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

TECHNICAL TERMS 199 Power Since long, the word varga has been used for the second power; the word also stands for square (Uttarādhyayana Sūtra¹, B. C. c. 300). The third power is similarly known as ghana, the fourth power as varga-varga (square-square), the sixth power as ghana-varga (cube-square) and the twelfth power as ghana-varga- varga (cube-square-square). In later days, the fifth power was called vargaghana ghāta (here the word ghāta means product; the term means product of cube and square). The former system was multiplicative, rather than additive; wheras the latter was on the additive system. The seventh power on the additive system was known as varga-varga-ghana-ghāta (product of square-square and cube). Brahmagupta, however, uses a more scientific system for expressing the powers more than four. He calls the fifth power as Pañca gata (literally meaning, raised to the fifth), the sixth power as ṣaḍ-gat (raised to the sixth) and so on, thus adding the suffix gata to the name of the number indicating that power.² Bhāskara II has followed the system of Brahmagupta almost consi- stently for powers one and upwards. Equation Perhaps Brahmagupta has for the first time used the term samakaraṇa or samīkaraṇa (literally meaning making equal) or simply sama (equal or equation)³. Pṛthūdaka Svāmī (860) employs the term sāmya (equality or equation) for equation⁴. The equation is said to possess two Pakṣas⁵ (sides) Itara-Pakṣa and apara-pakṣa. Absolute Term Brahmagupta uses the term rūpa (literally meaning appear- ance) for an absolute term. It represents the visible or known

1. Chapter XXX, 10, 11.
2. अव्यक्तवर्गे घनवर्ग-वर्ग-पंचगत-षड्गतादीनाम् । सदृश द्विवधो वर्गस्त्यादि वधस्तद्गतोऽन्य जातिवधः ॥
3. वर्णं प्रमाण भावित-घातो भवतीष्टवर्ण संख्यैवम् । सिध्यति विनाऽपि भावित-समकरणात् किं कृतं तदतः ॥ अव्यक्तान्तर भक्तं व्यस्तं रूपान्तरं समेऽव्यक्तः ।
4. Siśe. XIV, 19.
5. Bījagaṇita.

200 BRAHMAGUPTA AS AN ALGEBRAIST portion of the equation whilst its other part is practically invisible or unknown¹. Unknowns and Symbolism Āryabhaṭa I (499 A. D.) probably used coloured gulikās or shots for represtnting different unknowns. Brahmagupta men- tions varṇa as the symobls for unknown. He has, however, not indicated how these varṇas or colours were used as symbols for unknowns. Perhaps we might conculude from this that the method of using colours as symbols for unknown quantities was very common and familiar to the algehraists. Datta and Singh say that the Sanskrit word varṇa denotes colour as well as a letter of alphabet, and therefore, letters of alphabet came into use for unknown quantities : kālaka (black), nīlaka (blue), pītaka (yellow), lohita (red), haritaka (green), śvetaka (white), citraka (variegated), kapilaka (tawny), piṅgalaka (reddish-brown), dhūmraka (smoke- coloured), pāṭalaka (pink), śavalaka (spotted), śyāmalaka (blackish) mecaka (dark blue) etc². It should be further noted that the first unknown quantity yāvat-tāvat is not a varṇa or colour. It thus clearly indicates that the use of colours as symbols came at a later stage, whilst the word yāvat-tāvat was in currency from much earlier times. Some authorities think that the term yāvat-tāvat is a corrupted form of yāvakastāvat (where yāvaka means red). Pṛthūdaka Svāmī has sometimes used the term yāvaka for an unknown quantity³. Laws of Signs Brahmagupta has in his Chaper XVIII devoted a special section entitled "Dhanarṇa Śunyānāṁ Saṁkalanam" or calcu- lations dealing with quantities bearing positive and negative signs and zero,

1. अव्यक्तान्तरं भक्तं व्यस्तं रूपान्तरं समेऽव्यक्तः । —BrSpSi XVIII. 43 वर्गे चतुर्गुणितानां रूपाणां मध्यवर्गे सहितानाम् ॥ —BrSpSi XVIII. 44
2. यावत्-तावत् कालको नीलकोऽन्यो वर्णः पीतो लोहितश्चैतदाद्याः । अव्यक्तानां कल्पिता मानसंज्ञास्तत्संख्यानं कर्तुंमाचार्यवर्यैः ॥ यावत्-तावत् कालक नीलक पीताश्च लोहितो हरितः । श्वेतक चित्रक कपिलक पाटलकाः पण्डु धूम्र शवलाश्च ॥ श्यामलक-मेचक-धवलक-पिशङ्ग-शारङ्ग-बभ्रु-गौराद्याः । —Nārāyaṇa, Bījagaṇita
3. BrSpSi XII, 15. (Com.) ; XII 18.

: Does it look like ऋ? No, it has a horizontal top bar, a round left belly with a notch, connected to a vertical stem on the right: THAT IS 'ध'! Wait, why does it have 'ध'? Could it be ऋणयोर्धनयोर्घातो धनं ऋणयोर्वधो...? Wait, let's re-read the first half: Does it say ऋणयोर्धनयोर्घातो धनं? Wait! Look at the space in: ऋणयोर्धनयोर्घातोIs there a space afterघातो? Yes! Then what is the next letter? ध! Then न! Wait, has it got an anusvara? Look at the top of न: Wait, does it have an anusvara? NO! There is no anusvara on that न! It's just धन! Wait, then मृ? Wait, look at: ऋणयोर्धनयोर्घातो धनमृणयोर्वधोWait, why would it sayधनमृणयोर्वधो धनं भवति`? That would mean product of positive and negative is positive! Wait, could it be: "ऋणयोर्धनयोर्घातो धनं, ऋणयोर्वधो धनं भवति"? Wait! If: "The product of two negatives is positive; positive multiplied

202 BRAHMAGUPTA AS AN ALGEBRAIST Brahmagupta lays down the rules regarding evolution and involution as follows : The square of a positive or a negative number is positive .....The (sign of the) root is the same, as was that from which the square was derived¹. As regards the latter portion of this rule, Pṛthūdaka Svāmī has the following comment to make : "The square-root should be taken either negative or positive, as will be most suitable for subsequent operations to be carried on." It will be interesting to observe the following observation of Mahāvīra (850 A. D.) regarding square-root of a negative quantity "Since a negative number by its own nature is not a square, it has no square-root."² So says Śrīpati : "A negative number by itself is non-square, so its square-root is unreal; so the rule (for the square-root) should be applied in the case of a posi- tive number."³ Algebraic Operations Brahmagupta and other algebraists recognise six operations as fundamental in algebra : addition, subtraction, multiplication, division, squaring and the extraction of the square-root. Regarding addition and subtraction Brahmagupta says : Of the unknowns, their squares, cubes, fourth powers, fifth powers, sixth powers, etc., addition and subtraction are (performed) of the like; of the unlike (they mean simply their) statement severally.⁴ In place of "of the like", Bhāskara II uses the term "of those of the same species (jāti) amongst unknowns" : Addition and subtraction are performed of those of the same species (jāti) amongst unknowns; of different species they mean their separate statement.⁵

1. खोद्धृत मृणं धनं वा तच्छेदं खमृणधनविभक्तं वा । ऋणधनयोर्वर्गः स्वं खं खस्य पदं कृतैर्यत् तत् ॥ —BrSpSi, XVIII. 35
2. GSS. I, 52.
3. Śiśe, XIV, 5.
4. अव्यक्तवर्ग घनवर्ग वर्ग पंचगत षड्गतादीनाम् । तुल्यानां संकलित व्यवकलिते पृथगतुल्यानाम् ॥ —BrSpSi.XVIII. 41.
5. योगोऽन्तरं तेषु समान जात्योर्विभिन्न जात्योश्च पृथक् स्थितिश्च । — Bhāskara II, Bījagaṇita.

ALGEBRAIC OPERATIONS 203 This means that the numerical coefficients of x cannot be added to or subtracted from the numerical coefficients of y or x² or x³ or xy and so on because these terms belong to different jāti or they do not belong to the category of the "like". Again, regarding multiplication, Brahmagupta says : The product of the two like unknowns is a square; the product of three or more like unknowns is a power of that designation. The multiplication of unknowns of unlike species is the same as the mutual product of symbols; it is called bhāvita (product or factum).¹ Having given the rules of the operations for addition, sub- traction and multiplication, Brahmagupta does not think it necessary to deal with other operations. His section on the calculations with zero, negative and positive quantities ends here. How is an Equation Formed? Pṛthūdaka Svāmī while commenting on a verse in Brāhma- sphuṭasiddhānta speaks as follows : In this case, in the problem proposed by the questioner, yāvat-tāvat is put for the value of the unknown quantity. Then performing multiplication, division etc. as required in the problem the two sides shall be carefully made equal. The equation being formed in this way, then the rule (for its solution) follows.² Plan for Writing Equations When in regards to a given problem, an equation has been formed, it has to be written down for further operations. This writing down of an equation is technically known as nyāsa Perhaps the oldest record of nyāsa is to be found in the Bakhshālī Manuscript. According to the procedure prescribed in this work, the two sides of an equation are put down one after the other in the same line without any sign of equality being inter- posed. Thus the equations : √(x+5)=s, √(x-7)=t appear as

1. सदृशद्विवधो वर्गस्त्र्यादिवधस्तद् गतोऽन्यजातिवधः । अन्योन्यवर्णघातो भावितकः पूर्ववच्छेदम् ॥ —BrSpSi.XVIII. 42.
2. BrSpSi. XVIII. 43 (com)

204 BRAHMAGUPTA AS AN ALGEBRAIST | 0 5 yu mū 0 | sa 0 7 + mū 0 | | 1 1 1 | 1 1 1 | Here yu (यु) stands for yuta (युत), meaning added, subtraction is+sign, derived from Kṣaya or (क्षय) meaning diminished, gu (गु) for guṇa or guṇita, meaning multiplied; bhā (भा) for division from bhājita and mū (मू) for square-root, from mūla meaning root; zero (०) was used to mark a vacant place. Again, the following equation x+2x+3×3x+12×4x=300 is represented as ; | 0 | 2 1 | 3 3 | 12 4 | dṛśya 300 | 1 | 1 1 | 1 1 | 1 1 | There is no sign for unknown in the Bakhshālī Manuscript. Later on this plan of writing equations as adopted in Bakha- śālī Manuscript was abandoned in India; a new one was adopted in which the two sides are written one below the other without any sign of equality. It must be stated that in this new plan the term of similar denominations are usually written one below the other and even the terms of absent denominations on either side are clearly indicated by putting zeros as their coefficients. We find a reference to this new plan in the algebra of Brahmagupta. From which the square of the unknown and the unkno- wn are cleared, the known quantities are cleared (from the side) below that¹. Here in this verse, the words “adhastāt” clearly indicate that one side of the equation is written below the other., As an illustration, Pṛthūdaka Svāmī represented the equation² :— 10x-8=x²+1 as : ya va 0 ya 10 ru 8̇ (x².0+x.10-8) ya va 1 ya 0 rū 1 (x².1+x. 0+1) which means, x² was written as yāvat-varga (yā va) and x was written as yāvat or yā. The minus sign was represented by a dot at the top of the number.(-8 was written as 8̇). We shall take another illustration from Pṛthūdaka Svāmī. He would write the equation 197x-1644 y-z=6302 as

1. BrSpSi. XVII. 43; compare also Bhāskara II, Bījagaṇita.
2. Br SpSi. XVIII. 49 (com.)

PLAN FOR WRITING EQUATIONS 205 yā 197 kā 1644̇ nī 1̇ rū 0 yā 0 kā 0 nī 0 rū 6302 Here the first unknown x is represented by yā(vat), the second unknown y by kā(laka) and the third unknown z by nī- (laka) and the term without unknown, a mere number is written by rū(paka). The two sides, one written below the other if written in the present form, would appear as : 197x—1644y—z+0=0x+0y+0z+6302. The Bījagaṇita of Bhāskara II also follows the same proce- dure. One instance from it would be quoted here to illustrate the method of expressing equations. 8x³+4x²+10y²x=4x³+12y²x or 8x³+4x²+10y²x=4x³+0x²+12y²x is written as follows on Bhāskara's or Brahmagupta's plan : x³ is ghana of yāvat (abbreviated as yā gha) x² is varga of yāvat (abbreviated as yā va) y² is varga of kālaka (abbreviated as kā va) the coefficients 10 and 12 are bhāvita (abbreviated as bhā). The equation is : yā gha 8 yā va 4 kā va yā. bhā 10 yā gha 4 yā va 0 kā va yā. bhā 12 Datta and Singh state that the use of the old plan of writ- ing equations is sometimes met with in later works also. For instance, in the MS. of Pṛthūdaka Svāmī's commentary¹ on the Brāhmasphuṭasiddhānta, an incomplete copy of which is preserv- ed in the library of the Asiatic Society of Bengal (No. I B6), we find a statement of equations thus : “first side yāvargaḥ 1 yāvakaḥ 200 rū 0; second side yāvargaḥ 0 yāvakaḥ 0 rū 1500. Śodhana or Clearance of an Equation After nyāsa or statement of an equation, the operation to be performed is known as śodhana (clearance) or saṁśodhana (equi-clearance or complete clearance). The nature of this clearance varies according to the kind of equation In the case of an equation in one unknown only, whether linear,

1. BrSpSi. XII. 15 (com.)

206 BRAHMAGUPTA AS AN ALGEBRAIST quadratic or of higher powers, one side of it is cleared of the unknowns of all denominations and the other side of it of the absolute terms, so that the equation is ultimately reduced to one of the form ax² + bx = c, where a, b, c may be positive or negative; some of them may even be zero. Thus Brahmagupta observes : From which the square of the unknown and the un- known are cleared, the known quantities (rūpāṇi) are cleared (from the side) below that.¹ On this Pṛthūdaka Svāmī comments as follows : This rule has been introduced for that case in which the two sides of the equation having been formed in accordance with the statement of the problem, there are present the square and other powers of the unknown together with the (simple) unknown. The absolute terms should be cleared off from the side opposite to that from which are cleared the square (and other powers) of the unknown and the (simple) unknown. When perfect clearance (saṁśodhana) has been thus made...² Śrīdhara and Bhāskara II have also given the rules of clearance almost on the same lines. Thus the equation yā va 0 yā 10 rū 8̇ yā va 1 yā 0 rū 1 after perfect clearance having been made will be (according to Pṛthūdaka Svāmī) yā va 1 yā 10 rū 9̇ i.e. the equation 10x - 8 = x² + 1 after clearance would become x² - 10x = -9. Classification of Equations Usually equations are classified as : simple equation : yāvat-tāvat quadratic : varga

1. वर्गाव्यक्ताः शोध्या यस्माद् रूपाणि तदधस्तात् ॥ — BrSpSi. XVIII. 43.
2. BrSpSi. XVIII. 43 (com.)

CLASSIFICATION OF EQUATIONS 207 cubic : ghana biquadratic : varga-varga Brahmagupta classified them as (i) equations in one unknown quantity : eka-varṇa samīkaraṇa. (ii) equations in several unknowns : aneka-varṇa samī- karaṇa. (iii) equations involving products of unknowns : bhāvita. Eka-varṇa samīkaraṇas (equations with one unknown) are further divided into (i) linear equations, and (ii) quadratic equa- tions (avyakta-varga samīkaraṇa). Pṛthūdaka Svāmī has classified equations in a different manner as follows : (i) linear equations with one unknown : eka-varṇa samīkaraṇa. (ii) linear equations with more unknowns : aneka- varṇa samīkaraṇa. (iii) equations with one, two or more unknowns in their second or higher powers : madhyamāharaṇa. (iv) equations involving products of unknowns : bhā- vita. As the method of solution of an equation of the third class (i.e. equations with one or several unknowns in their second or higher powers) is based upon the principle of the elimination of the middle term, that class is called by the term madhyama (middle) āharaṇa (elimination). The classification of Brahma- gupta and Pṛthūdaka Svāmī more or less received recognition by later writers on algebra as Bhāskara II and others. Linear Equations with One Unknown and Their Solutions The first solution of a linear equation with one unknown is obtainable in the Śulba Sūtras but not through an algebraic process,—the Śulba process is geometrical. It is said that there is a reference in the Sthānāṅga Sūtra (c. 300 B.C.) to a linear equation by its name yāvat-tāvat. There has been a good deal of

208 BRAHMAGUPTA AS AN ALGEBRAIST controversy regarding the date of the Bakhaśālī Manuscript where we have definitely a method of solving linear equations by the Rule of False Position. It would be interesting to give an account of this rule here by taking an illustration from the Bakhaśālī Manuscript. Problem : The amount given to the first is not known. The second is given twice as much as the first; the third thrice as much as the second; and the fourth four times as much as the third. The total amount distributed is 132. What is the amount of the first ? (BMS. Folio 23, recto) In modern algebraic language, the solution of the problem would be given by the equation x + 2x + 6x + 24x = 132 where x is the amount given to the first. The solution of this equation is given as follows in the Bakhaśālī Manuscript : ‘Putting any desired quantity in the vacant place’; any desired quantity is || 1 ||, ‘then construct the series’ | 1 | 2 | 2 | 3 | 6 | 4 | | 1 | 1 | 1 | 1 | 1 | 1 | ‘multiplied’ || 1 | 2 | 6 | 24 | ; ‘added’ 33. ‘Divide the visible quantity’ | 132 / 33 ; which) on reduction be- comes | 4 / 1 |. (This is) the amount given (to the first) (BMS. Folio 23, recto) The Rule of False Position may be regarded as an early stage of the development of the science of algebra, since no symbol could have been evolved for an unknown quantity. As soon as the system of notations was introduced, the application of this Rule was no longer considered as necessary. Thus we find that Āryabhaṭa I does not mention of this Rule. Āryabhaṭa I states as follows regarding the solution of linear equations : The difference of the known ‘amounts’ (rūpaka) relat- ing to two persons should be divided by the difference

LINEAR EQUATIONS 209 of the coefficients on the unknown (gulikā). The quotient will be the value of the unknown (gulikā), if their possessions be equal.¹ The original verse contains the term "gulikāntara" which has been here translated as the difference of the coefficients of the unknowns. We have already stated earlier that Āryabhaṭa uses the term gulikā or shot for an unknown quantity, (gulikāntara literally means only the difference of unknowns). This practice is also followed by other Indian algebraists. Pṛthudaka Svāmī rightly observed that according to the usual practice in this country, "the coefficient of the square of the unknown is called the square (of the unknown) and the coefficient of the (simple) unknown is called the unknown.² The rule given by Āryabhaṭa, then, contemplates a problem of this kind : Two persons, who are equally rich, possess respectively a, b times a certain unknown amount together with c, d units of money in cash. What is that amount ? Now if x be the amount unknown, then according to the problem ax + c = bx + d Thence x = (d - c) / (a - b) Āryabhaṭa has merely expressed this solution in his langu- age. Regarding the solution of linear equations, Brahmagupta says : In a (linear) equation in one unknown, the difference of the known terms taken in the reverse order, divided by the difference of the coefficients of the unknown (is the value of the unknown).³

1. गुलिकान्तरेण विभजेद् द्वयोः पुरुषयोस्तु रूपकविशेषम् । लब्धं गुलिकामूल्यं यद्यर्थं कृतं भवति तुल्यम् ॥ —Ārya. II. 30
2. BrSpSi, XVIII. 44 (com.)
3. अव्यक्तान्तरभक्तं व्यस्तं रूपान्तरं समेऽन्यवतः । वर्गाव्यक्ताः शोध्या यस्माद् रूपाणि तदधस्तात् ॥ —BrSpSi. XVIII. 43

210 BRAHMAGUPTA AS AN ALGEBRAIST Similar solutions have been offered by the other Indian algebraists who followed Brahmagupta like Śrīpati, Bhāskara II and Nārāyaṇa. Here again, we take a problem proposed by Brahmagupta in this connection : Problem : Tell the number of elapsed days for the time when four times the twelfth part of the residual degrees increased by one, plus eight will be equal to the residual degrees plus one.¹ Pṛthūdaka Svāmī has solved this problem as follows : Here the residual degrees are (put as) yāvat-tāvat, yā; increased by one, yā 1 rū 1; twelfth part of it, (yā 1 rū 1) / 12 ; four times this, (yā 1 rū 1) / 3 ; plus the absolute quantity eight, (yā 1 rū 25) / 3 . This is equal to the residual degrees plus unity. The statement of both sides tripled is yā 1 rū 25 yā 3 rū 3 This difference between the coefficients of the unknown is 2. By this the difference of the absolute terms, namely 22, being divided, is produced the residual of the degrees of the Sun, 11. These residual degrees should be known to be irreducible. The elapsed days can be deduced then, (proceeding) as before. If put in the modern notations, it means the solution of the equation : 4/12 (x+1)+8=x+1, from which we have x+25=3x+3 or 2x=22 or x=11. Rule of Concurrence or Saṁkramaṇa Brahmagupta has included this rule in algebra, whereas other Indian mathematicians included it in arithmetic. Saṁ-

1. सैकादंशकशेषाद् द्वादशभागश्चतुर्गुणोऽष्टयुतः । सैकांशशेषतुल्यो यदा तदाऽहर्गणं कथय ॥ —BrSpSi. XVIII. 46

RULE OF CONCURRENCE 211 kramaṇa is the solution of the simultaneous equations of the type : x + y = a x - y = b Brahmagupta's rule for solution is: The sum is increased and diminished by the difference and divided by two; (the result will be the two un- known quantities) : (this is) concurrence (Saṁkra- maṇa).¹ Brahmagupta restates this rule in the form of a problem and its solution : The sum and difference of the residues of two (heavenly bodies) are known in degrees and minutes. What are the residues? The difference is both added to and subtracted from the sum, and halved; (the results are) the residues.² Linear Equations with Several Unknowns The first mention of a solution of the problem with more than one unknown is found in the Bakhshālī Manuscript, and a system of linear equations of this type is solved in the Bakhshālī treatise substantially by the False Position Rule. A generalised system of linear equations will be b₁Σx - c₁x₁ = a₁, b₂Σx - c₂x₂ = a₂,......, bₙ Σx - cₙxₙ = aₙ Therefore Σx = Σ(a/c) / (Σ(b/c) - 1) Hence xᵣ = (bᵣ / cᵣ) × [Σ(a/c) / (Σ(b/c) - 1)] - aᵣ / cᵣ r = 1, 2, 3......, n One particular case, where b₁ = b₂ = b₃ = ...... = bₙ = 1 and c₁ = c₂ = c₃ = ...... = cₙ = c has been treated by Brahmagupta at one place. He gives the rule as follows :

1. योगोऽन्तरयुतहीनो द्विहतः संक्रमणमन्तरविभक्तं वा । वर्गान्तरमन्तरयुतहीनं द्विहृतं विषमकर्म ॥ — BrSpSi. XVIII. 36
2. भागकला विकलैक्यं दृष्ट्वा विकलान्तरं च के शेषे । ऐक्यं द्विधाऽन्तराधिक हीनं च विभाजितं शेषे ॥ — BrSpSi. XVIII. 96

212 BRAHMAGUPTA AS AN ALGEBRAIST The total value (of the unknown quantities) plus or minus the individual values (of the unknowns) multi- plied by an optional number deing severally (given), the sum (of the given quantities) divided by the num- ber of unknowns increased or decreased by the multi- plier will be the total value; thence the rest (can be determined).¹ Σx±cx₁=a₁, Σx±cx₂=a₂, Σx±cx₃=a₃,...... Σx±cx =an Therefore Σx = (a₁ + a₂ + a₃ + ...... + an) / (n ± c) Hence x₁ = 1/c ( ±a₁ ∓ (a₁ + a₂ + a₃ + ...... + an) / (n ± c) ); and so on for x₂, x₃ etc. Now we shall give the rule enunciated by Brahmagupta for solving linear equations involving several unknowns : Removing the other unknowns from (the side of) the

QUADRATIC EQUATIONS 213 be performed all operations conformably to the state- ment of the example and thus should be carefully framed two or more sides and also equations. Equi-clear- ance should be made first between two and two of them and so on to the last : from one side one un- known should be cleared, other unknowns reduced to a common denominator and also the absolute numbers should be cleared from the side opposite. The residue of other unknowns being divided by the residual coefficient of the first unknown will give the value of the first unknown. If there be obtained several such values, then with two and two of them, equations should be formed after reduction to common denominators. Proceeding in this way to the end find out the value of one unknown. If that value be (in terms of) another unknown then the coefficients of those two will be reciprocally the values of the two unknowns. If, however, there be present more un- knows in that value, the method of the pulveriser should be employed. Arbitrary values may then be assumed for some of the unknowns. Datta and Singh have said that the above rule of Brahma- gupta, and also the one indicated in the commentary of Pṛthudaka Svāmī, embraces the solution of indeterminate as well as the determinate equations. In fact, all the examples given by Brahmagupta in illustration of the rule are of indeterminate character. So far as the determinate simultaneous equations are concerned, Brahmagupta's method for solving them will be easily recognised to be the same as our present one. Quadratic Equations The geometrical solution of a quadratic equation in this country would take us to the Vedic Śulba period. The Bakha- śālī Manuscript also contains certain problems which need the solving of quadratic equations. I shall quote one out of the numerous available : A certain person travels s yojana on the first day and b yojana more on each successive day. Another who travels at the uniform rate of S yojana per day, has a start of t days. When will the first man overtake the second ?

214 BRAHMAGUPTA AS AN ALGEBRAIST This problem would today be expressed in terms of the following equation : S(t+x)=x {s+((x-1)/2)b}, where x is the number of days after which the first overtakes the second. We may write this equation as bx²-{2(S-s)+b}x=2tS whence the value x would be after solving the quadra- tic : x= (√{2(S-s)+b}²+8bts+{2(S-s)+b}) / 2b The Bakhasālī Manuscript gives this solution as follows : The daily travel (S) diminished by the march of the first day (s) is doubled; this is increased by the common increment (b). That (sum) multiplied by itself is designated (as the kṣepa quantity). The product of the daily travel and the start (t) being multiplied by eight times the common increment, the kṣepa quantity is added. The square-root of this (is increased by the kṣepā quantity; the sum divided by twice the common increment will give the required number of days). (BMS. Folio 5, recto) Āryabhaṭa I (499 A.D.) is regarded as the founder of algebra, since he gives the solutions of a few quandratic problems. For example, to find the number of terms of an arithmetical pro- gression (A.P.), he gives the following rule : The sum of the series multiplied by eight times the common difference is added by the square of the dif- ference between twice the first term and the common difference: the square-root (of the result) is diminished by twice the first term and (then) divided by the com- mon difference : half of this quotient plus unity is the number of terms.¹ In the modern notations of algebra, the solution would be expressed as follows :

1. गच्छोऽष्टोत्तर गुणिताद् द्विगुणाद्युत्तर विशेषवर्गयुतात् । मूलं द्विगुणाद्यूनं स्वोत्तर भजितं सरूपार्थं ॥ —Ārya. II, 20

QUADRATIC EQUATIONS 215 n = 1/2 { [√(8bs + (2a - b)²) - 2a] / b + 1 } There is another certain interest problem¹, the solution of which has been provided in the Āryabhaṭīya as x = [√(Apt + (p/2)²) - p/2] / t which is the solution of the quadratic equation : tx² + px - Ap = 0 Āyabhaṭa I has thus given the solutions of a few quadratic equations, but he nowhere gives the procedure of solving these equations. We give here the Rules of Brahmagupta for the solution of quadratic equations. He undoubtedly is not the discoverer of these rules; but perhaps for the first time in the history of algebra we find the process of solving a quadratic equation so clearly indicated. First Rule : The quadratic : the absolute quantities multiplied by four times the coefficient of the square of the unknown are increased by the square of the coefficient of the middle (i.e. unknown); the square-root of the result being diminished by the coefficient of the middle and divided by twice the coefficient of the square of the unknown, is (the value of) the middle."² This expressed in the modern notations would mean x = [√(4ac + b²) - b] / 2a It would be noted that in this rule, Brahmagupta has emp- loyed the term madhya (middle) to imply the simple unknown as well as its coefficient. The origin of the term is doubtless connected with the mode of writing the quadratic equation in the form ax² + bx + 0 = 0x² + 0x + c so that there are three terms on each side of the equation.

1. मूलफलं सफलं कालमूलं गुणमर्धमूल कृति युक्तम् । मूलं मूलाधोनं कालहृतं स्यात्स्वमूलफलम् ॥ —Ārya. II. 25.
2. वर्गं चतुर्गुणितानां रूपाणां मध्यवर्गसहितानाम् । मूलं मध्येनोनं वर्ग द्विगुणोद्धृतं मध्यः ॥ —BrSpSi. XVIII. 44.

216 BRAHMAGUPTA AS AN ALGEBRAIST Second Rule : The absolute term multiplied by the coefficient of the square of the unknown is increased by the square of half the coefficient of the unknown; the square-root of the result diminished by half the coefficient of the unknown and divided by the coefficient of the square of the unknown is the unknown.² This when expressed in the modern algebraic notations would be x = [√(ac+(b/2)²) - (b/2)] / a Here if the quadratic equation is ax² + bx + c = 0 the 'absolute term' is c (the one without the unknown x), 'the coefficient of the square of unknown' means the coefficient of x², i.e. a, and the 'coefficient of the unknown' means the coefficient of x, i.e. b. The above two methods of Brahmagupta are exactly the same as were suggested by Āryabhaṭa I. The root of the quadratic equation for the number of terms of an arithmetic progression (A.P.) is given by Brahmagupta according to the first rule ² : n = [√(8bs+(2a-b)²) - (2a-b)] / 2b Third Rule : Brahmagupta also suggests a Third Rule which is very much the same as is used commonly now. Though it has not been expressedly suggested as a new rule, we find its application in a few instances. For example this rule has been suggested in connection with the following problem on interest : A certain sum (p) is lent out for a period (t₁); the interest accrued (x) is lent out again at this

1. वर्गाहत रूपाणामव्यक्ताधैंकृति संयुतानां यत् । पदमव्यक्ताधोनं तद्वर्ग विभक्तमव्यक्तः ॥ —BrSpSi. XVIII. 45
2. उत्तरहीनद्विगुणादि शेषवर्गं धनोत्तराष्टवधे । प्रक्षिप्य पदं शेषोनं द्विगुणोत्तरहृतं गच्छः ॥ —BrSpSi. XII. 18

QUADRATIC EQUATIONS 217 rate of interest for another period (t₂) and the total amount is A. Find x. The equation for determining x is (t₂ / pt₁) x² + x = A. The solution of this equation would be : x = √((pt₁ / 2t₂)² + (A / t₂) · pt₁) - pt₁ / 2t₂ Brahmagupta has stated the result in exactly the same form. Pṛthūdaka Svāmī has illustrated it in solving the following pro- blem of interest : Problem : A sum of five hundred paṇas (p) is lent out for a period of 4 months (t₁); the interest accrued (x) is lent out again at this rate of interest for another period of 10 months (t₂) and the total amount is 78 (A). Give the pramāṇa-phala, i.e., the interest accrued x. Here pramāṇa-kāla (t₁) = 4 months pramāṇa-dhana (p) = 500 paṇas para-kāla (t₂), the subsequent period = 10 months miśra dhana or the total interest accrued (A) = 78 paṇas. Brahmagupta states his solution of such quadratics like this : Take the product of the pramāṇa-dhana (p) or the sum originally lent out and pramāṇa-kāla, i.e. the period for which originally lent out (t₁); and divide by the para- kāla or the subsequent time (t₂); place this result at two places. Multiply the one placed at the first place with the miśra-dhana (A), that is with the total inter- est accrued; in this product add the square of half the one placed in the second place; now take the square- root of it, and from it subtract half of the one placed at the second place.¹

1. कालप्रमाणघातः परकालहृतो द्विधाऽऽप्तमिश्रवधात् । अन्यार्धकृति युतात् पदमन्यार्धोनं प्रमाण फलम् ॥ —BrSpSi. XII. 15.

218 BRAHMAGUPTA AS AN ALGEBRAIST Thus in the above example the product of pramāṇa-dhana and pramāṇa kāla divided by parakāla is (pt₁/t₂)—is (500 × 4) / 10 = 200. This is first multiplied by the total interest accrued (A); it becomes 200 × 78 = 15600. To this is now added square of half of 200 (which is 10000) ; it becomes 15600 plus 10000 = 25600. Its square-root is taken which is 160. From this is subtracted half of the quantity (i.e. half of 200 which is 100). Thus 160–100 = 60, which is the answer. It was the interest which first accrued (x). Another Quadratic Problem : Brahmagupta refers to an astronomical problem which involves the quadratic equation (72 + a²)x² ∓ 24 apx = 144 ( R² / 2 - p² ), where a = agra (the sine of the amplitude of the Sun), b = palabha (the equinoctial shadow of a gnomon 12 aṅguli long), R = radius, and x = koṇaśaṅku (sine of the altitude of the Sun when his altitude is 45°). Dividing out by (72 + a²) we have x² ∓ 2mx = n, where m = 12 ap / (72 + a²), n = 144(R²/2 - p²) / (72 + a²) Therefore we have x = √(m² + n) ± m, as stated by Brahmagupta. We find the same result in the Sūrya-siddhānta and in the text of Śrīpati. Āryabhaṭa II (1150) also followed the method of Āryabhaṭa I and Brahmagupta in solving a quadratic equation in connection with finding out the number of terms in an arithmetical progression (A.P.) whose first term is (a), common difference is b and the sum is s. The number of terms n is given by¹ n = [√(2bs + (a - b/2)²) - a + b/2] / b Two Roots of a Quadratic Equation and Brahmagupta A quadratic equation has two roots. This must have been known to Indian algebraists even at a very early stage. Bhāskara II in his Bījagaṇita has quoted a rule ascribed to an ancient writer Padmanābha whose works are not available now :

1. Mahāsiddhānta. Bhāskara II, XV. 50