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QuickRef: Science of the ancient Hindus
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Here is a compilation from the book
Hindu Achievements in Exact Science by Benoy Kumar Sarkar (published in 1918: downloadable in tiff format from
Hindu Achievements In Exact Science : Kumar Sarkar.B : Free Download & Streaming : Internet Archive )
To facilitate easy perusal, I shall post this compilation in four or five instalments.
Arithmetic
Two foundations discovered by Hindus: the symbol of numbers (numerals as they are called) and the decimal system of notation.
• Numerals were in use in India since 3rd century BCE. They were employed in the Minor Rock Edicts of Asoka the Great (256 BCE)
• The decimal system was known to AryabhaTTa (476 CE) and Brahmagupta (598-660 CE), and fully described by BhAskarAchArya (1114).
‣ In Subandhu's
vAsavadattA, a SaMskRta prose romance (550-606 CE?), the stars are described as zero.
• The decimal system was therefore known to the Hindus long before its appearance in the writings of the Arabs or Grco-Syrians.
• The Saracens learnt from the Hindus both the system of numeration and the method of computation. Alberuni (1033) wrote: "The numeral signs which we use are derived from the finest forms of the Hindu signs."
• It was probably in the 12th century that the Europeans learnt this Hindu science from their Saràcen masters.
• At the commencement of the Christian era, the Chinese "adopted the decimal system of notation introduced by the Buddhists, and changed their ancient custom of writing figures from top to bottom for the Indian custom of from left to right."
Algebra
• The mathematician who systematized the earlier algebraic knowledge of the Hindus and thus became the founder of a new science is AryabhaTTa.
• The Hindu algebra was the principal feeder of Saracen algebra through Yakub and Musa, and indirectly ifiuenced to a certain extent medieval European mathematics.
• The Hindu discoveries in algebra may be thus summarized from the recent investigatons of Nalin behari Mitra:
1. The idea of an absolutely negative quantity.
2. The first exposition of the complete solution of the quadratic equation: Brahmagupta.
3. Rules for finding permutations and combinatics BhAskara. These were unknown to the Greeks.
4. Indeterminate equations: “The glory of having invented general methods in this most subtle branch of mathematics belongs to the Indians.”
5. Indeterminate equations of the second degree.
• BhAskara invented the art of placing the numerator er the denominator in a fraction. He invented also √ (the racical sign). This was not known in Europe before Chuquet and Rudolf in the sixteenth century.
Geometry
• The earliest geometry of the Hindus is to be fouid in the
shulba-sUtras of BaudhAyana and Apastamba. In these treatises, which form parts of the Vedic literature, we get the application of mathematical knowledge to the exigencies of religious life, sacrifices, rituals, construction of altars, etc.
• At this stage Hindu geometry was quite independent of Greek influence. The following are some of the problems, which were solved by the mathematicians of the Vedic cycle:
1. The so-called Pythagorean theorem: The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.
2. Construction of squares equal to the sum or difference of two squares;
3. Conversion of oblongs into squares, and vice versa;
4. Drawing of a perpendicular to a given straight line at a given point of it;
5. Construction of lengths equal to quadratic surds: The approximate value of ╥--pi.
6. Circling of squares;
7. Squaring of circles,-—”that rock upon which so many reputations have been destroyed,” both in the East and West. The earliest Hindus got ╥ = 3.0044;
8. Construction of successive larger squares from smaller ones by addition;
g. Determination of the area of a trapezium, of an isosceles trapezium, at any rate, when the lengths of its parallel sides and the distance between them are known.
• We find AryabhaTTa solving the following among other problems:
1. The area of a triangle;
2. The area of a circle;
3. The area of a trapezium;
4. The distance of the point of intersection of the diagonals of a trapezium from either of the parallel sides;
5. The length of the radius of a circle.
AryabhaTTa gave also the accurate value of ╥ (62832/20000) and the area of the circle as ╥r^2. The Saracens learnt this from the Hindus. Probably Yakub (eighth century) was the first to get it when the astronomical tables were imported to Bagdad from India. The correct value of ╥ was not known in Europe before Purhach (1423—61).
• Brahmagupta made other contributions to geometry. BhAskara (1114) summarized and methodized the results of all previous investigators, e.g., Lata, AryabhaTTa, Lalla (499), VarAhamihira (5o5), Brahmagupta, ShrIdhara (853), MahAvIra (8o), AryabhaTTa the Younger (970), and Utpala (970).
• Among BhAskara's original contributions may be mentioned the fact that he gave two proofs of the so-called Pythagorean theorem. One of them was “unknown in Europe till Wallis (1616—1793) rediscovered it.”
Trigonometry
• The mathematicians of India devised (1) the table of sines, and (2) the table of versed sines. The term 'sine' is an Arabic corruption from SaMskRta
shinjini.
• The Hindu table of sines exhibits them to every twenty-fourth part of the quadrant, the table of versed sines does the same. In each, the sine or versed sine is expressed in minutes of the circumference, neglecting fractions.
• The astronomical tables of the Hindus prove that they were acquainted with th principal theorems of spherical trigonometry.
Co-ordinate Geometry
• VAchaspati (850 CE), commentator of NyAya (logic), anticipated in a rudimentary way the principle of co-ordinate (solid) geometry eight centuries before Descartes (1596—165o).
• VAchaspati’s claims are thus presented by Seal:
‣ To conceive position in space, VAchaspati takes three axes, one proceeding from the point of sunrise in the horizon to that of sunset, on any particular day (roughly speaking, from the east to the west);
‣ a second bisecting this line at right angles on the horizontal plane (roughly speaking, from the north to the south);
‣ and the third proceeding from the point of their section up to the meridian section of tho. sui on that day (roughly speaking, up and down).
‣ The position of any point in space, relatively to anothr point, may now be given by measuring distances, along these three directions, i.e., by arranging in a numerical series the intervening points of contact, the lesser distance being that which comes earlier in this serie, and the greater which comes later.
‣ The position of any single atom in space with reference to another may be inlicated in this way with reference to the three axis.
‣ But this gives only a geometrical analysis of the conception of three-dimensioned space, though it must be admitted in all fairness that by dint of clear thinking it anticipates in a rudimentary manner the foundations of solid (co-ordinate) geometry.
Differential Calculus
• BhAskarAchArya anticipated Newton (1642—1727) by over five hundred years (i) in the discovery, of the principles of differential calculus and (2) in its application to astronomical problems and computations.
• According to Seal, BhAskara’s claim is indeed far stronger than Archimedes’ to the conception of a rudimentary process of integration.
“BhAskara, in computing the instantaneous motion of a planet compare its successive positions, and regards its motion as constant during the interval (which of course cannot be greater than a
truti of time, i.e., 1/3375th part of a second, though it may be infinitely less).”
This process is not only “analogous to, but virtually identical with, that of the differential calculus.” As Spottiswoode remarks, mathematicians in Europe will be surprised to hear of th.existence of such a process in the age of Ehaskara (twelfth century).
!! Now we have four estimates, 0.637, 1.276, 0.2133, or 0.04, which is it?
