KK¯aty¯ayana ´Sulvas¯utra : Some Observations
S.G. Dani
Abstract
The K¯aty¯ayana ´Sulvas¯utra has been much less studied or discussed from amodern perspective, even though the first English translation of two adhy¯aya s(chapters) from it, by Thibaut, appeared as far back as 1882. Part of the rea-son for this seems to be that the general approach to the ´Sulvas¯utra studieshas been focussed on “the mathematical knowledge found in them”; as theother earlier ´Sulvas¯utras, especially Baudh¯ayana and ¯Apastamba substan-tially cover the ground in this respect the other two ´Sulvas¯utras, M¯anavaand K¯aty¯ayana, received much less attention, the latter especially so.On the other hand the broader purpose of historical mathematical studiesextends far beyond cataloguing what was known in various cultures, ratherto understand the ethos of the respective times from a mathematical pointof view, in their own setting, in order to evolve a more complete picture ofthe mathematical developments, ups as well as downs, over history.Viewed from this angle, a closer look at K¯aty¯ayana ´Sulvas¯utra assumessignificance. Coming at the tail-end of the ´Sulvas¯utras period, after whichthe ´Sulvas¯utras tradition died down due to various historical reasons that arereally only partly understood, makes it special in certain ways. What it omitsto mention from the body of knowledge found in the earlier ´Sulvas¯utras wouldalso be of relevance to analyse in this context, as much as what it chooses torecord. Other aspects such as the difference in language, style, would alsoreflect on the context. It is the purpose here to explore this direction ofinquiry.
The performance of the yajna s in the Vedic period involved construction of al-tars ( vedi ) and fireplaces ( citi or agni ) in a variety of intricate shapes, such as birds,tortoise and others, of quite large sizes; the dimensions of the vedi s often extendedto over 100 feet and the agni s could be 15 feet and 20 feet, or more, in width andlength. This warranted detailed description of procedures for their construction,1 a r X i v : . [ m a t h . HO ] J un hich is the subject of the ´Sulvas¯utras, which are parts of the Kalpas¯utra s associ-ated with the yajurveda . Apart from the direct aspect of step by step description inthe form of manuals, the ´Sulvas¯utras also include enunciation of various geometricprinciples involved, thereby setting up a body of geometric ideas and framework.The different ´s¯akh¯a s (branches) of the Vedic people had their respective ver-sions of ´Sulvas¯utras, though, as may be expected, a degree of intrinsic unity maybe seen in their overall contents. Notwithstanding the fact that there were a largenumber of ´s¯akh¯a s, possibly in hundreds, only eight (or nine?) ´Sulvas¯utras withmathematical content have been known in our times. Baudh¯ayana, ¯Apastamba,M¯anava, Maitr¯ayan¯iya, Var¯aha, Saty¯as.¯ad. ha, and V¯adul associated with the
Kr. s. nayajurveda , and a sole K¯aty¯ayana ´Sulvas¯utra associated with ´sukla yajurveda areknown. Of these, ¯Apastamba, Var¯aha, and V¯adul are literally the same [6]. (???)M¯anava and Maitr¯ayan¯iya are understood to be versions of each other (though a de-tailed comparison does not seem to have been made yet). Baudh¯ayana, ¯Apastamba,M¯anava, and K¯aty¯ayana ´Sulvas¯utras are independent in overall character, eventhough, as noted above, there are many commonalities.The dates of the ´Sulvas¯utras are uncertain; according to Kashikar, as quotedin [7], the following ranges may be associated with the composition of the re-spective ´Sulvas¯utras : Baudh¯ayana and V¯adul (800 - 500 BCE), ¯Apastamba andM¯anava (650 - 300 BCE) and K¯aty¯ayana, Saty¯as.¯ad. ha and Var¯aha (300 BCE -400 CE). There have also been other suggestions in this respect (see [11] for a dis-cussion on this), placing Baudh¯ayana around 5th or 6th century BCE, ¯Apastambaaround 5th and 4th century BCE, M¯anava between them, and K¯aty¯ayana around350 BCE. However all dates seem to be quite speculative, and there do not seemto be dependable inputs on the issue.The ´Sulvas¯utras are composed in the s¯utra (aphoristic) style, mostly in proseform, though parts of some of M¯anava and K¯aty¯ayana ´Sulvas¯utras are found to bein verse form. The texts have been divided by later commentators into convenientsegments, treated as individual s¯utras , with numbers attached, and grouped intoChapters. As presented in [11] Baudh¯ayana has 21 Chapters adding to 285 s¯utras,¯Apastamba has 21 Chapters adding to 202 s¯utras, M¯anava has 16 Chapters addingto 228 s¯utras, and K¯aty¯ayana has 6 Chapters adding to 67 s¯utras. It is an enigma in the subject that the Baudh¯ayana ´Sulvas¯utra which is theoldest happens to be the most systematic and comprehensive one in many ways.While the others do have some things in addition in certain respects, there are An extra chapter of K¯aty¯ayana is found in the version given in [6].
Pandit ,published from Benaras, during 1874 - 76. Edmund Burk brought out a trans-lation of the ¯Apastamba ´Sulvas¯utra in German in 1901. Translation of M¯anava´Sulvas¯utra was produced by J.M. van Gelder in 1964. There have also been vari-ous subsequent studies of these ´Sulvas¯utras, by western as well as Indian scholars.An English translation of all the four ´Sulvas¯utras, with commentaries, was broughtout by S.N. Sen and A.K. Bag [11] in 1983.On the whole the K¯aty¯ayana ´Sulvas¯utra has been much less studied or discussedin modern writings, even though the first English translation of two adhy¯aya s (chap-ters) from it, by Thibaut, appeared as far back as 1882. Part of the reason for thisseems to be that the general approach to the ´Sulvas¯utra studies has been focussedon the mathematical knowledge from the tradition as a whole; indeed, many writ-ers do not make adequate distinction between individual ´Sulvas¯utras in the overalldiscourse. Since the earlier ´Sulvas¯utras, especially Baudh¯ayana and ¯Apastambasubstantially cover the ground in this respect the other two ´Sulvas¯utras, M¯anavaand K¯aty¯ayana, received much less attention, the latter especially so. On theother hand, for a fuller understanding from a historical point of view it would beimportant to study the ´Sulvas¯utras with attention to their individual identities,comparisons between them etc.Our aim here will be to discuss the K¯aty¯ayana ´Sulvas¯utra in this overall con-text, concentrating on its specialities in relation to the earlier ´Sulvas¯utras, espe-cially Baudh¯ayana ´Sulvas¯utra. Special significance is lent to this by the fact thatK¯aty¯ayana is from substantially later times, towards the fag end of the Vedic pe-3iod, after which the yajna s lost their sheen, as a historical phenomenon, thoughsome feeble remnants of the idea are seen embodied in the later day Hindu ritualpractices, including in our times. We shall however content ourselves with com-parisons of the technical mathematical aspects, with some modest hints of theirpossible significance. The differences in various individual aspects would perhapshave various possible explanations, or they could even be coincidental. Howeverit seems worthwhile to “identify” them so that eventually a more comprehensivepicture may emerge, throwing light on broader historical issues relating to thetransformation of the ´Sulvas¯utra literature over its period, and in turn of the Vedicpeople.
One of the most striking distinctive features of the K¯aty¯ayana ´Sulvas¯utra is itsrather small size compared to the others (as the alert reader may have noted fromthe figures mentioned above in this respect). What is this economy in aid of?Broadly speaking the difference in size is accounted for by the fact that K¯aty¯ayanadoes not go into much detail of the construction of the individual citi s, on whichBaudh¯ayana, for instance, expends considerable amount of space, many chapters interms of the later organization of the text. K¯aty¯ayana ´Sulvas¯utra is largely focussedon “theory”, though there are some parts touching upon arrangement of the vedi sand specific features of certain citi s. Why was such a policy adopted? It wouldseem that the practice of yajna s become too diffuse for it to be worthwhile to go intothe details, and it was considered best to confine to discussing “general principles”.Notwithstanding the small size, one does not find “stinginess” in the discussion ofthe mathematical parts. In fact in some cases there is a lingering discourse on whatmay be considered quite elementary. An example, though rather an extreme one,is the part with the s¯utra translated as “Square on a side of 2 units is 4, on 3 unitsit is 9 and on 4 units it is 16.” which is then followed by the general statement forintegers and then again separately for select fractions , and for the length ofthe side. 4 Presentation of measures
The Baudh¯ayana ´Sulvas¯utra gives at the outset (in s¯utra 1.3, right after two s¯utraswhich are in the nature of an “Abstract” for what is to follow) names of lengthmeasures with various magnitudes. There are 18 of them, the smallest being tila (sesame seed) is th of an a ˙ngula (finger) and the largest one is ¯is. ¯a (pole) whichis 188 a ˙ngula s; the commonly occurring large unit however is purus. a (man), whichis 120 a ˙ngula s. The a ˙ngula measure was about th of an inch, or about 1.9centimetres. Many of the units occur infrequently, only in the description of specific vedi s.K¯aty¯ayana ´Sulvas¯utra involves only 9 units, all from Baudh¯ayana’s list, ex-cept that a term vitasti is used for a measure of 12 angula s, which is pr¯ade´sa inBaudh¯ayana, and pada is also taken to be same measure (12 angula s). The unitsrange only between the a ˙ngula to ¯is. ¯a , thus excluding in particular the fine unit mea-sures an. u and tila ; these are also absent in ¯Apastamba and M¯anava ´Sulvas¯utras.Unlike in the Baudh¯ayana ´Sulvas¯utra no systematic listing of the units is found inK¯aty¯ayana ´Sulvas¯utra. On the face of it this may suggest a greater standardisationof units over the period, but on the other hand it could also be because K¯aty¯ayanadoes not deal with many practical situations as involved in Baudh¯ayana. The factthat the small units do not appear in K¯aty¯ayana, as also in ¯Apastamba and M¯anava´Sulvas¯utras, shows that they were not part of the practical need in the Vedic prac-tices, and feature in Baudh¯ayana on account of specific theoretical preoccupations,corroborating Thibaut’s hypothesis connecting tila to a term in the expression for √
2. The theoretical inclination seems to have been lost over a period.
Unlike the earlier ´Sulvas¯utras K¯aty¯ayana gives explicitly a prescription for locatingand fixing the cardinal directions, over any day. The east-west line is obtained asthe line joining the two points on a circle drawn around the base of a pole wherethe shadow of the tip of the pole falls on the circle in the course of a day. Thenorth-south line is then obtained through a process of drawing a perpendicular to It was postulated by Thibaut (see [12], page 15) that the unit owes its origin to the fact thatthey had a formula for √ . Many of the intermediate units do not bear a simple fractional relation with purus. a however;e.g. a b¯ahu is 36 a ˙ngula s, a yuga is 86 a ˙ngula s, etc..
Kr. ttik¯a, ´Sravan. a or Pus. ya or as the midway of the directions of rising of
Citr¯a and
Sw¯at¯i . Presumably bothprocedures coexisted, and used for confirmation of each other; one wonders how-ever why then they were not mentioned together. Notwithstanding the reasons forthis and whatever their mutual role in practice, the procedure as above marks asignificant advance from a broader mathematical point of view.
Towards construction of the basic figures needed to be drawn, viz. rectangles,isosceles triangles, symmetric trapezia, with prescribed sizes, the ´Sulvas¯utras prin-cipally describe the steps for drawing perpendiculars to the line of symmetry (sucha line was a common feature of the figures involved, it being along the east-westdirection); these are however packaged into complete procedures, as in a manual,for drawing the desired figures; see [1] for a discussion on this.Baudh¯ayana’s well-known construction of the square involves the method ofdrawing a perpendicular that is now a familiar compass construction in school ge-ometry; given a line and a point on it, at which the perpendicular is to be drawnto the line, one picks two points on the line that are equidistant from the pointand located on opposite sides, and draws arcs with centres at the points with ra-dius greater than the distance from the point - the arcs intersect in two points,one on each side of the line, and joining them provides the desired perpendicularto the line passing through the given point. In this form this method is absent inK¯aty¯ayana, though a variation may be said to be involved in K¯aty¯ayana’s prescrip-6ion for locating the north-south direction, after the east-west direction is drawn, asmentioned above. On the whole during the entire ´sulva period it was not commonto use the compass construction as above for drawing perpendiculars, and it seemsto have disappeared by the time of K¯aty¯ayana. A method, known as
Nyancana (also called
Niranchana ) method, was more prevalent, and in K¯aty¯ayana it appearsas the “canonical” method for drawing perpendiculars. The method is based onthe converse of Pythagoras theorem, that in a triangle with sides of lengths a, b, c if c = a + b then the sides with lengths a and b are perpendicular to each other; see [1] for details on the method and its convenience as a tool. For the a, b, c asabove one uses what we now call Pythagorean triples, the three being integers suchthat c = a + b . The same of two such triples, (3 , ,
5) and (5 , , Nyancana .Baudh¯ayana includes a list with 5 primitive Pythagorean triples, including theabove and also (8 , , , ,
37) and (7 , , Nyancana method (they are noted right after the statement of thePythagorean theorem, and presumably meant as illustrations of the theorem - see[1]). In K¯aty¯ayana there is no mention of any of these other triples (or of other newones), though in ¯Apastamba we find four of the above triples, excluding (7 , , Mah¯avedi by the
Nyancana method.
The most notable feature of the ´Sulvas¯utras in terms of geometric theory is thestatement of the so called Pythagoras theorem. This stands out especially in thecontext of the fact that some of them, especially Baudh¯ayana, possibly predatePythagoras. Actually neither the notion of a right angle nor of a right angledtriangle are found in the ´Sulvas¯utras, as concepts; of course right angled trianglesappeared as parts of various figures, and were implicit in the
Nyancana operations,but were not identified separately. Thus the statement of the Pythagoras theoremoccurs not with respect to right angled triangles, but rather with reference torectangles. A close translation of how it is stated in Baudh¯ayana would be “thediagonal of a rectangle makes as much (area) as (the areas) made separately by thebase and the side put together”. The same statement also appears in K¯aty¯ayana It was believed at one time that the ancient Egyptians also adopted such a method, but ithas subsequently been discounted - see [3]. In the case of the ´Sulvas¯utras however such a methodis seen all over the place. iti ks. etrajn¯anam ”. The term ks. etra involved in this has been translated as “area” by Thibaut, but as “figure” byDatta [2]. It is argued in [11] that ´Sulvas¯utras use the term bh¯umi for area, so theexpression as above means “this is the knowledge about plane figures”. Whateverbe precise nuance of the meaning, the clause is evidently intended to emphasizethe importance of the statement to the reader. In this respect it has a pedagogicalvalue which seems significant.The Pythagoras theorem is also applied in the ´Sulvas¯utras for constructingsquares with area equal to the sum of areas of two given squares (including doublingof a square, called dvikaran. i which is described separately), and the difference ofareas of two given squares (with unequal areas). The constructions, which areof course a direct application of the Pythagoras theorem, are illustrated in thefollowing Figure; see [1] for details.Figure 1: The thick lines give the sides of the squares with areas equal to the sumand difference, respectively, of two given squares, with bases PA and PB as in thefigure.The procedure for “squaring” of the difference of two squares is also used in the´Sulvas¯utras, except M¯anava ´Sulvas¯utra, for squaring a rectangle, by first expressingit as a difference of two squares (by moving around half of the extra part on thelonger side); see [1] for details.The augmentation of squares was used systematically in the ´Sulvas¯utras forreplicating given figures in larger size, by simply enhancing the size of the referenceunit by the desired amount; e.g. to produce a replica of a figure with area 7 purus. a to one with area 8 purus. a , as was required, the unit would be changed8n a way that the area will increase by a factor of 1 + ; the side for this wouldbe obtained as a combination of the original unit with a square of area th ofit. In fact this problem may have been the inspiration for their discovery of thePythagoras theorem; see [1] for a discussion on how they may have arrived at thetheorem.While the conceptual framework in this respect is common to all the ´Sulvas¯utras,K¯aty¯ayana is seen to deal with some of the features involved more dexterously thanin the earlier ´Sulvas¯utras. The use of the method combining squares and rectan-gles seems to have become by now an art, with the individual steps dealt withalmost casually. In the construction of the dron. aciti (s¯utras at the beginning ofChapter 4) for instance, it is quite casually prescribed to divide the square into 10parts and to make one of them into a square and the rest into another square. Onthe problem of augmenting the unit for the purpose as described above, K¯aty¯ayanaintroduces a “rule of thumb” method; it however does not seem to conform to thestandard stipulations accurately; apparently it was decided to pay a price in termsof accuracy in aid of simplicity of execution.There is one especially notable mathematical observation found in K¯aty¯ayanain this context. It is the procedure to produce a square with area equal to anydesired multiple of the unit. In the general ´Sulvas¯utras spirit this could be doneby augmentation of squares of smaller squares, starting with complete squares.K¯aty¯ayana proposes a direct method, that it be constructed as the altitude of anisosceles triangle whose base is one less than the desired number (of area multiple)and the two equal sides add to one more than the desired number. This is aninteresting application of Pythagoras theorem and the identity ( n + 1) − ( n − = n . While there is indeed nothing quite like an abstract “variable” n inthe modern sense involved, the identity seems to have been realised in terms of thedesired number of unit squares to be combined ( y¯avatpram¯an. ¯ani samacaturasr¯an. i ),presumably by inspection of square grids. K¯aty¯ayana ´Sulvas¯utra gives the same method as Baudh¯ayana for “transforming”a square into a circle, namely for producing a circle with area equal to that of the In [11], both in the translation of the s¯utra for this (4.2 on page 123) and the commentaryon it page 268, it is said that the square is to be divided into 100 parts; that interpretation ishowever incorrect, as can be seen from [6] and [7].
P R as seen in the Figure 2 (see [1] for details).Figure 2: Circling the square: the thick line PR is given as the radius of the desiredcircle with area equal to that of the square ABCD.While the procedure is interesting (especially in its application of an intuitive“mean-value principle”) it is not very accurate. The area of this circle for the unitsquare works out to be 1 . ... , about 1 .
7% more than that of the originalsquare, and a computation of the value of π with it comes to 3 . ... ). Despitethis, there seems to have been no change made from Baudh¯ayana to K¯aty¯ayana. Onthe other hand the M¯anava ´Sulvas¯utra, though less “sophisticated” than either ofthese, and substantially older than K¯aty¯ayana, seems to contain a more accurateprocedure for producing a circle with the area of a given square; see [1]. TheMaitr¯aya ˙n¯iya ´Sulvas¯utra, which is akin to M¯anava ´Sulvas¯utra gives a constructionwhich involves taking the radius of the desired circle to be times the side of thegiven square; see [4]. Both of these involve only about percent error.The converse problem of “squaring the circle” viz. finding a square with thearea of a given circle is also considered in the ´Sulvas¯utras. Typically the treatmentis not geometric, but by assigning a numerical relation between the side of thedesired square to the diameter of the given circle; according to [5], sutra 3.2.1010rom M¯anava is in fact a geometric construction for squaring the circle, but weshall not go into it here; see [1]. Baudh¯ayana ´Sulvas¯utra gives two formulae forsquaring the circle. The first one gives the value for the side of the square witharea equal to that of the circle with unit radius to be78 + 18 × − × × × × × . ( ∗ )The area of the square with that as the side works out to be 3 . ... , about 98 . th of the diameterof the given circle for the side of the desired square. The second is qualified byBaudh¯ayana as an “incidental” ( anitya ) method for squaring, signifying that notwishing to use the cumbersome formula one could do with this approximate one.Curiously, only this crude formula, giving a value that is smaller by 4% has beendescribed in K¯aty¯ayana ´Sulvas¯utras (as also ¯Apastamba) for the purpose. Three of the four ´Sulvas¯utras, Baudh¯ayana, ¯Apastamba and K¯aty¯ayana, describea formula for √ × − × × . In decimal expansion the value works out to be 1 . . . . , and is noted to beaccurate upto 5 decimal places. The s¯utra giving the formula is followed by “ savi´ses. ah. ” in Baudh¯ayana, “ savi´ses. ah. ” in ¯Apastamba and “ sa vi´ses. a iti vi´ses. ah. ” in K¯aty¯ayana. The word vi´ses. ah. means “extra”. However it does not refer to any comparison with an accuratevalue of √
2. It is known from the tradition of ´sulvavid s that vi´ses. ah. was used as atechnical term for the difference between the dvikaran. i , namely √
2, and the unit(signifying what comes up as extra in terms of the side, while doubling a square),and in conjunction with it savi´ses. ah. stood for the dvikaran. i itself. Occurrence ofthe phrase following the s¯utra is what tells us what the number in the s¯utra standsfor (the rest of the s¯utra only provides a number and contains no reference to whatit is). It may be recalled here that the Babylonians also had a similarly close approximation for √ . . . . , expressed in the sexagesimal system that they used.
11n [11] the second part “ iti vi´ses. ah. ” has been translated as “this is approxi-mate.” (curiously, reference the s¯utra is missing from the commentary section inthe book); see also [9], pp. 21. However, there does not seem to be adequatejustification for interpreting or connecting the part with an assertion about thevalue being approximate. Khadilkar [6] translates the part, in Marathi, as “Ha dvikaran. i t.haravin. yaca niral.a prak¯ara.”, or “This is a different method of deter-mining dvikaran. i ”. The Karka bh¯as.ya commentary seems to confirm this; see [6]for the commentary. The overall context seems to favour this interpretation. Fromall indications the close to accurate value of √ ∗ ) asabove; see [10] and [1] for detailed argument in this respect. However, subsequentto Baudh¯ayana somehow no one seems to have been interested in that formula,they being content to use the simple proportion of 13:15 for the desired ratio. Theformula for √ dvikaran. i , which actually for their purpose they couldsimply measure out from the diagonal with a rope.There is also another interpretation possible. As noted above vi´ses. ah. standsfor the excess of the dvikaran. i over the unit. The s¯utra thus seems to say that vi´ses. ah. is such that sa vi´ses. a is given by the previous expression. This is suggestedby the translation in [7], where iti vi´ses. ah. is translated, in Hindi, as “Yaha vi´ses.aki vy¯akhy¯a hai”. It may also be recalled here that the word “ iti ” in Sanskritcorresponds to “in this manner”, “thus” or “as you know” (see [8]) which fits wellwith this interpretation.Of course it would have been known, at least when the formula √ My translation from the Marathi version. Concluding remarks
On the whole one notices that in the directions which were applicable to the prac-tical issues they met with, in terms of constructing various rectilinear figures withconditions on the area etc., there was progress in terms of simplifying the proce-dures and devising new ones. However not much attention seems to have been paid,collectively, to preserving interesting findings, even those with high aesthetic qual-ities, that were not directly involved in regular practice. In some ways this couldbe the result of a diffused organisation, with feeble communication and inadequateopportunities for intellectual interaction.
Acknowledgement : The author would like to thank Kim Plofker for some usefulcomments on an earlier version of the manuscript and Manoj Choudhury for pro-ducing soft versions of the figures.
References [1] S.G. Dani, Geometry in the ´Sulvas¯utras, Studies in the history of Indian mathemat-ics, 9 - 37, Cult. Hist. Math. 5, Hindustan Book Agency, New Delhi, 2010.[2] Bibhutibhusan Datta, Ancient Hindu Geometry: The Science of the ´Sulba, CalcuttaUniv. Press. 1932; reprint: Cosmo Publications, New Delhi, 1993.[3] R.J. Gillings, Mathematics in the Time of Pharaohs, Dover Publications, New York,1972.[4] R.C. Gupta, ´Sulvas¯utras: earliest studies and a newly published manual, Indian J.Hist. Sci. 41 (2006), 317 - 320.[5] Takao Hayashi, A new Indian rule for the squaring of a circle: M¯anava´Sulvas¯utra3.2.9-10, Ganita Bharati 12 (1990), 75-82.[6] Shri. Da. Khadilkar,
K¯aty¯ayana ´sulba s¯utre (in Marathi), Maharashtra Rajya SahityaSanskrity Mandal, Mumbai, 1974.[7] Raghunath P. Kulkarni,
Char Shulbas¯utra (in Hindi), Maharshi Sandipani RashtriyaVedavidya Pratishthana, Ujjain, 2000.[8] Monier Monier-Williams, A Sanskrit-English Dictionary, Motilal Banarasidass,2002. S.G. DaniUM-DAE Centre for Excellence in Basic SciencesVidyanagari Campus of University of MumbaiKalina, Mumbai 400098IndiaE-mail: [email protected]@cbs.ac.in