Indian Mathematics

Front matter

INDIAN MATHEMATICS

G. R. KAYE

CALCUTTA & SIMLA

Introduction

I.

1. The orientalists who exploited Indian history and literature about a century ago were not always perfect in their methods of investigation and consequently promulgated many errors. Gradually, however, sounder methods have obtained and we are now able to see the facts in more correct perspective. In particular the early chronology has been largely revised and the revision in some instances has important bearings on the history of mathematics and allied subjects. According to orthodox Hindu tradition the Sūrya Siddhānta, the most important Indian astronomical work, was composed over two million years ago! Bailly, towards the end of the eighteenth century, considered that Indian astronomy had been founded on accurate observations made thousands of years before the Christian era. Laplace, basing his arguments on figures given by Bailly considered that some 3,000 years B.C. the Indian astronomers had recorded actual observations of the planets correct to one second; Playfair eloquently supported Bailly's views; Sir William Jones argued that correct observations must have been made at least as early as 1181 B.C.; and so on; but with the researches of Colebrooke, Whitney, Weber, Thibaut, and others more correct views were introduced and it was proved that the records used by Bailly were quite modern and that the actual period of the composition of the original Sūrya Siddhānta was not earliar than A.D. 400.

It may, indeed, be generally stated that the tendency of the early orientalists was towards antedating and this tendency is exhibited in discussions connected with two notable works, the Śulvasūtras and the Bakhshālī arithmetic, the dates of which are not even yet definitely fixed.

2. In the 16th century, A.D., Hindu tradition ascribes the invention of the nine figures with the device of places to make them suffice for all numbers 'to the beneficent Creator of the universe'; and this was accepted as evidence of the very great antiquity of the system! This is a particular illustration of an attitude that was quite general, for early Indian works claim either to be directly revealed or of divine origin. One consequence of this attitude is that we find absolutely no references to foreign origins or foreign influence. We have, however, a great deal of direct evidence that proves conclusively that foreign influence was very real indeed—Greek and Roman coins, coins with Greek and Indian inscriptions, Greek technical terms, etc., etc.; and the implication of considerable foreign influence occurs in certain classes of literature and also in the archæological remains of the north-west of India. One of the few references to foreigners is given by Vahrāha Mihira who acknowledged that the Greeks knew something of astrology; but although he gives accounts of the Romaka and the Pauliśa siddhāntas he never makes any direct acknowledgment of western influence.

The Sulvasūtras

II.

3. For the purpose of discussion three periods in the history of Hindu mathematics may be considered:

(I) The S'ulvasūtra period with upper limit c. A.D. 200;

(II) The astronomical period c. A.D. 400—600.

(III) The Hindu mathematical period proper, A.D. 600—1200.

Such a division into periods does not, of course, perfectly represent the facts, but it is a useful division and serves the purposes of exposition with sufficient accuracy. We might have prefixed an earlier, or Vedic, period but the literature of the Vedic age does not exhibit anything of a mathematical nature beyond a few measures and numbers used quite informally. It is a remarkable fact that the second and third of our periods have no connection whatever with the first or S'ulvasūtra period. The later Indian mathematicians completely ignored the mathematical contents of the S'ulvasūtras. They not only never refer to them but do not even utilise the results given therein. We can go even further and state that no Indian writer earlier than the nineteenth century is known to have referred to the S'ulvasūtras as containing anything of mathematical value. This disconnection will be illustrated as we proceed and it will be seen that the works of the third period may be considered as a direct development from those of the second.

4. The S'ulvasutra period.—The term S'ulvasūtra means 'the rules of the cord' and is the name given to the supplements of the Kalpasūtras which treat of the construction of sacrificial altars. The period in which the S'ulvasūtras were composed has been variously fixed by various authors. Max Müller gave the period as lying between 500 and 200 B.C.; R. C. Dutt gave 800 B.C.; Bühler places the origin of the Apastamba school as probably somewhere within the last four centuries before the Christian era, and Baudhāyana somewhat earlier; Macdonnell gives the limits as 500 B.C. and A.D. 200, and so on. As a matter of fact the dates are not known and those suggested by different authorities must be used with the greatest circumspection. It must also be borne in mind that the contents of the S'ulvasūtras, as known to us, are taken from quite modern manuscripts; and that in matters of detail they have probably been extensively edited. The editions of Āpastamba, Baudhāyana and Kātyāyana which have been used for the following notes, indeed, differ from each other to a very considerable extent.

The S'ulvasūtras are not primarily mathematical but are rules ancillary to religious ritual—they have not a mathematical but a religious aim. No proofs or demonstrations are given and indeed in the presentation there is nothing mathematical beyond the bare facts. Those of the rules that contain mathematical notions relate to (1) the construction of squares and rectangles, (2) the relation of the diagonal to the sides, (3) equivalent rectangles and squares, (4) equivalent circles and squares.

5. In connection with (1) and (2) the Pythagorean theorem is stated quite generally. It is illustrated by a number of examples which may be summarised thus:

| Āpastamba. | | | | Baudhāyana.

| | |

Kātyāyana gives no such rational examples but gives (with Āpastamba and Baudhāyana) the hypotenuse corresponding to sides equal to the side and diagonal of a square, i.e., the triangle , and he alone gives , and . There is no indication that the S'ulvasūtra rational examples were obtained from any general rule. Incidentally is given an arithmetical value of the diagonal of a square which may be represented by

This has been much commented upon but, given a scale of measures based upon the change ratios 3, 4, and 34 (and Baudhāyana actually gives such a scale) the result is only an expression of a direct measurement; and for a side of six feet it is accurate to about th of an inch; or it is possible that the result was obtained by the approximation by Tannery's R-process, but it is quite certain that no such process was known to the authors of the S'ulvasūtras. The only noteworthy character of the fraction is the form with its unit numerators. Neither the value itself nor this form of fraction occurs in any later Indian work.

There is one other point connected with the Pythagorean theorem to be noted, viz., the occurrence of an indication of the formation of a square by the successive addition of gnomons. The text relating to this is as follows:

"Two hundred and twenty-five of these bricks constitute the sevenfold agni with aratni and pradesa."

"To these sixty-four more are to be added. With these bricks a square is formed. The side of the square consists of sixteen bricks. Thirty-three bricks still remain and these are placed on all sides round the borders."

This subject is never again referred to in Indian mathematical works.

The questions (a) whether the Indians of this period had completely realised the generality of the Pythagorean theorem, and (b) whether they had a sound notion of the irrational have been much discussed; but the ritualists who composed the S'ulvasūtras were not interested in the Pythagorean theorem beyond their own actual wants, and it is quite certain that even as late as the 12th century no Indian mathematician gives evidence of a complete understanding of the irrational. Further, at no period did the Indians develop any real theory of geometry, and a comparatively modern Indian work denies the possibility of any proof of the Pythagorean theorem other than experience.

The fanciful suggestion of Bürk that possibly Pythagoras obtained his geometrical knowledge from India is not supported by any actual evidence. The Chinese had acquaintance with the theorem over a thousand years B.C., and the Egyptians as early as 2000 B.C.

6. Problems relating to equivalent squares and rectangles are involved in the prescribed altar constructions and consequently the S'ulvasūtras give constructions, by help of the Pythagorean theorem, of

(1) a square equal to the sum of two squares;

(2) a square equal to the difference of two squares;

(3) a rectangle equal to a given square;

(4) a square equal to a given rectangle;

(5) the decrease of a square into a smaller square.

Again we have to remark the significant fact that none of these geometrical constructions occur in any later Indian work. The first two are direct geometrical applications of the rule ; the third gives in a geometrical form the sides of the rectangle as and ; the fourth rule gives a geometrical construction for and corresponds to Euclid, II, 5; the fifth is not perfectly clear but evidently corresponds to Euclid, II, 4.

7. The Circle.—According to the altar building ritual of the period it was, under certain circumstances, necessary to square the circle, and consequently we have recorded in the S'ulvasūtras attempts at the solution of this problem, and its connection with altar ritual reminds us of the celebrated Delian problem. The solutions offered are very crude although in one case there is pretence of accuracy. Denoting by a the side of the square and by d the diameter of the circle whose area is supposed to be the rules given may be expressed by Neither of the first two rules, which are given by both Āpastamba and Baudhāyana, is of particular value or interest. The third is given by Baudhāyana only and is evidently obtained from by utilising the value for given in paragraph 5 above. We thus have which, neglecting the last term, is the value given in rule . This implies a knowledge of the process of converting a fraction into partial fractions with unit numerators, a knowledge most certainly not possessed by the composers of the S'ulvasūtras; for as Thibaut says there is nothing in these rules which would justify the assumption that they were expert in long calculations; and there is no indication in any other work that the Indians were ever acquainted with the process and in no later works are fractions expressed in this manner.

It is worthy of note that later Indian mathematicians record no attempts at the solution of the problem of squaring the circle and never refer to those recorded in the S'ulvasūtras.

Circ. A.D. 400–600

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. 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 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 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 , with r=60 (b) The Pauliśa sine or , with r=120 (c) The Aryabhata sine or , with r=3438′ To obtain (c) the value of actually used was

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 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. 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., ; but it is rather extraordinary that Āryabhata himself never utilised this value, that it was not used by any other Indian mathe 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 . 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."

Circ. A.D. 600–1200

IV.

A.D. 600—1200.

11. Āryabhata appears to have given a definite bias to Indian mathematics, for following him we have a series of works dealing with the same topics. Of the writers themselves we know very little indeed beyond the mere names but some if not all the works of the following authors have been preserved:

Bhāskara is the most renowned of this school, probably undeservedly so, for Brahmagupta's work is possibly sounder mathematically and is of much more importance historically. Generally these writers treat of the same topics—with a difference—and Brahmagupta's work appears to have been used by all the others. Bhāskara mentions another mathematician, Padmanābha, but omits from his list Mahāvīra.

One of the chief points of difference is in the treatment of geometry. Brahmagupta deals fairly completely with cyclic quadrilaterals while the later writers gradually drop this subject until by the time of Bhāskara it has ceased to be understood.

The most interesting characteristics of the works of this period are the treatment of:

(i) indeterminate equations;

[(ii) the rational right-angled triangle;

and (iii) the perfunctory treatment of pure geometry.

Of these topics it will be noted that the second was dealt with to some extent in the S'ulvasūtras; but a close examination seems to show that there is no real connection and that the writers of the third period were actually ignorant of the results achieved by Baudhāyana and Āpastamba.

12. Indeterminate Equations. The interesting names and dates connected with the early history of indeterminates in India are:

cir. A.D.

That we cannot fill up the gap between Diophantus and Āryabhata with more than the mere name of Hypatia is probably due to the fanatic ignorance and cruelty of the early Alexandrian Christians rather than the supposed destruction of the Alexandrian library by the Muhammadans. It would be pleasant to conceive that in the Indian works we have some record of the advances made by Hypatia, or of the contents of the lost books of Diophantus—but we are not justified in indulging in more than the mere fancy. The period is one of particular interest. The murder of Hypatia (A.D. 415), the imprisonment and execution of Boethius (A.D. 524), the closing of the Athenean schools in A.D. 530 and the fall of Alexandria in 640 are events full of suggestions to the historian of mathematics. It was during this period also that Damascius, Simplicius (mathematicians of some repute) and others of the schools of Athens, having heard that Plato's ideal form of government was actually realised under Chosroes I in Persia, emigrated thither (c. A.D. 532). They were naturally disappointed but the effect of their visit may have been far greater than historical records show.

13. The state of knowledge regarding indeterminate equations in the west at this period is not definitely known. Some of the works of Diophantus and all those of Hypatia are lost to us; but the extant records show that the Greeks had explored the field of this analysis so far as to achieve rational solutions (not necessarily integral) of equations of the first and second degree and certain cases of the third degree. The Indian works record distinct advances on what is left of the Greek analysis. For example they give rational integral solutions of

p|aj}The solution of (A) is only roughly indicated by Āryabhata but Brahmagupta's solution (for the positive sign) is practically the same as where t is zero or any integer and p/q is the penultimate convergent of a/b.

The Indian methods for the solution of may be summarised as follows:

If and then will where r is any suitable integer.

Also where n is any assumed number.

The complete integral solution is given by a combination (a) and (b) of which the former only is given by Brahma , while both are given by Bhāskara (five centuries later). The latter designates (a) the 'method by composition' and (b) the 'cyclic method.' These solutions are alone sufficient to give to the Indian works an important place in the history of mathematics. Of the 'cyclic method' (i.e., the combination of (a) and (b)) Hankel says, "It is beyond all praise: it is certainly the finest thing achieved in the theory of numbers before Lagrange." He attributed its invention to the Indian mathematicians, but the opinions of the best modern authorities (e.g., Tannery, Cantor, Heath) are rather in favour of the hypothesis of ultimate Greek origin.

The following conspectus of the indeterminate problems dealt with by the Indians will give some idea of their work in this direction; and although few of the cases actually occur in Greek works now known to us the conspectus significantly illustrates a general similarity of treatment.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30).

14. Rational right-angled triangles.—The Indian mathematicians of this period seem to have been particularly attracted by the problem of the rational right-angled triangle and give a number of rules for obtaining integral solutions. The following summary of the various rules relating to this problem shows the position of the Indians fairly well.

| | | A | | B | | | | Authorities.

| i | | n | | | | | | Pythagoras (according to Proclus)

| ii | | | | | | | | Plato (according to Proclus)

| iii | | | | | | | | Euclid and Diophantus.

| iv | | | | | | m | | Diophantus.

| v | | 2mn | | | | | | Brahmagupta and Mahāvīra.

| vi | | | | | | | | Brahmagupta, Mahāvīra, and Bhāskara.

| vii | | m | | | | | | Bhāskara.

| viii | | | | | | m | | Bhāskara.

| ix | | 2lmn | | | | | | General formula.

Mahāvīra gives many examples in which he employs formula (V) of which he terms m and n the 'elements.' From given elements he constructs triangles and from given triangles he finds the elements, e.g., "What are the 'elements' of the right-angled triangle (48, 55, 73)? Answer: 3, 8."

Other problems connected with the rational right-angled triangle given by Bhāskara are of some historical interest: e.g., (1) The sum of the sides is 40 and the area 60, (2) The sum of the sides is 56 and their product , (3) The area is numerically equal to the hypotenuse, (4) The area is numerically equal to the product of the sides.

15. The geometry of this period is characterised by:

(1) Lack of definitions, etc.; (2) Angles are not dealt with at all; (3) There is no mention of parallels and no theory of proportion; (4) Traditional inaccuracies are not uncommon; (5) A gradual decline in geometrical knowledge is noticeable.

On the other hand, we have the following noteworthy rules relating to cyclic quadrilaterals—where x and y are the diagonals of the cyclic quadrilateral (a, b, c, d). This (ii) is sometimes designated as 'Brahmagupta's theorem'.

16. The absence of definitions and indifference to logical order sufficiently differentiate the Indian geometry from that of the early Greeks; but the absence of what may be termed a theory of geometry hardly accounts for the complete absence of any reference to parallels and angles. Whereas on the one hand the Indians have been credited with the invention of the sine function, on the other there is no evidence to show that they were acquainted with even the most elementary theorems (as such) relating to angles.

The presence of a number of incorrect rules side by side with correct ones is significant. The one relating to the area of triangles and quadrilaterals, viz., the area is equal to the product of half the sums of pairs of opposite sides, strangely enough occurs in a Chinese work of the 6th century as well as in the works of Ahmes, Brahmagupta, Mahāvīra, Boethius and Bede. By Mahāvīra, the idea on which it is based—that the area is a function of the perimeter—is further emphasized. Āryabhata gives an incorrect rule for the volume of a pyramid; incorrect rules for the volume of a sphere are common to Āryabhata, S'rīdhara and Mahāvīra. For cones all the rules assume that . Mahāvīra gives incorrect rules for the circumference and area of an ellipse and so on.

17. Brahmagupta gives a fairly complete set of rules dealing with the cyclic quadrilateral and either he or the mathematician from whom he obtained his material had a definite end in view—the construction of a cyclic quadrilateral with rational elements.—The commentators did not fully appreciate the theorems, some of which are given in the works of Mahāvīra and S'rīdhara; and by the time of Bhāskara they had ceased to be understood. Bhāskara indeed condemns them outright as unsound. "How can a person" he says "neither specifying one of the perpendiculars, nor either of the diagonals, ask the rest? Such a questioner is a blundering devil and still more so is he who answers the question."

Besides the two rules (i) and (ii) already given in paragraph 15, Brahmagupta gives rules corresponding to the formula

(iii) , etc., and

(iv) If and then the quadrilateral is cyclic and has its diagonals at right angles.

This figure is sometimes termed "Brahmagupta's trapezium." From the triangles (3, 4, 5) and (5, 12, 13) a commentator obtains the quadrilateral (39, 60, 52, 25), with diagonals 63 and 56, etc. He also introduces a proof of Ptolemy's theorem and in doing this follows Diophantus (iii, 19) in constructing from triangles (a, b, c) and new triangles and and uses the actual examples given by Diophantus, namely (39, 52, 65) and (25, 60, 65).

18. An examination of the Greek mathematics of the period immediately anterior to the Indian period with which we are now dealing shows that geometrical knowledge was in a state of decay. After Pappus (c. A.D. 300) no geometrical work of much value was done. His successors were, apparently, not interested in the great achievements of the earlier Greeks and it is certain that they were often not even acquainted with many of their works. The high standard of the earlier treatises had ceased to attract, errors crept in, the style of exposition deteriorated and practical purposes predominated. The geometrical work of Brahmagupta is almost what one might expect to find in the period of decay in Alexandria. It contains one or two gems but it is not a scientific exposition of the subject and the material is obviously taken from western works.

Indian Mathematical Works—

V.

19. We have, in the above notes, given in outline the historically important matters relating to Indian mathematics. For points of detail the works mentioned in the annexed bibliography should be consulted; but we here briefly indicate the other contents of the Indian works, and in the following sections we shall refer to certain topics that have achieved a somewhat fictitious importance, to the personalities of the Indian mathematicians and to the relations between the mathematics of the Chinese, the Arabs and the Indians.

20. Besides the subjects already mentioned Brahmagupta deals very briefly with the ordinary arithmetical operations, square and cube-root, rule of three, etc,; interest, mixtures of metals, arithmetical progressions, sums of the squares of natural numbers; geometry as already described but also including elementary notions of the circle; elementary mensuration of solids, shadow problems, negative and positive qualities, cipher, surds, simple algebraic identities; indeterminate equations of the first and second degree, which occupy the greater portion of the work, and simple equations of the first and second degrees which receive comparatively but little attention.

Mahāvīra's work is fuller but more elementary on the whole. The ordinary operations are treated with more completeness and geometrical progressions are introduced; many problems on indeterminates are given but no mention is made of the 'cyclic method' and it contains no formal algebra. It is the only Indian work that deals with ellipses (inaccurately).