Indian Mathematics/Circ. A.D. 400–600

by George Rusby Kaye

III.

A.D. 400 to 600.

8. There appears to be no connecting link between the S'ulvasūtra mathematics and later Indian developments of the subject. Subsequent to the S'ulvasūtras nothing further is recorded until the introduction into India of western astronomical ideas.^[1] In the sixth century A.D. Varāha Mihira wrote his Pañcha Siddhāntikā which gives a summary account of the five most important astronomical works then in use. Of these the Sūrya Siddhānta, which was probably composed in its original form not earlier than A.D. 400, afterwards became the standard work. Varāha Mihira's collection is the earliest and most authentic account we have of what may be termed the scientific treatment of astronomy in India. "Although," writes Thibaut, "not directly stating that the Hindus learned from the Greeks, he at any rate mentions certain facts and points of doctrine which suggest the dependence of Indian astronomy on the science of Alexandria; and, as we know already from his astrological writings, he freely employs terms of undoubted Greek origin."

Varāha Mihira writes:—"There are the following Siddhāntas—the Pauliśa, the Romaka, the Vasishtha, the Saura and the Paitamaha.…… The Siddhānta made by Pauliśa is accurate, near to it stands the Siddhānta proclaimed by Romaka, more accurate is the Sāvitra (Sūrya). The two remaining ones are far from the truth."

9. The Pañcha Siddhāntikā contains material of considerable mathematical interest and from the historical point of view of a value not surpassed by that of any later Indian works. The mathematical section of the Pauliśa Siddhānta is perhaps of the most interest and may be considered to contain the essence of Indian trigonometry. It is as follows:—

"(1) The square-root of the tenth part of the square of the circumference, which comprises 360 parts, is the diameter. Having assumed the four parts of a circle the sine of the eighth part of a sign [is to be found].

"(2) Take the square of the radius and call it the constant. The fourth part of it is [the square of] Aries. The constant square is to be lessened by the square of Aries. The square-roots of the two quantities are the sines.

"(3) In order to find the rest take the double of the arc, deduct it from the quarter, diminish the radius by the sine of the remainder and add to the square of half of that the square of half the sine of double the arc. The square-root of the sum is the desired sine."

[The eighth part of a "sign" (=30°) is 3° 45' and by "Aries" is indicated the first "sign" of 30°.]

The rules given may be expressed in our notation (for unit radius) as ${\begin{aligned}\\&\scriptstyle {(1)\quad \pi ={\sqrt {10}}\quad (2)\ {\text{Sin }}30^{o}={\frac {1}{2}},\ {\text{Sin }}60^{o}={\sqrt {1-{\frac {1}{4}}}},}\\&\scriptstyle {(3)\quad {\text{Sin}}^{2}\gamma =({\frac {{\text{Sin }}2\gamma }{2}})^{2}+({\frac {1-{\text{Sin }}(90-2\gamma )}{2}})^{2}}\\\end{aligned}}$ They are followed by a table of 24 sines progressing by intervals of 3° 45' obviously taken from Ptolemy's table of chords. Instead, however, of dividing the radius into 60 parts, as did Ptolemy, Pauliśa divides it into 120 parts; for as $\scriptstyle {\sin {\frac {a}{2}}={\frac {{\text{chord }}a}{2}}}$ this division of the radius enabled him to convert the table of chords into sines without numerical change. Āryabhata gives another measure for the radius (3438′) which enabled the sines to be expressed in a sort of circular measure.

We thus have three distinct stages:
(a) The chords of Ptolemy, or $\scriptstyle {{\text{ch'd }}\alpha }$, with r=60
(b) The Pauliśa sine or $\scriptstyle {\sin {\frac {\alpha }{2}}={\frac {{\text{ch'd }}\alpha }{2}}}$, with r=120
(c) The Aryabhata sine or $\scriptstyle {\sin {\frac {\alpha }{2}}={\frac {3}{\pi }}\cdot {\frac {{\text{ch'd }}\alpha }{2}}}$, with r=3438′
To obtain (c) the value of $\scriptstyle {\pi }$ actually used was $\scriptstyle {{\frac {600}{191}}(=3.14136)}$

Thus the earliest known record of the use of a sine function occurs in the Indian astronomical works of this period. At one time the invention of this function was attributed to el-Battâni [A.D. 877—919] and although we now know this to be incorrect we must acknowledge that the Arabs utilised the invention to a much more scientific end than did the Indians.

In some of the Indian works of this period an interpolation formula for the construction of the table of sines is given. It may be represented by $\scriptstyle {\Delta _{n+1}=\Delta _{n}-{\frac {{\text{Sin }}n.\alpha }{{\text{Sin }}\alpha }}}$ where $\scriptstyle {\Delta _{n}={\text{Sin }}n.\alpha -{\text{Sin }}(n-1)\alpha }$. This is given ostensibly for the formation of the table, but the table actually given cannot be obtained from the formula.

10. Āryabhata.—Tradition places Āryabhata (born A.D. 476) at the head of the Indian mathematicians and indeed he was the first to write formally on the subject.^[2] He was renowned as an astronomer and as such tried to introduce sounder views of that science but was bitterly opposed by the orthodox. The mathematical work attributed to him consists of thirty-three couplets into which is condensed a good deal of matter. Starting with the orders of numerals he proceeds to evolution and involution, and areas and volumes. Next comes a semi-astronomical section in which he deals with the circle, shadow problems, etc.; then a set of propositions on progressions followed by some simple algebraic identities. The remaining rules may be termed practical applications with the exception of the very last which relates to indeterminate equations of the first degree. Neither demonstrations nor examples are given, the whole text consisting of sixty-six lines of bare rules so condensed that it is often difficult to interpret their meaning. As a mathematical treatise it is of interest chiefly because it is some record of the state of knowledge at a critical period in the intellectual history of the civilised world; because, as far as we know, it is the earliest Hindu work on pure mathematics; and because it forms a sort of introduction to the school of Indian mathematicians that flourished in succeeding centuries.

Āryabhata's work contains one of the earliest records known to us of an attempt at a general solution of indeterminates of the first degree by the continued fraction process. The rule, as given in the text, is hardly coherent but there is no doubt as to its general aim. It may be considered as forming an introduction to the somewhat marvellous development of this branch of mathematics that we find recorded in the works of Brahmagupta and Bhāskara. Another noteworthy rule given by Āryabhata is the one which contains an extremely accurate value of the ratio of the circumference of a circle to the diameter, viz., $\scriptstyle {\pi =3{\frac {177}{1250}}(=3\cdot 1416)}$; but it is rather extraordinary that Āryabhata himself never utilised this value, that it was not used by any other Indian mathematician before the 12th century and that no Indian writer quotes Āryabhata as recording this value. Other noteworthy points are the rules relating to volumes of solids which contain some remarkable inaccuracies, e.g., the volume of a pyramid is given as half the product of the height and the base; the volume of a sphere is stated to be the product of the area of a circle (of the same radius as the sphere) and the root of this area, or $\scriptstyle {\pi {\frac {3}{2}}\gamma ^{3}}$. Similar errors were not uncommon in later Indian works. The rule known as the epanthem occurs in Āryabhata's work and there is a type of definition that occurs in no other Indian work, e.g., "The product of three equal numbers is a cube and it also has twelve edges."

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1. This has a somewhat important bearing on the date of the S'ulvasūtras. If, for example, the date of their composition were accepted as 500 B.C. a period of nearly 1,000 years, absolutely blank as far as mathematical notions are concerned, would have to be accounted for.
2. Although Āryabhata's Ganita, as first published by Kern, is generally accepted as authentic, there is an element of doubt in the matter.