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

144 GREEK AND INDIAN METHODS It is almost needless to say that neither in the method nor in the rules is there any agreement between the Indian and Greek spherical astronomy in the solution of this problem. Kaye’s view¹ As to Kaye, it appears that he has not been able to find a method in the translation of the Sūyrasiddhānta by Burgess. The figures of his paper referred to before do not show the “Akṣakṣet- ras” even in their projections on the meridian place. He refers to Braunmühl’s History of Trigonometry but does not appear to have been able to follow him in his “Methode der indischen Trigonometrie.” Kaye, however, is not slow in belittling Indian trigonometry when he says :—The Indian astronomers employed the sine function principally and the versed sine occasionally ; they never employed the tangent function; and generally, but not always, preferred to employ the sine of the complementary angle rather than the cosine functions.’’ It is evident that Kaye never understood the meaning of the Indian functions of ‘sine’ and ‘cosine.’ These functions are fully explained by Bhāskara² when he says :— “Of that point the dis- tance from the east-west line is the sine and the distance of the point from the north-south line is the “cosine”. [Fig. 14: Circle with points N, S, E, W; P₁, P₂, P₃, P₄; N₁, N₂, N₃, N₄; M₁, M₂, M₃, M₄, A] Fig. 14 In (Fig. 14), of the arc AP₁, P₁M₁ is the “sine” and P₁N₁ is the “cosine” of AP₂, P₂M₂ is the “sine” and P₂N₂ is the “cosine”; of AP₃, P₃M₃ is the “sine” and P₃N₃ is the “cosine”; etc. It is evident that a better definition of these fun- ctions was never given. We have thus seen that some of the solutions of Āryabhaṭa

1. J. A. S. B., N. S., XV, p. 154.
2. तस्य बिन्दोः प्राच्य-परायाश्च यदन्तरं सादोर्ज्या । बिन्दोर्याम्योत्तरायाश्च यदन्तरं सा कोटिज्या ॥ —(Bhāskara, Grahagaṇita, commentary. II. 88-21)

KAYE'S VIEW 145 are imperfect, of Brahmagupta the solutions are more accurate, while those of Bhāskara are generally mathematically correct. The date of the scientific ancient Indian Astronomy is indeed 499 A. D., while that of the Syntaxis is about 150 A. D. It is by these shortcomings and differences in the methods, new ideas (e.g., the idea of the differential calculus)* and the like, that we can safely say that Indian Astronomy in its scientific form, although of a later date than the "Syntaxis" of Ptolemy, is origi- nal and not borrowed from foreign source. There is evidence that some crude form of Greek astronomy was transmitted to India and went by the name of the "Romaka" or the "Pauliśa" Siddhānta, prior to the time of Āryabhaṭa but our great Indian astronomers, Āryabhaṭa with his pupils, Varāha-mihira and Brahmagupta, had to construct a new science altogether. (This Chapter is almost a reproduction of the paper by P. C. Sengupta, as acknowledged earlier). —: o :— Reference P.C. Sengupta : The Khaṇḍakhādyaka, 1934

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CHAPTER VII Epicyclic Theory of Ancient Indians We shall give here some details of the Indian concepts regarding the motion of planets or wandering bodies among the stars. The Vedāṅga Jyotiṣa (1400 B. C. or earlier) does not speak of this, A comparison of the astronomical constants of the Greek and the ancient Indian systems, points unmistakably to the conclusion that the Indian constants as determined by Āryabhaṭa I and his successors, are almost in all cases different from those of Greeks. Indian astronomers were highly original in their con- cepts and treatment. The originality of Āryabhaṭa I and other astronomers would be seen from what we are discribing below. Apparent Motions of the Sun and Moon We have the following passages from Āryabhaṭa : All planets move in eccentrics to their orbits at the mean rates of angular motion, in the direction of the signs of the zodiac from their apogees (or aphelia) and in the opposite directions from their Śīghroccas. The eccentric circles of planets are equal to their concentrics and the centre of the eccentric is removed from the centre of the Earth. The distance between the centre of the Earth and the centre of the eccentric is equal to the radius of the planet's epicycle; on the circumference (whether of the epicycle or of the eccentric) the planet undoubtedly moves with the mean motion. Here the central idea was that undou- btedly planets moved unifomly in circles round the Earth; if the motion appeared to be variable, it was due to the fact that the centres of such circle (i. e. the con- centric circles) did not coincide with the centre of the Earth. Fig. 15

148 EPICYCLIC THEORY OF ANCIENT INDIANS Let E represent the centre of the Earth(Fig.15).APM the Sun's circular orbit or concentric; let A and P be the apogee and the perigee respectively. From EA, cut off EC equal to the radius of the Sun's epicycle. With centre C and radius equal to EA describe the eccentric A'P'S cutting AP and AP produced at P' and A'. Here A' and P' are the real apogee and perigee of the Sun's orbit. Let PM and P'S be any two equal arcs measured from P and P'. The idea is that the mean planet M and the apparent Sun S move simultaneously from P and P' in the counterclockwise direction along the concentric and the eccentric circles. They move with the same angular motion and arrive simultaneously at M and S. Here EM and CS are parallel and equal, hence MS is also equal and parallel to EC. Let SH be drawn perpendicular to EM. The angle PEM is the mean anomaly and the angle P'ES the true anomaly; the angle SEM is the equation of the centre, is readily seen to be plus (+) from P' to A' and minus (—) from A' to P'. Thus as regards the character of the equation, the eccentric circle is quite right. We now turn to exmine how far it is true as to the amount. Let the angle SEM denoted by E and the angle ∠PEM =∠P'CS=θ; EP=CP'=a; EC=MS=p, then tan E = SH / HE = p sin θ / (a - p cos θ) ∴ E = (p / a) sin θ - (p² / 2a²) sin 2θ + (p³ / 3a³) sin 3 θ......... Now the true value of E in elliptic motion is given by E = ( 2e - e³/4 ) sin θ + (5/4) e² sin 2θ + (13e³ / 12) sin 3θ*;... It we now put p / a = 2e - e³/4, as a first approximation p / a = 2e. Hence p² / 2a² = 2e², which is greater than 5/4 e² by 3/4 e². In the case of the Sun if the value of p be correctly taken the error in the coefficient of the second term becomes +3'; similarly in the case of the Moon, the corresponding error becomes +8'. *Godfray's Astronomy, p. 149.

APPARENT MOTION OF THE SUN AND MOON 149 Again if p/a=2e, what is the centre of the eccentric circle is the empty focus of the ellipse or that the ancient astronomers practically took the planets to be moving with uniform angular motion round the empty focus. This was not a bad approxi- mation. Also ES=r=EH approximately, ∴ r=a ( 1 - p/a cos θ ) but in the elliptic motion r=a (1-e cos θ).* Hence the error is not very considerable here also. This is the way in which the ancient astronomers, both Greek and Hindu, sought to explain the inequalities in the motion of the Sun and the Moon. In the case of the Moon, these astro- nomers took the coefficient 2e - e³/4 = 300' nearly; the modern value is 377' nearly. The reason for this has been pointed out to be that the Moon was observed correctly only at times of eclipses. At the eclipses of syzygies, the evection term of the Moon's equa- tion diminishes (numerically) the principal ecliptic term by about 76'. We have thus far explained the idea of planetary motion of the ancients under the eccentric circle costruction. The same, however, is explained under the epicyclic construction. Let AMP be the circular orbit of the Sun, having E the centre of the earth for the centre. (Fig. 16) Let the diameter AEP be the apse line. A the apogee and P the perigee. Let M be the mean position of the Sun in the orbit. With M as the centre, describe the epicycle UNS. Let EM cut the epicycle at N and U. Now the construction for finding S the apparent Sun is thus given:— Fig. 16 Make ∠UMS= ∠MEA, the arc US is measured clockwise whereas the arc A to M is measured counterclockwise. From this construction MS is parallel to EA. If EC be measured equal to MS, the radius of the epicycle, along EA to-

• Godfray's Astronomy, p. 149.

150 EPICYCLIC THEORY OF ANCIENT INDIANS wards the apogee, then CS is a constant length and C is a fixed point. Hence the locus of S is an equal circle with the centre at C. Thus both the eccentric, and the epicycle and the concentric combined, led to the same position. It was thus usual to explain the planetary motion under both the assumed constructions; and both gave the position for a planet. The eccentric circle construction appears to be the earlier in the history of astronomy and the latter was later. If the former construction can be traced to Apollonius of Perga who did so much to develop the "conic sections" as science, the reason why he preferred the eccentric circle to the ellipse, appears to be that either that this planetary construction was always deep-rooted in the minds of men or that he was carried by the idea that "the circle was the most perfect curve." We are inclined to the view that the eccentric circle idea was transmitted from Babylonia to Greece. We now pass on to consider the Indian construction for the possition of superior and inferior planets. Superior Planets With regard to the five planets, Mercury, Venus, Mars, Jupiter and Saturn, the Indian astronomers give only one constru- ction for finding the apparent geocentric position. Each of these "star planets" is conceived as having twofold planetary inequalities : (i) the inequality of apsis, (ii) the inequality of the śīghra. With regard to the superior planets, the śīghra apogee or the śīghrocca coincided with the mean position of the Sun. As Varāhamihira observed, of the other planets beginning with Mars, the Sun is the so-called śīghra. (PSi. XVII. 1) Let AMSP be the concentric of which the centre E is the same as that of the earth(Fig.17); A'M₁P' the eccentric circle of apsis of a superior planet, of which the centre is C;A,M,S,P be respectively the apogee, the Mars .planet, the direction of the śīghra and the perigee of the concentric; A',M₁,P' be the apogee, the planet as corrected by the equation of apsis, P' the perigee in the eccentric. The arc Fig. 17

INFERIOR PLANETS 151 AM=arc A'M₁; MM₁ is parallel and equal to EC. As used be- fore, both the concentric and the eccentric are of the same radius. Here the mean planet M in the concentric is taken to be, deflected to M₁ due to the true motion in the eccentric circle. Join EM₁ cutting the concentric at M₂. Now let ES be joined and let S' be taken along ES, such that ES' / ES = (Śīghra periphery of the planet in degrees) / 360 = (Sun's mean distance from the Earth) / (Planets mean distance from the Sun or the Earth); ES' thus determined is called the radius of the śīghra epicycle of the superior planet. With S' as the centre and the radius equal to ES or EA describe another circle which is called the śīghra eccentric cutting ES produced at S''. Now measure the arc S''M₃ in the concen- tric=SM₂ in the concentric. The apparent superior planet is seen in the direction EM₃ from the Earth. This is the construc- tion used in Hindu astronomy calculating the geocentric longitude of any star planet. It is evident in the case of a superior planet that the eccen- tric having S' for the centre and whose radius=EA=R the standard radius for any circular orbit, is the mean orbit of the planet and S' the mean position of the Sun. In other words, in the case of a superior planet, the śīghra eccentric represents the mean orbit round the Sun. If the parallelogram CES'C' be constructed, then an equal circle described with C' as the centre is the apparent eccentric orbit of the superior planet. In the actual method of calculating the geocentric longitude of a “star planet” there are four operations given, the first two of which have the effect of changing the arc MA or rather the point A. The last two operations relate to the two displacements MM₁ and M₂M₃. We have here followed solely the construction of the eccentric circles; the same geocentric position of a superior planet could be equally well obtained by the epicyclic cons- truction. In describing the constructions for finding the position, of an inferior planet we shall follow the epicyclic construction only Inferior Planets Let E be the centre of the Earth (Fig. 18), AMS the orbit of

152 EPICYCLIC THEORY OF ANCIENT INDIANS a mean inferior planet or the mean Sun. EA the direction of the apogee of apsis and ES that of the śīghra. The inequality of the apsis takes the mean geocentric planet from M to M₁, such that MM₁ is parallel to EA. Let EM₁ be joined cutting the con- centric at M₂ ; M₂ is taken as the centre of the śīghra epicycle or the real circular orbit in which the appa- rent planet moves. Fig. 18 With M₂ as the centre and the radius of the inferior pla- nents' śīghra epicycle as radius, describe the circle NVU which is here the śīghra epicycle or the real circular orbit. In it draw the radius M₂V parallel to ES; then EV is the geocentric position of the inferior planet. Here the first displacement MM₁ is due to the inequality of apsis and is for finding the position of M₂ the centre of the real circular orbit. The idea was that the apparent planet moved in a circular orbit of which the centre was very near the mean position of the Sun, the first operation in this construction was calculated to determine the centre of this so-called circular orbit of an inferior planet. The śīghra of an inferior planet moves round the Earth at the same mean rate in which the inferior planet moves round the Sun; hence the line ES in this figure is always parallel to the line joining the Sun to the mean heliocentric inferior planet, and in our construction, it is parallel to M₂V. This in brief is an outline of the Indian idea of planetary motion as taught by Āryabhaṭa I, Brahmagupta and Bhāskara II. — : o : — Reference P.C. Sengupta : The Khaṇḍakhādyaka, 1934.

CHAPTER VIII Brahmagupta and Arithmetic Scope of Gaṇita The word Gaṇita means the science of calculation. The term occurs in the Vedāṅga Jyautiṣa (c. 1200 B. C.) : Just as the crest is to the peacocks, and just as the head- gem is to the snakes, so the Gaṇita among the Vedāṅga Śāstras stands at the head.¹ (VJ. 4) In the ancient Buddhistic literature, we find mention of three classes of gaṇita : (i) mudrā (finger arithmetic). (ii) gaṇana (mental arithmetic), and (iii) saṅkhyāna (higher arith- metic in general). In the Brāhmasphuṭasiddhānta, Brahmagupta uses the word gaṇita in the sense of entire calculations. His gaṇitādhyāya (Chapter XII) includes ; (i) Miśraka (mixtures). (ii) Śreḍhī (series). (iii) Kṣetra plane figures), (iv) Vṛtta-kṣetra (circles), (v) Khāta (ex- cavations), (vi) Citi (piles of bricks), (vii) Krākacika (sawn pieces of timber), (viii) Rāśi (heaps or mounds of grain, and (ix) Chāya (shadow). Brahmagupta also uses the term Dhūlīkarma (literally meaning “ʇsnpwork”) for higher mathematics : The one learned man who knows the dhūlīkarma or the science of mathematics as propounded by Brahmagupta would far excell them in learning who are taught the calculations according to Āryabhaṭa, Viṣṇucandra and others.² In these ten chapters of the Brahmasiddhānta has been given the dhūlīkarma or the science of entire calculations

1. यथा शिखा मयूराणां नागानां मणयो यथा । तद् वद् वेदांग-शास्त्राणां गणितं मूर्धनि स्थितम् ॥ VJ. 4.
2. नावायों ज्ञातेरपि तन्त्रैरार्यभटविष्णुचन्द्राद्यैः । यो ब्रह्म धूलिकर्मविदाचार्यत्वं भवति तस्य ॥ —BrSpSi. X. 62.

154 BRAHMAGUPTA AND ARITHMETIC which is faultless.¹ This science of dhūlīkarma has not been imparted by great teachers for blasphemy. One who would be using it for this purpose would lose all good name.² Brahmagupta uses the term gaṇita only for those calculations which are of arithmetical in nature. The science of algebra, the foundations of which was laid by Āryabhaṭa I, was named as kuṭṭaka or kuṭṭākāra by Āryabhaṭa, and in the Brāhmasphuṭa- siddhānta also it is separately dealt with under Kuṭṭādhyāya or kuṭṭakādhāya (Chapter XVIII). Later on the term bījagaṇita was specifically given to the science of algebra. The Kuṭṭādhyāya of the Brāhmasphuṭasiddhānta deals with the (i) concept of kuṭṭaka (pulveriser), addition of positive and negative as well as zero quantities, equations in one unknown (eka-varṇa samīkaraṇa), equations in several unknowns (aneka- varṇa samīkaraṇa), equations involving products of unknowns (bhāvita) and quadratic equations (varga-prakṛtiḥ) (Chapter XVIII of the Brāhmasphuṭasiddhānta). Āryabhaṭa, Bhāskara and Brahmagupta use Place Value Notations. In Europe the first definite traces of the place-value nume- rals are found in the tenth and eleventh centuries, but the numerals came into general use in mathematical text books only in the seventeenth century. In India, however, Āryabhaṭa I (499), Bhāskara I (522), Lalla (c. 598) and Brahmagupta (628) all use the place value numerals. There is no trace of any other system in their works. Perhaps in this country we had the place value system as early as 200 B.C. if not earlier. The use of a symbol for zero is found in Piṅgala's Chandaḥ Sūtra (perhaps of 200 B.C.). In literature, we have an indication of the place value from about 100 B.C. and later in the Purāṇas from the second to the fourth century A.D. The Bakhāśalī Manuscript (perhaps of 200 A.D.) uses the place-value notations. The earliest use of the place value principle with the letter numerals

1. ग्रहयोगोत्र ग्रहयुतिरार्यांत्रिशतियुताष्टसप्तत्या । अध्याब्दैरभिभू लिकर्म वोच्चैर्विना भङ्गै ॥ —BrSpSi. X. 66.
2. गुरुणा न धूलिकर्म प्रतिकंचुककारिणे प्रदातव्यम् । दत्तं सुकृतप्रणाशं कुरुते प्रतिकंचुकं यस्य ॥ —BrSpSi. X. 67

OPERATIONS AND DETERMINATIONS IN PĀṬĪGAṆITA 155 is found in the works of Bhāskara I about the beginning of the sixth century A.D. Thus for 3179, the expressive words are _Navādrirūpāgni_¹ (nava 9, adri 7. rupa 1 and agni 3). Similarly in the Brāhmasphuṭasiddhānta, for a large number like 2296828522, the expressive terms are DVIYAMAŚARĀṢṬAPAKṢAVASŪRA- SANAVADVIYAMĀḤ(Dviyama two twos 22, Śara 5, aṣṭa 8, pakṣa 2, vasū 8, rasa 6. nava 9, dviyamāḥ 22).² Such usages are to be found in all works, which clearly state the place value concept was popular as a routine. From India, this system reached Arabia. During the reign of the Khalif Al-Mansur (753-774 A.D.) there came embassies from Sindh to Baghdad, and among them were scholars, who brought along with them several works on mathe- matics including the Brāhmasphuṭasiddhānta and the Khaṇḍa- khādyaka of Brahmagupta: With the help of these scholars, Al- fazari, perhaps also Yakub ibn Tarik, translated them into Arabic. Both works were largely used and exercised great influence on Arab mathematics. It was on that occasion that the Arabs first became acquainted with a scientific system of astronomy. It is acceptable to all writers on the subject that it was at that time that the Hindu numerals were first definitely introduced amongst the Arabs. Arabs at first adopted the ghobar form of num- erals which they had already obtained (but without zero) from the Alexandrians or from the Syrians. This they continued for about two centuries, but since they were not suited to their right- to-left script, they gave them up and adopted the more convenient ones. For a detailed discussion on how numerals went to the west from India and spread in Europe one is referred to this dis- cussion in the History of Hindu Mathematics. Part I by Datta and Singh (1935, Single volume Edition, 1962, pp. 83-104). It is remarkable that Brahmagupta's works like the Brāhma- sphuṭasiddhānta and the Khaṇḍakhādyaka became instrumental in the spread of the place-value notation in the neighouring coun- tries of the Middle East, and from their this system spread into Europe. Operations and Determinations in Pāṭīgaṇita The word Pātīgaṇita is a compound formed from the words pāṭī, meaning 'board', and gaṇita, meaning 'science of calculation',

1. MBh. 1. 4;
2. BrSpSi 1-16.

156 BRAHMAGUPTA AND ARITHMETIC hence it means the science of calculation which requires the uses of writing material (the board). The word pāṭī is not Sanskrit (it originated in the non-Sanskrit literature in India); the oldest term in Sanskrit for the board is Phalaka or paṭṭa. However this term got currency in the Sanskrit literature also about the beginning of the seventh century. Brahmagupta does not use the term pāṭīgaṇita: he favours the use of the term dhūlikarma or writing figures on dust spread on a board or on the ground. The word pāṭīgaṇita was translated into Arabic as ilm-hisab-al-takht (calculation on board) and the word dhūlikarma as hisab-al-ghobār (calculation on dust). Brahmagupta, in the very first verse in the Chapter XII (Gaṇitādhyāya) refers to twenty operations (parikarma) and eight determinations : He who distinctly and severally knows the twenty logistics, addition etc.. and the eight determinations (vyavahāra) inculding (measurement by) shadow is a gaṇaka (mathe- matician).¹ The commentators have given the list of these logistics (parikarma) and determinations (vyavahāra) as follows; (A) Parikarma or logistics

1. Saṁkalitam (addition)
2. Vyavakalitam (subtraction)
3. Gaṇanam (multiplication)
4. Bhāgahāraḥ (division)
5. Vargaḥ (square)
6. Vargamūlam (square-root)
7. Ghanaḥ (cube)
8. Ghanamūlam (cube root)
9. 13. Five standard forms of fractions (Pañca-jāti)
10. Trairāśikam (the rule of three)
11. Vysta-trairāśikam (the inverse rule of three)
12. Pañca-rāśikam (the rule of five)
13. Sapta-rāśikam (the rule of seven)
14. Nava-rāśikam (the rule of n ine)

1. परिकर्म विंशतिं यः संकलिताद्यं पृथक् विजानाति । अष्टौ च व्यवहारान् छायान्तान् भवति गणकः सः ॥ —BrSpSi. XII. 1

MULTIPLICATION 157 19. Ekādaśa-rāśikam (the rule of eleven) 20. Bhāṇḍa-pratibhāṇḍam (barter and exchange) (B) Vyavahāra or determinations

1. Miśrakaḥ (mixture)
2. Śreḍhī (progession or series)
3. Kṣetram (plane figures)
4. Khātam (excavation)
5. Citiḥ (stock)
6. . Krākacikaḥ (saw)
7. Rāśiḥ (mound)
8. Chāyā (shadow) Of the operations enlisted here, the first eight have been considered fundamental by later writers as Mahāvīra. The opera- tions of duplation and mediation (doubling and halving) were considered fundamental by Arabs, Greeks and Egyptians; since they were not familiar with the place-value system. Mathematics in this country developed as an aid to astro- nomy, and therefore, for the first time we find Āryabhaṭa(499AD.) in his Āryabhaṭīya describing as a special section (Gaṇitapāda). Brahmagupta (628 A.D.) also followed Āryabhaṭa in this respect and gave the science of calculation (gaṇita) a special place in his treatise on astronomy. The Siddhānta treatises, earlier than those of Āryabhaṭa and Brahmagupta do not contain a chapter exclu- sively devoted to gaṇita (the Sūrya-Siddhānta and the Siddhāntas of Vasiṣṭha, Pitāmaha and Romaka are thus without gaṇita chapetrs). Later on Bhāskara I and Lalla also did not include gaṇita as a section or chapter in their treatises. It is said, howe- ver, that Lalla wrote a separate treatise on Pāṭīgaṇita. It may further be remarked here that Āryabhaṭa I gives the rules for finding the square and cube-roots only whilst Bra- hmagupta gives the cube-root rule only (BrSPSi. XII. 7). Multiplication Undoubtedly the common Indian name ‘multiplication’ is ‘guṇana’, this term occurs in the Vedic literature also. The other terms for this logistics are hanana, vadha, kṣaya etc., which all mean ‘killing’ or ‘destroying.’ The synonyms of ‘hanana’ (killing) for multiplication have been used by Āryabhaṭa I (499). Brahma- gupta (628), Śrīdhara (c.750) and later writers, and these terms

158 BRAHMAGUPTA AND ARITHMETIC also occur in the Bakhaśālī Manuscript. Āryabhaṭa I does not mention the everyday methods of multiplication in his Āryabhaṭīya probably because they were too elementary to be included in a Siddhānta work. Brahma- gupta, however, in a supplement to the section on mathematics in his Siddhānta, gives the names of some methods with very brief descriptions of the processes:— The multiplicand repeated, as in gomūtrikā as often as there are digits in the multiplier, is severally multiplied by them and (the results) added according to places; this gives the product. Or the multiplicand is repeated as many times as there are component parts in the multiplier.¹ (the word bheda occurring in the verse has . been translated as "integrant portions" by Colebrooke p. 319. Again by the term bheda are meant portions which added together make the whole, or aliquot parts which multiplied together make the entire quantity. The multiplicand is multiplied by the sum or the differ- ence of the multiplier and an assumed quantity and, from the result the product of the assumed quantity and the multiplicand is subtracted or added.² (Colebrooke thinks that this is a method to obtain the true product when the multiplier has been taken to be too great or too small by mistake.³ Datta and Singh think, however, that this is not correct.⁴ Thus Brahmagupta mentions four methods of multiplica- tion: (i) gomūtrikā, (ii) khaṇḍa, (iii) bheda, and (iv) iṣṭa. The com- mon and the well known method of kapāṭa-sandhi has been omitted by him.

1. गुणकारखण्डतुल्यो गुण्यो गोमूत्रिकाकृतो गुणितः । सहितः प्रत्युत्पन्नो गुणकारक्रमेदतुल्यो वा ॥ — BrSpSi XII. 55
2. गुण्यो राशिर्गुणकारराशिनेष्टाधिकोनकेन गुणः । गुण्योष्टवधो न युतो गुण्येऽभ्यधिकोनके कार्यः ॥ — BrSpSi. XII. 56
3. Colebrooke, T. H., Hindu Algebra, p. 320.
4. Datta, B. and Singh, A. N., History of Hindu Mathematics Pt. I (Arithmetic), p. 135 (1962).

MULTIPLICATION 159 (i) Gomūtrika-method or zig-zag method. The word gomū- trikā means "similar to the course of cow's urine", hence "zigzag". This method in all essentials is the same as the sthāna- khaṇḍa method. The following illustration is based on the commentary of Pṛthūdaka Svāmī : Example : To multiply 1223 by 235. The numbers are written thus : 2 1223 3 1223 5 1223 The first line of figures is t hen multiplied by 2, the process beginning at units place, thus : 2×3=6; 3 is rubbed out and 6 substituted in its place, and so on. After all the horizontal lines have been multiplied by the corresponding numbers on the left in the vertical line, the numbers on the pāṭī stand thus : 2446 3669 6115

287405 after being added together as in the present method. The sthāna-khaṇḍa and the gomūtrikā methods resemble modern plan of multiplication most closely. (ii) Khaṇḍa Method · or Parts Multiplicatian Method : Since the days of Brahmagupta, this method of multiplication also became very popular. We have two methods under this head : (i) The multiplier is broken up into two or more parts whose sum is equal to it. The multiplicand is then multiplied severally by these and the results added. To take an example : 13×158=(6+7)×158=(6×158)+(7×158) =948+1106 =2054 (ii) The multiplier is broken up into two more aliquot parts. The multiplicand is then multiplied by one of these, the resulting product by the second and so on till all the parts are exhausted. The ultimate product is the result.

160 BRAHMAGUPTA AND ARITHMETIC Thus for example : 96 × 237 = (4 × 4 × 6) × 237 = (4 × 237) × 4 × 6 = 948 × 4 × 6 = (4 × 948) × 6 = 3792 × 6 = 22752 These methods of multiplication are found among the Arabs and the Italians, having obtained from people of India. They were known as the “Scapezzo” and “Repiego” methods respectively amongst Italians. (iii) Iṣṭa-guṇana Method or the Algebraic Method. We have already quoted the relevant verse from the Brāh- masphuṭa-siddhānta in this connection; (XII. 56) : The multiplicand is multiplied by the sum or the difference of the multiplied and an assumed quantity and from the result the product of the assvmed quan- tity and the multiplicand is subtracted or added.¹ [L

SQUARE 161 bhājaka, bhāgahāra or simply hara; quotient is known as labdhi or labdha (or "what is obtained"). India never regarded this operation as a difficult one; in Europe, this operation was regarded as a tedious one till the 15th century or so. Division was such a common operation that Āryabhaṭa did not regard it as worth being included in his treatise. But since he has given the methods of extracting square-roots and cube-roots, which obviously depend on division, we conclude that the method of division was known to him. Most Siddhānta writers have followed Āryabhaṭa I in omitting this operation from their texts, this being regarded too elementary to be included. Brahmagupta does not give details of this operation. The later treatises on Arithmetic as Śrīdhara's Triśatikā and the Pāṭīgaṇita (I.20) and Āryabhaṭa II (c.950 A.D.) have given the details of this operation. Square The Sanakrit term for square is varga or kṛti (varga lite- rally means "rows" or "troops" of similar things). In mathema- tics, it usually means the square power and also the square figure or its area. Thus we find in the Āryabhaṭīya : A square figure of four equal sides (and the number representing its area) are called varga. The product of the two equal quantities is also varga¹. The term kṛti means "doing", "making" or "action". It carries with it the idea of specific performance probably the gra- phical representation. For the first time we have a definite rule for squaring in the writings of Brahmagupta. But it does not mean that prior to him it was not known. It must have been known to Āryabhaṭa I since he has given the square-root method. Brahmagupta gives his method of squaring briefly as follows : Combining the product, twice the digit in the less (lowest) place into the several others (digits) with its (i.e. of the digit in the lowest place) square (repeatedly) gives the square.²

1. वर्गस्समचतुरश्रः फलञ्च सदृशद्वयस्य संवर्गः। —Ārya. II. 3.
2. राशेरूनं द्विगुणं बहुतरगुणमूनकृतियुतं वर्गः। —BrSpSi. XII. 63.

162 BRAHMAGUPTA AND ARITHMETIC The method has been more clearly enunciated by Mahāvīra (850 A.D.) in the Gaṇitasārasaṁgraha : Having squared the last (digit), multiply the rest by the digits by twice the last, (which) is moved forward (by one place). Then moving the remaining digits con- tinue the same operation (process), This gives the square.¹ Brahmagupta's method of squaring is shown by the follow- ing example : To square 125. The number is written down 125 The square of the digit in the last place, i. e., 5²=25 is set over it thus : 25 125 Then, 2 x 5=10 is placed below the other digits, and 5 is rubbed out, thus : 25 12 10 Multiplying by 10 the rest of the digits, i.e., 12 and setting the product over them (the digits), we have. 1225 12 10 Then rubbing out 10 which is not required and moving the rest of the digits, i. e. 12 we, have 1225 12 Thus one round of operations is completed. Again as before, setting the square of 2 above it and 2×2=4 below 1. we have 1625 1 4

1. GSS. P. 12.

C U B E 163 Multiplying the remaining digit 1 by 4, and setting the product above it, we have 5625 1 Then moving the remaining digit 1, we obtain 5625 1 Thus the second round of operations is completed. Next setting the square of 1 above it the process is comple- ted for there are no remaining figures, and the result stands thus : 15625 Algebraic Method of Squaring Brahmagupta in his Brāhmasphuṭasiddhānta gives a minor method of squaring thus : The product of the sum and the difference of the number (to be squared) and an assumed number plus the square of the assumed number give square¹. This may be represented by the following identity : n²=(n—a) (n+a)+a² This identity has been used for squaring by most of the Indian mathematicians. Thus 15²=(15—5) (15+5)+5²=225 We are not giving here other identities which have been used by latter mathematicians of India in getting the squares of numbers; for example, when Mahāvīra says : The sum of the squares of the two or more portions of the number together with their products each with the others multiplied by two gives the square² : he obviously refers to the identity (a+b+c............)²=a²+b²+c²+......+2ab+...... Cube The Sanskrit term for cube is ghana. It when used in the geometrical sense also means the solid cube. In the arithmeti- cal sense, it means the continued product of the same number taken three times. Thus we have the definition in the Ārya-

1. राशेरिष्टयुतोनाद्वधः कृतिर्वेष्टकृतियुक्तः । — BrSpSi. XII. 63
2. GSS. p. 13.