Mathematical Achievments of Aryabhatas

Alexander Volodarsky

Institute of the History of Science and Technology, Moscow

Full fifteen centuries have passed in 1976 since the birth of Aryabhata, an outstanding Indian mathematician and astronomer.

Our knowledge of the scholar's life is very scarce. We know neither who his parents were, nor his teachers, nor even the exact time of his death. Aryabhata was just 23 years old when in 499 A.D. he completed the famous Aryabhatiya, the only work of his to be preserved till our time. Writes Aryabhata: "When sixty times sixty years and three quarter yugas (of the current yuga) had elapsed, twenty-three years had then passed since my birth". According to the Indian tradition, there are four epochs, or yugas — the Golden Age, the Silver age, the Bronze Age, and the Iron Age — and the last of these, the kaliyuga, began in 3102 B.C. It is from its beginning that sixty times sixty years had elapsed, i.e. Aryabhatiya was written in 499 A.D. by the twenty-three-years-old author, which permits fixing 476 a.d. as the year in which he was born.

The exact place of Aryabhata's birth is unknown. The treatise only mentions a major Indian scientific centre — Kusumapura (Pataloputra, modern Patna in Bihar), where the scholar may have worked: "Having bowed with reverence to Brahma, Earth, Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn and the asterisms, Aryabhata sets forth here the knowledge honoured at Kusumapura" (see ref, 1, part II, rule 1). Some authors believe him to be a native of Asmaka, a province in Southern India (p. 93), but that view is not shared by everybody.

Of his personal biography we know nothing more, but we have got something far more precious — the work which was indeed a turning point in the history of exact science in India. In a way, Aryabhatiya was an interface work which took of previous development and as far as was possible had imbibed the best achievements of preceding epochs. But on the other hand, it marked the start of a new scientific tradition in India and was studied and analysed over the centuries. Twelve commentaries to the work are on record, the earliest dating back to the first quarter of the 6th century and the latest to the mid-19th century. The commentators include famous Indian mathematicians and astronomers, notably Bhaskara I (7th century), Paramesvara (15th century) and Nilakantha (15th - 16th century). Quite a few manuscripts of some of the commentaries have been preserved which is an indication that Aryabhatiya was studied rather extensively. This is also indicated by commentaries in vernacular languages. The original Sanskrit treatise had been translated into vernacular Hindi, Telugu, and Malayalam and was studied thoroughly.

Apart from his main work, Aryabhata had written a work on astronomy, which was known Aryabhata-siddhanta (p. 36-42), but it has not been preserved.

Aryabhatiya is a relatively small work written in traditional Indian form of distinctly metrical verses made up into the four parts of the treatise: ?Dasagitika or the Ten Giti Stanzas; Ganitapada or Mathematics; Kalakriya or the Reckoning of Time; and Gola or the Sphere.

It is in Aryabhata's exposition that a number of mathematical rules have come down to us. Mathematical matter is given, not just in the special second part, but throughout all other chapters.

The treatise never mentions the ways by which rules were obtained and never sets forth proofs or conclusions. The presentation is as succinct as could be, with all rules stated in the form of advice or prescription. Aryabhatiya treats of diverse problems of arithmetic, algebra, geometry, theory of numbers, trigonometry and astronomy'.

One of the most significant contribution to world science which was made by Indian mathematicians is the establishment of decimal place-value system. Though there is an abundant literature on the time and place of origin of this numeration, and the components which led to its creation, these problems are still open to debate. The scientific proceed from such clues as the shape of figures, the first application of zero, the first record of the figures in this numeration, and the evidences of contemporaries. All these must undoubtedly be taken into consideration, but the most important is the rules for performing arithmetic operations according to decimal place-value numeration. The earliest arithmetic rules known to us in this system were described by Aryabhata in Aryabhatiya, namely the square-root and cube-root evolution.

Closely related to the decimal place-value system was the alphabetic numeration also given by Aryabhata. Such numerations were aimed at reducing the long strings of words arising when numbers are written in a verbal form.

A central part in the arithmetic part of all Indian works was held by the Rule of Three, teaching how to find a number x forming with three given numbers a, b, c the proportion $\frac{a}{b}=\frac{c}{x}.$ Many problems were reduced to an application of this rule. Indian scholars had coined a name for each term of the proportion and, in fact, gave its name to the rule itself.

From the Indian the Rule of Three passed over into Arabic and thence into West European mathematical writings.

The types of problem subject to the Rule of Three had been certainly known elsewhere — in China, Greece and Egypt, but it was only in India that the rule was singled out, translated into problem solving methods, and extended to the case of five, seven, etc. quantities.

These extensions seem to have been familiar to Aryabhata, even though he cites only the Rule of Three. In his commentary, Bhaskara I writes: "Here Acarya Aryabhata has described the rule of three only. How the well known rules of five, etc. are to be obtained? I say thus: The Acarya has described only the fundamental of anupata (proportion). All others such as the rule of five, etc. follow from the fundamental rule of proportion. How? The rule of five, etc. consists of combinations of the rule of three.. In the rule of five there are two rules of three, in the rule of seven, three rules of three, and so on".

The treatise considers several problems which reduce to solving a linear equation in one unknown. One problem, set forth in part II, rule 30, is to calculate the value of an object if it is known that two men having equal wealth possess a different number of objects, a1 a2 and different pieces of money remaining after the purchase, b1, b2. The problem reduces itself to solving the equation a1x + b1 = a2x + b2.

Aryabhata formulates the rule of solving the linear equation in this manner: "Divide the difference between the rupakas with two persons by the difference between their gulikas. The quotient is the value of one gulika, if the possessions of the two persons are of equal value" (See ref. 1, part II, rule 30). That is to say,
            $x = \frac{b_2-b_1}{a_1-a_2}.$

Another problem is the famous Problem of Messengers, which later peregrinated all over the world'. algebraic literature. It is to calculate the time of meeting of two planets moving in opposite directions, or in the same direction. Aryabhata formulates this rule: "Divide the distance between the two bodies moving in the opposite directions by the sum of their speeds, and the distance between the two bodies moving in the same direction by the difference of their speeds; the two quotients will give the time elapsed since the two bodies met or to elapse before they will meet" (See ref. 1, part II, rule 31).

Thus, if the distance S between the two bodies and their velocities V1 and V2 are known, the time of meeting is found as $t = \frac{S}{V_1 + V_2}$ when they are moving in opposite directions or as $t = \frac{S}{V_3-V_2}$ when they are moving in the same directions.

Noteworthily, Aryabhata formulates the solution in such a way as to avoid introducing negative numbers, which Indian scholars, beginning with Brahmagupta (7th century) later adopted and used quite regularly.

Several of problems in Aryabhatiya lead to quadratic equations, in particular, the finding of the number of terms in an arithmetical progression and the calculation of interest. In the latter case, the following problem is solved, which is quoted by one of Aryabhata's commentators: capital A yields an unknown monthly profit x, which is then itself lent for interest for T months. The initial profit added together with the new interest is equal to B. Find the initial interest rate. Aryabhata gives the solution of the equation Tx2 + Ax = AB in verbal form corresponding to this expression:
        $x = \frac{\sqrt{BAT + \left(\frac{A}{2}\right)^2}-\frac{A}{2}}{T}.$

Similar problem of compound interest are posed by many Indian authors. They also occur in European manuals belonging to modern history. For example, the first problem of quadratic equations in Elements d'algebre by A. Clairaut (1746) is for compound interest.

Beginning with Aryabhata, most Indian mathematical texts give rules and examples of arithmetical progression Aryabhata knew the rules for the general term, sum, and the number of terms of an arithmetic progression. The rules for the summation of an arithmetical progression are set forth by Aryabhata in the part II, rule 19: "Diminish the given number of terms by one, then divide by two, then increase by the number of the proceeding terms (if any), then multiply by common difference, and then increase by the first term of the (whole) series: the result is the arithmetic mean (of the given number of terms) This multiplied by the given number of terms is the sum of the given terms. Alternatively, multiply the sum of the first and last terms (of the series or partial series which is to be summed up) by half the number of terms" The first part of the rule finds the sum S of the terms of an arithmetical progression from the term p+1 to the term p+n:
      $S = n[a + \left(\frac{n-1}{2} + p\right)d].$
The second part of the rule gives the formula $S = \frac{a_1+a_n}{2}n.$

Aryabhata also formulates the rules for finding the number of terms of an arithmetical progression
      $n = \frac{1}{2}\left[\frac{\sqrt{8ds + (d-2a)^2}-2a}{d} + 1\right].$

Aryabhatiya states the rules for the summation of natural squares and cubes, as well as some other series, which, however, had been previously known to Babylonians and Greeks.

Aryabhata contributed enormously to the theory of numbers and its important chapter — the indeterminate equations. The problem first arose in India from calendar astronomical needs of determining the periods of repetition of certain relative positions of celestial bodies (the Sun, the Moon, and the planets) which had different revolution periods and from other related issues. The problem reduces itself to finding integer numbers which divide by given remainder, i.e. satisfying indeterminate linear equations and equation systems.

In the third century a.d. the Greek mathematician Diophantus was concerned with indeterminate equations, but he only was seeking for rational solutions. Beginning with Aryabhata, the Indians tried to solve these equations in positive integers, which was a far stronger proposition. Any direct Greek influence on the Indian scholars is unlikely here, for each school had arrived at problems of the theory of numbers proceeding from different needs and using different methods. One may rather suppose some contacts of Indians linking them to ancient Chinese mathematicians, who had likewise arrived at indeterminate equations proceeding from the needs of astronomy and the problems of remainder and, moreover, also were only seeking after integer solutions (See ref. 8 pp. 143-144). Aryabhata's contribution to the theory of numbers was very valuable indeed; he was the first in the world literature to formulate very elegant methods of integer solution of indeterminate equation of the first degree.

Aryabhata gives the pertinent rule in part II, rule 32-33 for the Solution of this problem: find a number N, which, when divided by given numbers a, c yields two known remainders p, q. The problem leads to these indeterminate equations of the first degree:
      ax + b = cy, if p > q (b = p — q)
      ax — b = cy, if p < q

Incidentally, the latter equation can be reduced to the former by substitution of the unknown.

Aryabhata's rule is stated in an extremely succinct formulation, which had given rise to a great deal of comment and debate.

Aryabhata's geometrical rules include several verbal formulas. For example, he defines the area of a triangle as the product of the height multiplied by a half of the base (See ref. 1, part II, rule 7) as a half of the circle's length multiplied by a half of the diameter.

The area of any plane figure, writes Aryabhata in part II, rule 9, can be found if we single out two sides and then multiply one by the other. The commentator Paramesvara explains that what is meant here is the mean length and width.

Aryabhata determines the volume of a pyramid as base area multiplied by half the height. This, rather rough approximation is refined by other mathematicians, and in particular by Sridhara, who finds the volume as the base area multiplied by a third of the height. Aryabhata calculates the volume of a sphere by the formula πr2πr2, which is equal to 1•47πr3. This is rather approximative as compared with the exact formula for the volume of the sphere, $\frac{4}{3}\pi r^3$ given in Bhaskara II.

An essential mathematical constant, which also had a great practical value, was π the number estimating the ratio of the length of a circle of its diameter. For his time, Aryabhata's estimation was rather accurate (ref. 1, part II, rule 10). The value which was given by Aryabhata is correct to four decimal places: π ≈ 3.1416.

In part II, rule 14, Aryabhata gives the Pythagorean theorem: "Add the square of the height of the gnomon to the square of its shadow. The square root of that sum is the semi-diameter of the circle of shadow".

In part II, rule 13, the scholar gives several geometrical definitions which are rather rare in Indian mathematical literature: "A circle should be constructed by means of a pair of compasses; a triangle and quadrilateral by means of the two hypotenuses. The level of ground should be tested by means of water; and verticality by means of plumb".

A look at some of the geometrical problems considered by Aryabhata shows that he knew the basic properties of similar triangles and proportions, had an idea about derived proportions, relations of the segments of two intersecting chords, and the properties of the diameter perpendicular to a chord.

The trigonometric problems expounded in Aryabhatiya axe interesting. The Indians seem to have lent in their trigonometric studies the works of early Hellinistic astronomers, who had a fairly developed trigonometry of chords. But the Indians replaced chords with sines, which enabled them to introduce various functions related with the sides and angles of the right-angled triangle. They considered the line of sine, the line of cosine, and the line which was later in Europe named the sinus-versus, or reversed sinus. The earliest sine table is found in Surya-siddhanta and in the Aryabhatiya [ref. 1, part I, rule 12]. The table is compiled with a step of 3°45' = 225', i.e. 1/24 of the quadrant arc.

Aryabhata, as well as other Indian mathematicians made a wide use of the shadow cast by a vertical pole, the gnomon, to determine heights and distances. A number of relevant rules and problems are given in the geometrical chapter. This anticipated the introduction of tangent and cotangent, which were introduced in the 9th century by mathematicians in Islamic countries; incidentally, these functions were described by the name of "shadows".

How far-reaching was the true mathematical contribution of Aryabhatiya? It contains the first description of the rules in the decimal place-value system; the first description of the alphabetic numeration; it contains the first Indian description of the evolution of the square and cubic roots; the treatise considers several very interesting problems, which had played the great role in the development of mathematics; Aryabhata was also the first to formulate the rule of integer solution of indeterminate equation of the first degree in two unknowns; he set forth the methods of finding the general term, the sum, and the number of terms of an arithmetical progression; for his time Aryabhata's estimation of π was very accurate; his methods of computing the sinus-table in trigonometry was an important contribution.

Those are just the principal mathematical innovations appearing in Aryabhata's treatise. But this rundown by no means fathoms the important role that Aryabhatiya played in the development of Indian and world's science.

Towards the end of the eighth century, the treatise was translated into Arabic under the title of Zij al-Arjabhar. About the same time, two works by Brahmagupta were also translated which carried some of Aryabhata's mathematical and astronomical innovations. Later, when Arabic scholarly tests were translated into Latin, some of Aryabhata's ideas were inherited by West European scientists.

            *             *             *

April 19, 1975. Soviet spaceport. National flags of the Soviet Union and Indi» waving at a top of the ground control station. Up dashes a Soviet carrier rocket launching into the Earth's orbit the first Indian sputnik. After a few minutes of suspense, the loud-speaker announces: "The main fairing is off .... The second stage is working.... The sputnik has separated itself".
    India has become a space power!
    The first Indian artificial Earth's satellite was given the name of Aryabhata.
    Antiquity and modernity intertwine.

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