Ptolemy, from which Tycho Brahe derived his equatorial armillary, consisted only of four circles: two placed at right angles to represent the ecliptic and solstitial colure; a third turning on the poles of the ecliptic and serving to mark longitudes; and a fourth, within the other three, furnished with sights to observe celestial objects and measure their latitudes and longitudes. Whether the ancient Greeks had any more complicated instrument formed on similar principles, and applicable to astro- nomical observations, is perhaps uncertain. We have no detailed description of the instrument which Archimedes is said to have devised to represent the phenomena and motions of the heavenly bodies; nor any sufficient hint of its construction; nor does Cicero’s account of the sphere exhibited by Posidonius suggest a distinct notion of its structure. Among the Arabs, no addition is at present known to have been made to the armillary sphere; between the period when the Almagest was translated and the time of Alhazen, who wrote a treatise of optics, in which a more complicated instrument than that of Ptolemy, is described; Alhazen’s armillary sphere is stated to have been the prototype of Tycho Brahe’s; but neither the original treatise, nor the Latin translation of it, are procurable and one is therefore unable to ascertain whether the sphere, mentioned by the Arabian author, resembled that described by Indian astronomers. At all events, says Colebrooke, he is more modern than the oldest of the Hindu writers. Here we give the literal translation of the passage on armi- llary sphere or Golayantra occurring in the Sūrya-Siddhānta : Let the astronomer frame the surprising structure of the terrestial and celestial spheres. Having caused a wooden globe to be made (of such size) as he pleases; to represent the Earth : with a staff for the axis passing through the centre, and exceeding the globe at both ends; let him place the supporting hooks, as also the equinoctial circle. Three circles must be prepared, (divided for signs and degrees), the radius of which must agree with the

342 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS respective diurnal circles, in proportion to the equino- ctial : the three circles should be placed for the Ram (Meṣa) and following signs, respectively, at the proper declination in degrees N. or S. ; the same answer contrariwise for the Crab (Karkaṭa) and other signs. In like manner three circles are placed in the southern hemisphere, for the Balance (Tulā) and the rest, and contrariwise for Capricorn (Mṛga) and remaining signs. Circles are similarly placed on both hoops for the asterisms in both hemispheres, as also for Abhijit and for the Seven Ṛṣis, Agastya, Brahmahṛdaya and other stars. In the middle of all these circles is placed the equinoctial. At the intersection of that and supporting hoops, the distant from each other half the signs, the two equino- xes should be determined; and the two solstices, at the degrees of obliquity from the equinoctial; and the the places of the Ram (Meṣa) and the rest, in the order of the signs, should be adjusted by the strings of the curve. Another circle thus passing from equinox to equinox, is named the ecliptic; and by this path, the Sun illuminating worlds, for ever travels. The Moon and other planets are seen deviating from their nodes in the ecliptic to the extent of their respective greatest latitudes (within the zodiac).¹

1. भूलोकस्य रचनां कुर्यादाश्चर्यकारिणीम् । अभीष्टं पृथिवीगोलं कारयित्वा तु दारवम् ॥ दण्डं तन्मध्यगं मेरोरुभयत्र विनिर्गतम् । आधारकक्षाद्वितयं कक्ष्यां वैषुवतीं तथा ॥ भगणांगुलैः कार्या दलितास्तिस्र एव ताः । स्वाहोरात्रार्धकर्णैश्च तत्प्रमाणानुपातः ॥ क्रान्तिविक्षेपभागैश्च दलिता दक्षिणोत्तरा । स्वैस्स्वैरपक्रमैः कार्या मेषाद् नामपक्रमात् । कक्षाः प्रकल्पयेत्ताश्च कर्क्यादीनां विपर्ययात् । तद्वत्तिस्रस्तुलादीनां मृगादीनां विलोमतः ॥ याम्यगोलाश्रिताः कुर्यात् कक्ष्यावारद्वयोपरि । याम्योद्ग्भाग संस्थानां भानामभिजितस्तथा ॥ सप्तर्षीणामगस्त्यस्य ब्रह्महृदीनां प्रकल्पयेत् । मध्ये वैषुवती कक्ष्या सर्वासामेव संस्थिता । [Cont. on page 341]

HOW TO OBSERVE PLACES OF STARS 343 The author of the Sūrya-Siddhānta then proceeds to notice the relation of the great circles before mentioned to the horizon, and observes that whatever place be assumed for the apex of the sphere, the middle of the heaven for that place is its horizon. He concludes by showing, that the instrument may be made to revolve with regularity, by means of a current of water; and hints, that the appearance of spontaneous motion may be given, by a concealed mechanism, for which quicksilver is to be employed. There is a hint of secrecy also in one of the lines, and it has, therefore, been stated that the construction and the mechanism of working should be learnt under the guidance of a teacher. How to Observe Places of Stars Details are not available in this connection. The Sūrya- Siddhānta only hints "that the astronomer should frame a sphere and examine the apparent longitude and latitude (sphuṭavikṣepa and sphuṭadhruvaka)". The commentators, however, describe the manner of making the observation. They direct a spherical instrument (Golayantra) to be constructed as described above. This instrument is very much similar to the armillary sphere. An additional circle graduated for degrees and minutes, is direct- ed to be suspended on the pins of the axis as pivots. It is named as Vedhavalaya or intersecting circle, and appears to be a circle of declination. After noticing this addition to the instrument, the instructions proceed to the rectifying of the Golayantra or armillary sphere which is said to be placed, so that the axis shall point to the pole: and the horizon be true by a water level. The instrument being thus placed, the observer is instructed to look at the star Revatī through a sight fitted to an orifice at the centre of the sphere; and having found the star, to adjust by it the end of the sign Pisces on the ecliptic. The observer is then to look through the sight, at the yoga star of Aśvinī, or at तदाधारयुतेः भार्धमयने विषुवद्वये । अयनाद्वयने चैव कक्ष्या तिर्यक्संस्थाऽपरा ॥ क्रान्तिसंज्ञा तया सूर्यः सदा सदापर्य्येति भासयन् । चन्द्राद्याश्च स्वरकैः पातैरपमण्डलमाश्रितैः ॥ ततोऽपकृष्टादृश्यन्ते विक्षेपाग्रे स्वपक्रमात् । SuSi. XIII. 3-12

344 BRAHMAGUPTA AND ASTRONOMICAL INSTRUMENTS some other proposed object; and to bring the moveable circle of declination over it. The distance in degrees, from the inter- section of this circle and ecliptic, to the end of Mīna or Pisces, is its longitude (dhruvaka) in degrees; and the number of degrees on the moveable circle of declination, from the same intersection to the place of the star, is its latitude (vikṣepa) North or South. The commentators have rightly remarked that the 'latitude so found is sphuṭa or apparent, being the place intercepted bet- ween the star and the ecliptic, on a circle passing through the poles; but the true latitude (asphuṭa) is found on a circle hung upon the poles of the celestial sphere as directed in another place". (From Colebrooke's Paper on the Indian and Arabian Divisions of the Zodiac. Miscellaneous Essays, Vol. II, 324-326). For the details of the Golayantra, readers are requested to refer to the description in the Siddhānta-Śiromaṇi of Bhās- kara II. —: o :— Reference Brahmagupta : yantrādhyāya in —BrSpSi. K.S. Shukla : The Mahābhāskarīya. Bhāskara I : Commentary on the Āryabhaṭīya. Paramādīśvara : Bhaṭadīpikā, a commentary on the Āryabhaṭīya, H.T. Colebrooke : Miscellaneous Essays, Vol. II., 1872.

१ ॥ श्रीगणेशायनमः ॥ अथ ब्राह्मस्फुट सिद्धान्तस्य पूर्वावशाध्यायां मध्यमाधिकारः जयति प्रणतसुरासुरकिरीटरत्नप्रभाछुरितपादः । कर्ता जगदुत्पत्तिस्थितिविलयानां महादेवः ॥१॥ ब्रह्मणोक्तं ग्रहगणितं महता कालेन यत् खिलीभूतम् । अभिधीयते स्फुटं तज्जिष्णुसुतब्रह्मगुप्तेन ॥ २ ॥ ध्रुवताराप्रतिबद्धं ज्योतिश्चक्रं प्रतिक्षणगमादौ । पौष्णाश्विन्यन्तस्थैः সহ ग्रहैर्ब्रह्मणा सृष्टम् ॥ ३ ॥ चैत्रसितादेरुदयाद्भानोर्दिनमासवर्ष युगकल्पाः । सृष्ट्यादौ लंकायां समं प्रवृत्ता दिनेऽर्कस्य ॥ ४ ॥ प्राणैर्विनाडिकाक्षैः षड्भिर्घटिका विनाडिका षष्ट्या । घटिका षष्ट्या दिवसो दिवसानां त्रिशता मासाः ॥ ५ ॥ (ग) १. श्री नमः ।। परमात्मने । श्री रामा

( २ ) मासा^१ द्वादशवर्षं विकलालिप्तांशराशिभगणांतः^२ क्षेत्रविभागस्तुल्यः कालेन विनाडिकाद्येन^३ ॥ ६ ॥ स्वचतुष्टयरदवेदा^१ रविवर्षाणां चतुर्युगं भवति । ४३२००००^२ सन्ध्या सन्ध्यांशै^३ सह चत्वारि पृथक्कृतादीनि ॥७॥ युगदशभागो गुणितः कृतं चतुर्भि^२स्त्रिभिर्गुणास्त्रेता^३ । द्विगुणो^४ द्वापरमेकेन संगुणः कलियुगं भवति ॥ ८ ॥ १७२८०००।१२९६०००।८३४०००^५।४३२००० युगपादानार्य^१भटश्चत्वारि समानि^२ कृतयुगादीनि । १०८००००^३ यदभिहितवान्^४ न तेषां स्मृत्युक्तसमानमेकमपि^५ ॥ ६ ॥

६. (ख) १. मास for (मासा)
[L1

( ३ ) मनुरेक सप्ततियुगः² कल्पो³ मनवश्चतुर्दश मनूनाम् । आद्यंतरांतसंधिषु⁵ कृतकालो⁴ऽस्माद्द्युगसहस्रम्⁶ ॥ १० ॥ ४३२०००००००¹ आद्यंतरातसंधिषु⁴ कल्पमनूनां⁵ कृताब्दसमकालम्⁶ । नेच्छंति ये षडूनं¹ तेषां कल्पो युगसहस्रम्² ॥ ११ ॥ ४२६४०८०००० मनुसंधियुग⁵मिच्छत्यार्यभटस्तन्मनुर्यतस्त्वयुगः । कल्पश्चतुर्युगानां² सहस्रमष्टाधिकं³ तस्य ॥१२॥ ४३५४५६००००⁴

१०. (घ) २. ४३२०००००० for (४३२०००००००) (क) संख्या लुप्त है। टीका में
अंकित है।
(ग) २ + ७१ + (च)
३ कल्पो नवचतुर्दश १४ (च) for (कल्पोमनवश्चतुर्दश

( ४ ) युग्मन्वन्तरकल्पाः कालपरिच्छेदकाः² स्मृतावुक्ताः । यस्मान्न¹ रोमके ते स्मृति³ बाह्यो रोमकस्तस्मात् ॥ १३ ॥ कालर्क्ष¹देशयोगाद्भू²यो ग्रहमन्दशीघ्रपातानाम्⁵ । कल्पेन यतो योगस्ततः³ स्फुटं ग्रहयुगं⁴ कल्पः ॥ १४ ॥ कल्पेऽर्कबुध¹ सितानां भगणाः⁴ शून्यानि सप्तरदवेदाः । प्राग्व्रजता² कुजगुरुशनिशीघ्रोच्चानां स्वकक्षासु⁵ ॥ १५ ॥ ४३२०००००००³ १३. (घ) १. न रोमके ते (च) न रौनुके ते for (न्न रोमके ते) (ख) २. कलापरिच्छेदकाः for (कालपरिच्छेदकाः) (च) ३. वि० इस प्रति में “……परिच्छे” के पश्चात् लेखक ने भूल से फिर “मनुरेक सप्तति युग” से लिखना आरंभ कर दिया । इस प्रकार ५½ पंक्तियां पुनः लिखी गईं ! उसके पश्चात् फिर क्रमशः लिखता गया । ४. स्मृतबाह्यो for (स्मृतिबाह्यो) १४. (घ) १. कालक्ष्यं (ख) कालक्ष (च) कालक्ष्यं for (कालर्क्ष) २. दूद्भूयो (च) for (द्भूयो) ३. तत (च) for (ततः) ४. ग्रहयुगकल्पः (ग) ग्रहयुतं कल्पः for (ग्रहयुगं कल्पः) (ग) ५. ग्रहशीघ्रमंद पातानां (क) (ख) for (ग्रहमन्द शीघ्रपातानां) (च) ४. ग्रहयुग कल्पः for (ग्रहयुगं कल्प.) १५. (घ) १. बुध (च) for (बुध) २. व्रजनां (ग) व्रजती (ख) व्रजतां for (व्रजता) (क) २. प्राग्व्रजतां for (प्राग्व्रजता) ३. संख्या लुप्त है । टीका में अंकित है । (ख) ४. भगराः for (भगणाः) ५. सा for (सु) (च) स कक्षासु for (स्व कक्षासु)

( ५ ) ³ ⁴ पंचांबराणि गुणरामंपंच सप्तस्वरेषवः शशिनः । १ ५७७५३३००००० ५ भौमस्य द्वियमशराष्टपक्षवसुसरत्नवद्वियमाः ॥ १६ ॥ २ २२६६२८२५२२ १ ४ ६३६६६८६८४ ३ २ कृतवसुनवःष्टनवनव षडूनवागेन्द्वो ज्ञशीघ्रस्य । ५ १७६३६६६८६८४ ३ जीवस्य शरेष्वद्वि षड्यक्षि द्विकृतसप्तरामाः ॥ १७ ॥ ६ ३६४२२६४५५ ६ सितशीघ्रस्य यमलंगो वेदनवाष्टाग्नि पक्षयमखनगाः । ४ ७०२२३८६४६२ २ ३ अष्टनवपक्षमुनिरसशररसमनवोऽर्कपुत्रस्य ॥ १८ ॥ ५ १४६५५६७२६८

१६. (क) १. संख्या लुप्त है । टीका में अंकित है ।
२. संख्या लुप्त है । टीका में अंकित है ।
(ख) ३. पंच वाराणि for (पंचांबराणि)
४. गुणरामं for (गुणराम)
(च) ५. वृद्धि for (वद्वि)
१७. (घ) १. १७६३६६६८६४ for (१७६३६६६८६८४) (च)
२. वांगे (ग) षड्ित्रनव (क) षट्त्रिन (ख) षट्त्रिंनवा for (षडूनवागेन्द्वो)
३. पद्यक्ष (ग) षट्द्वयक्षि (क) पट्पक्ष (ख) पट्ट्यक्ष for (षड्यक्षि)
(ग) ४. + २२६६८२८५२२ (क) ‘ग’ में अंकित संख्या १६वें श्लोक की टीका के
अन्त में है ।
(क) ५. संख्या मूल में नहीं, टीका में अंकित है ।
६. संख्या मूल में नहीं, टीका में अंकित है ।
(ख) ७. वसू for (वसु)
(च) २. षड्नवागेदवो for (षडूनवागेवो) ३ पद्यक्ष for (पड्यक्षि)
१८. (घ) १. लागो for (लगो) (च) यमलागो for (यमलंगो)
२. यक्ष for (पक्ष)
३. ऽक्र्कं (ग) ऽर्क्क for (ऽर्क)
(क) ४. मूल में संख्या लुप्त है। टीका में अंकित है।
५, मूल में संख्या लुप्त है। टीका में अंकित है।
(च) ६. चाष्टाग्नि for (वाष्टाग्नि)

( ६ ) खाष्टाब्धयो^१ ४८० वसुशर^७ वसुपंचखचन्द्रवसुवसुसमुद्राः । ४८८१०५८५८ द्विनवयमा^४ २९२ द्वित्रिगुणा^५ ३३२ शरेषुवसवं^६ ८५५^८ स्त्रिपञ्चरसाः^२ । १६ ।। ६५३ ।। १६ ।। शशिवेदा ४१ मन्दानामर्कादीनां^१ विलोमपातान्तम् वसुरसरुद्रेन्द्रगुण द्वित्रियमा^२ २३२३१११६८^३ सप्तरसपक्षाः २६७^४ ।। २० ।।

१६. (घ) १. षष्ठौदधया (ग) खाष्टोदधयो for (खाष्टाब्धयो)
२. स्त्रिपंचसाः (च) स्त्रिपंचरसाः (स्त्रिपञ्चरसा)
(वि० चिन्हित संख्याएं यहां उपलब्ध नहीं हैं)
(ग) ४. द्विनवयमाः for (द्विनवयमा)
५. द्वित्रिगुणाः for (द्वित्रिगुणा