The Kriyākramakarī’s Integrative Approach to Mathematical Knowledge

The purpose of this paper is to review the general organization of knowledge in the Kriyākramakarī, a sixteenth-century treatise of Kerala mathematics. Specifically, I will argue that the authors' interest in justification or proof is integrative, rather than hierarchical or cumulative. In other words, the purpose of proofs in the Kriyākramakarī is to connect various different aspects of mathematics, rather than just establish results by means of previously known results.

1 The title can be translated as something like "the performer of activity in due order," but this literal translation is not very meaningful here. The three words in the title are derived from the root √kṛ, which is associated with the semantic field of doing, acting, and performing ("create" is an English derivative of this Indo-European root). The title is therefore a sort of pun, centered round the word krama, which, in a mathem-atical, context could mean something like the correct sequence of an algorithmic procedure or calculation rule. 2 For details about the authors see Sarma 1972: 130, 169;Joseph 2009: 21-2. 3 For surveys of Kerala school mathematics see : Sarma 1972;Joseph 2009;Plofker 2009: ch. 7;Puttaswamy 2012: ch. 13; for translated editions see Sarma et al. 2009;Ramasubramanian and Sriram 2010. history of science in south asia 6 (2018) 84-126 roy wagner 85 The Kriyākramakarī is, to risk an anachronistic expression, an encyclopedic work. It follows the verses of the Līlāvatī, but, unlike a typical commentary, is not content with an interpretation of the verses and illustrative examples. It complements Bhāskara's verses by related verses drawn from other treatises (named and unnamed -the latter may sometimes be original additions), and expands the scope by including relevant methods and topics that were not covered by Bhāskara (for a summary of the content of the Kriyākramakarī and a specification of its most important additions see Appendix A, p. 99). But even more important than its extended scope is the fact that the Kriyākramakarī includes detailed justifications (upapatti or yukti) of most of the methods and statements which it includes, whether Bhāskara's or others'.
I believe that even after Nārāyaṇa's supposed completion of the treatise, the Kriyākramakarī should not be considered a complete work -at least not based on the four manuscripts used for its critical edition (Sarma 1975). The manuscripts indicate various substantial gaps in the source material (Sarma 1975: 99, 145, 153,164, 177, 211, 295). These gaps sometimes correspond to what appears to be unfinished treatment of the subject matter, and this is sometimes the case even where no gap is indicated. This may be due to a defective archetype of the manuscripts, but some of the summary verses that conclude the various sections appear in more than one variant, and some summary verses are appended at the end of the treatise, rather than where their subjects are covered, which suggests that Śaṅkara's text itself contained lapses. Nārāyaṇa did complete the treatise in the sense of adding commentary to verses 200-269 of the Līlāvatī, but he does not seem to have intervened in the gaps and hiccups left in Śaṅkara's draft.
Several publications treat specific portions of the Kriyākramakarī, including some translations of extracts. Among these, one can find the approximation of (Katz 2007: 481-93), Govindasvāmin's arithmetic rules quoted in the Kriyākramakarī (Hayashi 2000), and Citrabhānu's 21 questions (Hayashi and Kusuba 1998;Mallayya 2011;cf. Wagner 2015). Some of the mathematical contents of the book is also covered (without translating the source material as such) in Vrinda (2014), Mallayya (2002), Gupta (1987), Sarasvati Amma (1979), and the general surveys of the Kerala school quoted above.
The purpose of this paper is not to review the treatment of any specific mathematical subject in the Kriyākramakarī, but rather to review its general organization of knowledge. Specifically, I will argue that Śaṅkara's presentation of justification or proof is integrative, rather than hierarchical or cumulative. In other words, the purpose of proofs in the Kriyākramakarī is, among other things, to connect various different aspects of mathematics, rather than just to convincingly establish or explain mathematical claims by means of previously known claims.
The next section provides general background on proofs in medieval Indian mathematics. This is followed by a section that surveys sources of knowledge history of science in south asia 6 (2018) 84-126 86 the kriyākramakarī 's integrative approach used in the proofs of the Kriyākramakarī. The subsequent section will present the evidence for the Kriyākramakarī's integrative approach.

P ROO F I N M E D I E VAL I N D IA N M AT H E M AT I C S
S ome (but not all!) historians of mathematics used to claim that Indian mathematics had no interest in proofs (Sarma et al. 2009: 267-70). Part of the problem was that non-commented Sanskrit mathematical texts are mostly succinct verse summaries of mathematical algorithms and results, which include no justifications. However, these verses are highly elliptic, and can hardly be deciphered without the aid of a qualified teacher or a detailed commentary. The existence of Indian commentaries that include justifications was already made known to English language readers by Whish (1834), and then again by Indian scholars publishing in English since at least the 1940s (Marar and Rajagopal 87 obtain these goals, one relies on perception (pratyakṣa) and inference (anumāna), supported by authentic tradition (śabda, see Sarma et al. 2009: 286 f.). Proofs are therefore not meant to be part of a purely logical system founded on axioms, but they are not simple empirical generalization from examples either. In Indian mathematics, sense, reason, and authenticated traditions combine to form mathematical proofs just as they do in other Indian scientific contexts. Mathematics is thus not grounded in an exceptional epistemology, when compared to natural sciences.
If we follow this thread, then a proof is not supposed to meet a-priori criteria for absolute truth (as in the classical Greek proof architecture or later logical reconstructions), but only to answer those doubts, misunderstandings and dissents that happen to arise in actual practice. Moreover, according to Raju (2007: ch. 2), Srinivas (2015) and Divakaran (2016), some of the logical traditions most relevant for mathematical proofs are not bivalent, and may therefore allow some forms of contradictions (for instance, regarding fictional entities). Since imaginary, nonobservable and idealized entities are part of Indian mathematical-astronomical calculation and reasoning, Indian astronomers were sometimes ready to accommodate inexplicable or even seemingly contradictory procedures as component part of their models. (Srinivas 2015: 232) Despite this lack of interest in an absolute ground, one can find in Sanskrit mathematics statements that might seem to be foundational (but later I will suggest a different interpretation). These statements usually assign to the rule of three and the "Pythagorean Theorem" the role of a foundation, at least in the context of astronomical mathematics. The Yuktibhāṣa, an important treatise of the Kerala school, states that, most of mathematical computations are pervaded by this … 'rule of three' and the … 'rule of base height and hypotenuse'…. All arithmetical operations like addition etc. function as adjuncts to the above. (Sarma et al. 2009: 30) Nīlakaṇṭha, another proponent of the same school, makes a similar statement in the context of astronomical calculations (Śāstrī 1930: 100). A similar exaltation of the rule of three is available also in the Līlāvatī's verse 241 (Sarma 1975: 434). However, these statements should not be taken too seriously as foundational statements. Indeed, the "Pythagorean Theorem" and, to an extent, also the rule of three are themselves subject to justification, and obviously cannot have all Sanskrit mathematical reasoning reduced to them.
Following on this question of foundation, I would like to qualify one of the characterizations of medieval Indian mathematical proofs made by Srinivas history of science in south asia 6 (2018) 84-126 88 the kriyākramakarī 's integrative approach (2015): that proofs proceed from the known or established to that which is to be established. This view might be interpreted as restricting proof to synthetic reasoning. But medieval Indian mathematical proofs (as well as those of many other historical and contemporary mathematical cultures) are often enough heuristic or analytic, in the sense of starting from that which is to be established, and deriving its necessary conditions. A synthetic verification that those conditions are also sufficient is sometimes lacking, and the very distinction between deduction and abduction is not salient in many proofs.
Indeed, when reading a mathematical justification, the sequence of written statements does not necessarily correspond to their inferential order (compare "I came because you called" and "you called, so I came"). The ambiguity of some Sanskrit adverbials and the free word order allowed in Sanskrit verse make it sometimes difficult to decipher the intended logical order -assuming that a clear-cut order was actually intended. Modern reconstructions of proofs in Sanskrit mathematical literature often suppress this ambiguity and impose logical clarity where the sources are ambiguous (note that this is not something special to Sanskrit -translations sometimes prefer -or are forced to -translate an ambiguous term in the original by a univocal term).
The traditional Sanskrit term for proof, upapatti, stems from the grammatical root √pad, with the prefix upa, and has to do with approaching, reaching and occurrence and production, which may suggest a linear advance from established knowledge to new knowledge, or an account of how knowledge is born. But the other term, yukti, which has become more popular in Kerala mathematics (Divakaran 2016), has a different semantic field. The root √yuj has to do with tying or connecting things together (hence the English "yoke" and Sanskrit yoga; yukta is one of the terms for the arithmetical adjective "added"). This semantic field does not suggest a directionality of reason, but the integration of different pieces of knowledge. 5 Regardless of whether this etymology reflects a conscious choice or just the unconscious vicissitudes of language, I will show how a principle of integration of knowledge, rather than one of linear progression, manifests itself in the Kriyākramakarī. 6 5 This interpretation is already suggested in a different context by Wujastyk (2003: 25). 6 I note also that the above terms sometimes accompany what we would consider an illustrative example, rather than a proof. Fur-thermore, these terms do not explicitly accompany all proofs or justifications in the text. Therefore, identifying these terms with contemporary mainstream notions of mathematical proof may be problematic. history of science in south asia 6 (2018) 84-126 roy wagner 89 2. CO M P O N E NT S O F P RO O F I n this section i will describe the main basic sources of knowledge used in the Kriyākramakarī. I did not find sufficiently detailed explicit discussions of the components of mathematical proof in the Kriyākramakarī itself, so, as a tentative ersatz, I will apply the above quoted epistemological division into perception, tradition and inference. Note, however, that this division is not made salient in the Kriyākramakarī or other related mathematical texts, and therefore might not be the authoritative way to organize this material. Moreover, I do not restrict my use of the above categories to their indigenous theoretical meaning. My purpose here is to describe the sources of mathematical knowledge used in the Kriyākramakarī, not how its authors would have classified them. perception An example for an application of perceptual knowledge that stands out most clearly for a reader versed in contemporary mathematical standards is the following. Two parallel bamboo reeds have strings connecting the root of each to the tip of the other.
When there's equality [in the size of the reeds], the intersection of the strings is in the middle of the space between them; when there's no equality, the intersection of the reeds is near the smaller cane. 7 But this kind of observational justification referring to non-mathematical entities is not used frequently in the Kriyākramakarī. 8 A more obvious and prevalent perceptual source of knowledge is cut-andpaste geometric arguments. 9 In the history of mathematics, geometric diagrams are most strongly associated to the classical Greek tradition, but they are found in most other mathematical cultures as well. While classical Greek mathematical reasoning depended on a complementary relation between a lettered diagram and a highly formulaic text (Netz 1999), in other cultures diagrams did not depend on letters or on a strongly regimented deductive system. 7 Sarma 1975: 311, verse 2: सा े वे वोर रालभू म े सू ऽयोयु ितः। सा ाभावे ऽ वं श िनकटे सू ऽयोयु ितः॥ 8 The Yuktibhāṣa has a striking observational explanation that uses interlocking slanting beams in a roof structure to justify the similarity of some triangles (Sarma et al. 2009: 184 f.). 9 One reviewer suggested that this would not be considered perceptual knowledge by the authors of the Kriyākramakarī, due to the absence of actual diagrams in the text. Since the authors do not explicitly analyze proofs using this term, it is hard to decide on this issue. history of science in south asia 6 (2018) 84-126 90 the kriyākramakarī 's integrative approach A famous anecdote from the Indian tradition is Bhāskara II's diagrammatic proof of the "Pythagorean Theorem" accompanied by a single word: "behold!" But surviving sources do accompany such diagrams with instructions that clarify their meaning. In fact, one is more likely to find a geometric cut-and-paste argument without a diagram than a diagram meant to justify a general claim without the accompaniment of a textual argument. The Kriyākramakarī contains quite a few arguments of this kind, especially in the context of quadratic and cubic identities, the summation of progressions, and, of course, geometric area calculations. A less standard diagrammatic argument involves cutting a circle into sectors, and fitting them together to form an approximate rectangle to justify the formula for the area of a circle (Sarma et al. 2009: 264, see Figure 1).
Other sources of knowledge that may arguably be placed under the heading of "perception" (and here we are being anachronistic) include elementary arithmetic rules and combinatorial reasoning. For example, what we would call "the distributivity of multiplication" is introduced in the following verse: When a multiplier subtracted by something is multiplied, the multiplicand should be multiplied by that something and subtracted [from the product of the original multiplier and multiplicand]. If something is added, it [the product] should be added. 10 This could be read as an observation of what happens when one performs a multiplication according to its definition as repeated addition (later this is also explained by cut-and-paste geometry). 10 Sarma 1975: 17, verse  In the context of combinatorics, the counting of the number of meters with exactly 8 syllables, of which exactly are to be long syllables, is explained as follows: successively choose places for the long syllables, and then cancel the repetitions by dividing the total number of arrangements obtained by the number of repetitions of each arrangement. Here again, the number of repetitions is observed, rather than derived. Note that this is true for contemporary combinatorics classes as well. 11 authority Commentaries can be polemical and argumentative, as was often the case in the Greek, Arabic and Latin cultures. Indian mathematical commentaries tend, however, to be more respectful toward their sources. Even when information has to be complemented or made more precise, they often (but not always!) present it as the fault of the uneducated reader, rather than that of the master, who simply took things for granted. Nevertheless, authority was not followed blindly, and Srinivas (2015) and Divakaran (2016) show that when observation contradicts authority the latter has to retreat or, at least, find excuses (perhaps things have changed since the time of the old masters…).
The Kriyākramakarī accepts the authority not only of its source, the Līlāvatī, but of many other authors, listed in appendix A. The verses of these authors are sometimes brought simply to state the rules of the Līlāvatī in different words, and sometimes to provide additional methods and introduce problems not covered in the Līlāvatī. The accumulation of different authorities echoing and complementing each other obviously serves to strengthen the reader's sense of conviction.

inference (and other forms of reasoning)
The Kriyākramakarī includes one substantial reference to the anumāna logic system, relying on it to explain the rule of three: Just as smoke on a mountain indicates a fire, so does smoke in the kitchen; similarly, a given ratio between the price and quantity of some fruit is preserved when a different quantity is to be bought. In both cases there's a common "law" that connects the cause and effect in the two compared situations (see Hayashi 2000: section 3.2, and the translated verses in Appendix E below). the kriyākramakarī 's integrative approach In a broader sense (and again, risking anachronism), one might use the same kind of framework for applying known results in justifications. For example, just as the product of sum and difference equals the difference of squares in one context, so does the equality apply in another. But such inferences are not explicitly presented as applications of anumāna in the Kriyākramakarī, and are perhaps better treated as naïve deductions that are not theoretically grounded, or, to use Keller's terms, a form of re-interpretation.
Another form of reasoning used in the Kriyākramakarī is calculation. This is obvious when it comes to actual calculation with specific numbers in solved examples, but perhaps not so obvious when it comes to calculations applied to undetermined quantities. This is not proper algebra, as it does not use Bhāskara II's algebraic terminology from his Bījagaṇita. But calculation algorithms are applied rhetorically to undetermined quantities (e.g., root extraction applied to a term containing an arbitrary square -see the first rule in Appendix B below) in order to derive conclusions (e.g., that it has a rational square root).
Next comes inversion. If a certain procedure leads one from the datum to a result, then reversing the procedure and applying it to the result should retrieve the datum. This is brought up explicitly as a problem solving method (vyastavidhi) in the Sanskrit tradition. The Kriyākramakarī also uses it to justify some calculation procedures, such as root extraction, which is an inversion of the squaring procedure.
Mathematical induction can also be found in the Kriyākramakarī, for example, in the proof that the sum of squares up to equals ( + 1) (2 + 1) 6 .
The proof starts by multiplying everything by 6. Then, it considers the last term of the sum, 6 2 . This term can be interpreted as the number of blocks in the external half-shell (two levels of base and two adjacent walls) of a box of size × ( + 1) × (2 + 1) Applying the same reasoning to the previous terms fills up the entire box (see Katz 2007: 493-8 for Nilakaṇṭha's version from Śāstrī 1930, which is very similar to that of the Kriyākramakarī). 12 While mathematical induction is never stated explicitly as an independent principle, it is used in several geometric proofs of summations.
12 One might object that this is not proper induction because it goes backwards (from down) rather than forward. But in fact, induction should go backward. The differ-ence is imperceptible in proofs of simple integer formulas, but can lead to confusion in more complex situations. history of science in south asia 6 (2018) 84-126 roy wagner 93

TH E O RGA N I Z AT I O N O F K NOW L E D G E
T his section will characterize the Kriyākramakarī's approach to the organization of knowledge, arguing that it presents an integrative, rather than a hierarchical view of mathematical justification. I will start with the way specific problem types are treated, and then move to the organization of the treatise as a whole.
permuting givens and unknowns Consider, for example, the treatment of arithmetic progressions. In the Āryabhaṭīya (fifth century), we are taught how to sum a progression given its initial term, difference and number of terms, and how to derive the number of terms from the other data. The Līlāvatī also adds procedures to determine the first term and difference from the other terms (this is easily obtained by inversion). We then get a system of four parameters, any three of which determines the fourth. For the Kriyākramakarī, this form of completion is a general principle in organizing the treatment of problems.
For example, in the case of the Līlāvatī's barter problems, one is given the price of a certain amount of one commodity and the price of another amount of another commodity, and is required to compute the exchange rate between the two commodities. The Kriyākramakarī adds problems where the price of one commodity is given together with the exchange rate of the two commodities, and one has to calculate the price of the other commodity. A similar approach is taken in the context of the rule of 5, where each parameter, rather than just the yield (icchāphala), is separately set as unknown (Sarma 1975: 195 f., 204).
Elsewhere, the Līlāvatī discusses the following problem: given two rightangled triangles sharing a given side, find their bases and hypotenuses from Consider, for example, the following false theorem: "All simple graphs without isolated vertices are connected". Here is the inductive "proof." For the case of 1 vertex, the theorem is true by default, and for 2 vertices the only simple graph without isolated vertices is an edge, which is indeed connected. Now suppose the theorem is true for vertices, so that every graph of vertices without isolated vertices is connected. Add one more vertex. By hypothesis, it is not isolated, so it is connected by at least one edge to the rest of the connected graph. Therefore, the entire graph connected.
Nevertheless, the theorem is wrong for 4 vertices (two disjoint edges have 4 vertices, none of which is isolated). A correct proof should have started with a graph of + 1 vertices, none of which is isolated. Then, removing one vertex and its edges, the remaining graph might have an isolated vertex (as in the example of the two disjoint edges above), so the inductive hypothesis could not have be applied, and the proof would fail. The moral of this is that an inductive proof should consider a general ( + 1) th case and cite cases of size or less to derive the conclusion. history of science in south asia 6 (2018) 84-126 94 the kriyākramakarī 's integrative approach the difference of their bases and the difference of their hypotenuses. The Kriyākramakarī complements this problem by considering all combinations of sums and differences of the bases and hypotenuses (Sarma 1975: 423-28).
So far, this is not unique to the Kriyākramakarī or to medieval Indian mathematical commentaries. The view of a problem as a set of connected quantities, each of which should be obtained from the others, can be found in many mathematical cultures. What emerges as a feature specific to Kerala mathematics is that it becomes an explicit strategy with a generic title: the so-and-so many questions (praśna) or questions and answers (praśnottara).
A famous example is Citrabhānu's 21 questions and answers, mentioned above, which survives in the Kriyākramakarī (Sarma 1975: 108-26;Hayashi and Kusuba 1998). Here given any two of seven possible combinations of two unknowns (involving sums, products, squares and cubes), the other quantities are to be reconstructed. The Yuktibhāṣa too has a section called 10 questions and answers, based on five out of those seven quantities (Sarma et al. 2009: 20-22). Note that in Arabic algebra, the treatment of such questions would be very different: the problems would be reduced to one of the canonical equations in one variable (e.g., 2 = + ), and solved using the appropriate procedure. Here, instead of reducing to a basic canonical form, the system of different problems is explored.
This approach applies to astronomical-mathematical problems as well. The Tantrasaṅgraha applies the same organization to five spherical-trigonometric quantities, of which any three are given (Ramasubramanian and Sriram 2010: 200-28), and the Yuktibhāṣa introduces a set of 15 problems based on six spherical-trigonometric quantities, any two of which are given (Sarma et al. 2009: 533-40). When the Kriyākramakarī introduces the problem of the shadow of a sphere on a surface, it is organized as a system of four quantities, such that given any three, the fourth is to be determined (Sarma 1975: 435-37).
We see that an integrative view of problems based on permuting their givens and unknowns becomes an explicit principle for the organization of knowledge in Kerala mathematics. This practice has antecedents elsewhere, and is related to the method of inversion. But the thematization of this practice as a general approach does seem to be specifically endorsed by Kerala mathematicalastronomers. Our next step is to see how this plays out on a larger scale.
relating distant pieces of knowledge should instead connect disparate pieces of mathematical knowledge in various directions.
A first example is the treatment of the formula As one would expect, it is indeed derived from algebraic manipulations (applying the distributive law to the left hand side) as well as from cutting and pasting rectangular diagrams (Sarma 1975: 31-36). What is more surprising is that it is also derived from the fact that the sum of odd integers up to 2 −1 equals 2 (Appendix D). This line of reasoning explicitly connects arithmetic progression, the discussion of squares and roots, and the above quadratic identity. It also justifies a rather simple, and already established quadratic identity by means of a more advanced summing of an arithmetic progression, which is only justified later in the book (Sarma 1975: 241). Another example involves the proof of Heron's formula for the area of a triangle. The proof relies on knowledge about triangles already established in the treatise in order to analyze various proportions in a triangle, eventually combined to produce the formula. But it seems that the author runs into difficulties when attempting to conclude the proof (if my reading is correct -and I am not certain that it is -the author confuses √ 2 + 1 with + 1, and then abandons the original line of proof). At that point, the author notes an analogy between the system of proportions that had been established, and the proportions relating arrows, chords and radii in intersecting circles, which goes back to Āryabhaṭa (see Appendix F). This analogy allows to clinch the proof without going through the entire argument. Note that we don't have here an abstract theorem on proportions that is applied to two different situations (circles and triangles). What we have here is an analogy relating two different geometrical problems, one concerning triangles, and the other concerning circles, using one to enhance our understanding of the other. But the most striking example for my claim is the rich discussion of solutions of quadratic Diophantine variations of the form: "find two squares such that their sum and/or difference together with some given perturbation is a square." Given a solution for such a problem, the most obvious justification would be to plug in the suggested solution, use algebraic identities to rearrange the resulting algebraic expression and verify that the result is a square (we may call this "algebraic synthesis"). While for some rules this is precisely the course taken (rules 0b and 12 as well as the vargaprakṛti method in Appendix B), it seems that the Kriyākramakarī is intent on exploring as many courses of justification as possible.
Other methods of justification include the following (see Appendix B for details): history of science in south asia 6 (2018) 84-126 96 the kriyākramakarī 's integrative approach • Apply the procedure for root extraction of numbers to the general algebraic term provided by the rule, to show that the resulting sum has a square root (Rule 0a) • Use cut-and-paste geometry to show that the resulting sum can be rearranged as a square (rule 1) • In the context of the above, use heuristic reasoning to reconstruct the coefficient that would allow the above procedure to succeed (rule 1) • Heuristically suggest a form for the solution, find circle segments (Rsines, Rcosines, arrows and radii) that model this solution, and use them to specify solutions that fit the heuristic model (rule 2) • Heuristically suggest a form for the solution, and use algebraic identities and manipulation to specify solutions that fit the heuristic model (rules 4-6, 10-11, all using different heuristic models; we may call this "algebraic analysis") • Brahmagupta's vargaprakṛti method (a quadratic variant of the kuṭṭaka method) • Deriving difference equations for solutions from specific applications of the last method (rules 7, 8).
We see here an attempt to connect algebraic analysis and synthesis, rectilinear geometry, circle geometry, root extraction, quadratic Diophantine equations, and their recursive solutions by the vargaprakṛti method. The distinction between these forms of reasoning is not an anachronism -many of them are indigenously understood as distinct, as reflected in the internal divisions of the Kriyākramakarī (see Appendix A). The message, I believe, is one of an underlying unity of mathematics. The same kind of problem is revisited when discussing the construction of rational right-angled triangles (which, in arithmetic terms, means finding two squares that sum to a square, see Appendix C). Here one uses some algebraic manipulations similar to those already used when the algebraic problem was solved by a geometric model. The Kriyākramakarī applies the same treatment, but from an opposite point of view: going from geometry to algebra, rather than vice-versa.
A second solution to the algebraic problem, which is more sophisticated, is justified both by Mādhava's formula for the Rsine and Rcosine of a sum of angles as well as algebraically. Again, we see an attempt to relate as many different points of view to a given family of problems, and no hesitation in applying heavy guns to solve relatively simple problems. At the end of the discussion of the various solutions, they are all united as applications of the rule of three to the same general solution.
We do not have any explicit statements declaring that the authors specifically intend to integrate different aspects of mathematics. One may therefore claim history of science in south asia 6 (2018) 84-126 roy wagner that the purpose is simply to explore many different solutions for each problem (with no intention of integrating mathematical knowledge), or simply to demonstrate virtuosity.
The latter claim seems unlikely, as the Kriyākramakarī often cites and credits others, without ever explicitly claiming anything to be an original invention. The former interpretation, however, is not as easy to rule out. To argue against it, recall that some of the justification and solutions presented are highly inefficient and contrived. If the authors simply wanted to propose different kinds of solutions, they could have achieved this goal by simpler means. Moreover, the "directionality" of knowledge (going from the simple or established to the advanced or not yet known) is not preserved. As this might raise more suspicion than conviction, it's unlikely to assume that the authors' goal was simply variety of justifications, and even if it was, the effect generated is that of an integration of mathematical knowledge, where distant components end up relating to each other. I therefore conclude that the authors sought to present mathematics as a unified whole, rather than as a system of separate problems, each having several different solutions.
I do not claim that integrating distant pieces of knowledge is the only purpose of proofs in the Kriyākramakarī. It is clear that the proofs were meant to convince and explain, and it is clear that the diversity of methods presented in the treatise teaches the reader different approaches to problems. Such approach can be attested elsewhere, for example in the context of Pṛthūdaka's nineth-century mathematical commentary Vāsanābhāṣya, which Keller (forthcoming) characterizes as trying, to make sense of the variety of possible understandings of the text, taken in itself. Such an attitude has often been noted in other scholarly disciplines of South Asia as well.
But in the case of Kriyākramakarī, the overall picture suggests an attempt to relate distant aspects of mathematics in the presentation of mathematical knowledge. 4. CO NC LU S IO N I n this paper i studied the system of justifications in the Kriyākramakarī. The variety of proof methods and the organization of knowledge suggest that the authors are interested in presenting an integrative view of mathematical knowledge, which emphasizes the links between different domains of mathematics.
Within this context, the statements that appear to "reduce" mathematics to the rule of three or the "Pythagorean theorem" turn out to attempt not some naïve version of a foundational approach, but rather an integration of mathemathistory of science in south asia 6 (2018) 84-126 98 the kriyākramakarī 's integrative approach ics. This integration is achieved not only by framing many forms of mathematical reasoning in terms of the two tools above. It is achieved by exploring how we can move back and forth between various parameters that exchange the roles of data and solution in given problems, and by linking disparate areas of mathematics through mutual justifications.
We must note, however, that the findings presented here are highly restricted. In order to present a more general picture concerning the organization of knowledge in Kerala mathematics, we should compare the Kriyākramakarī to other Kerala based commentaries. The Yuktibhāṣa, for instance, is not interested in ingenious justifications of simple mathematics, and glosses over well established mathematics rather quickly. It is clear that its main interest is to provide justifications for the most advanced pieces of mathematical knowledge applicable to astronomy. Nevertheless, even the Yuktibhāṣa sometimes presents several justifications for a single result.
How does this picture relate to other proponents of the Kerala school (e.g., Nīlakaṇṭha)? What about practitioners of other sciences in the same social system or elsewhere? Is a concern with integrating knowledge reflected in indigenous philosophical-logical debates? And how does this relate to vernacular or practical mathematics? All these questions will require further research.

A PPENDI X A : S U M M A RY O F T H E CO N T E N T O F T H E L Ī L ĀVAT Ī AND T HE A D D I T I O N S O F T H E KR I YĀ K R A M A K A R Ī ( K K K )
I n the following table, the verse numbering in the second column follows Sarma 1975. In the third column, when the name of a treatise is not specified, the treatise is unknown. The entries are derived from Sarma's edition, corrections of one of the reviewers and additional observations in Vrinda 2017. Additional examples; summing an arithmetic progression starting from its th term, the notion of a "middle term" when there is an even number of terms. Justification of procedures by symmetry arguments, inversion, and, for the quadratic procedure for retrieving the number of terms, a geometric cut-and-paste argument. Āryabhaṭīya, Śrīdhara Cut-and-paste proof of the "Pythagorean" theorem. The root approximation algorithm (see Appendix A, rule 0a). Algebraic justification of rescaling for root approximation.
Sides of rational right angle triangles: history of science in south asia 6 (2018) 84-126 108 the kriyākramakarī 's integrative approach 6. A P P EN D I X B D isclaimer: the reconstructions here and in the following appendices misrepresent several aspects of the original text. First, they use modern notation, so they do not reflect the internal logic of the original terms and what these terms do and do not render salient. Second, these reconstructions do not represent the rhetorical structure of the original argument, in terms of omission, emphasis and patterns of expression. Finally, they are not even completely true to the logical structure of the original, as the reconstructions turn elliptic verses into a linear chain of arguments, imposing on the text my interpretations concerning the logical order of statements and some gaps (but when I fill in the larger gaps, I do note it explicitly). I allow myself all this because the purpose of this paper is to explore the organization of relations between different aspects of mathematical knowledge, and not the detailed structures of specific proofs. But the reader should be careful when drawing conclusions on a fine grained scale from this bird's-view reconstruction.
I use Sarma's (1975) numbering of the rules. The first, unnumbered rules are marked as rules 0a and 0b. Rules 0-1 are cited from the Līlāvatī. A reviewer noted that rules 2-11 (with the exception of two items under rule 3) are quoted from Parameśvara's commentary on the Līlāvatī, and that rule 12 is due to the Brāhmasphuṭasiddhānta.
2. The proof that 2 + 2 −1 has a root depends on the algorithm for extracting a root. This algorithm can be described as follows: in order to calculate the root of 2 + a. Define the divisor to be 2 . b. Define the dividend to be . c. Divide the dividend by the divisor to obtain a possibly approximate value . d. Update the divisor to 2 + . e. Update the dividend to 2 + − . f. Update the divisor to 2 + 2 . history of science in south asia 6 (2018) 84-126 roy wagner 109 g. Half the divisor is the approximate root. h. Repeat steps c-f; in even iterations of the procedure, 's are subtracted rather than added, and the subtraction in step e is reversed. i. If the difference in step e is zero, the exact root is obtained.
5. Updating the divisor and dividend according to step c-e, we obtain 2 + 4 and 4 respectively.
6. The next quotient is 2. Therefore, 2 equals a rectangle of length 8 4 and width 8 2 . Break it into two strips of width 4 2 each.
3. Arrange the strips around a square with side 8 4 . To complete the square, a corner of area (4 2 ) 2 is missing.
4. The square of consists of the square of side 8 4 added to 2 × 8 × 4 + 1.
history of science in south asia 6 (2018) 84-126 110 the kriyākramakarī 's integrative approach The former term equals the original square from step 3, and the latter term equals the missing corner +1.
6. For subtraction, take the strips away from the initial square, and proceed similarly.
7. Consider the coefficient (guṇaka) of 3 in and the coefficient of 4 in .
We need that = and /2 2 2 = , so that the equalities in step 4 will work.
8. So we need to find such that /2 2 2 = . If we choose = 5 or 6, the left hand side is a fraction. If we choose = 12, then we get . Then 2 + 2 has a root.
3. To find such numbers, consider a circle with a right angled triangle lying on the radius. The sides of the triangle are (bhuja) and (koṭi), the diagonal (karṇa, śruti) is , and the arrows (śara) that extend from the sides to the circumference are and respectively (see Fig. 2).
4. In a circle, 2 = 2 + 2 (this follows from the proportional intersection of chords in a circle, which is stated later in the Līlāvatī). Since = − , this fits the situation in step 2, with = and = .
6. To show that = is a rational number (given rational and ), note that analogously to the situation in step 4, 2 = 2 + 2 , and therefore The proof ends there with no further details. I suggest the following heuristic reconstruction: 1. We are looking for two numbers whose difference of squares is a square, say 2 − 2 = 2 .
Proof 1. The product has to be a square ±1.
3. The quotient of squares is a square, and the same goes for the quotient of a half square and a double square. The product of squares is a square, and the same goes for a product of double squares.
4. In step 2 we have a product of two double squares divided by a square, which is a square. The roots of + and − are 2 2 2 and 2 respectively. vargaprakṛti (158-167) Following the rules presented above, the text presents the Vargaprakṛti method dealing with problems of the form 2 = 2 ± . The solution follows Udayadivākara's commentary on Jayadeva's elaboration of Brahmagupta's work (Shukla 1954). Only the first seven of Jayadeva's twenty verses are included. These are enough to find rational solutions for = 1, but does not provide the full cakravāla method for finding integer solution of the general equation. The method presented is the following: • If 1 2 = 1 2 + 1 and 2 2 = 2 2 + 2 , then 1 2 + Ny 1 2 2 = 1 2 + 2 1 2 + 1 2 .
• Taking arbitrary 1 , one can choose 1 such that its square approximates 1 2 . The difference is set as 1 .
• Composing this solution with itself as above, we get a solution for 2 = 2 + 1 2 .
3. The cross product of five terms in the squares from steps 1 and 2 (after the former is multiplied by ) are identical. We are left with having to show that 4. Note that 2 = 2 − . The proof is not detailed any further, but one can see that we get on the left hand side 1 2 ( 2 2 − 2 ) + 2 2 ( 1 2 − 1 ) + 1 2 and on the right hand side 1 2 2 2 + ( 1 2 − 1 ) ( 2 2 − 2 ). The equality is now easy to justify. 7 . A P P E N D IX C T he following is the justification that for given rational and , the three numbers ,