Abstract
The Arthaśāstra of Kauṭilya (ca. first century BCE-third century CE) is the most important source on state administration from classical India. Its far-ranging instructions on state activities depict an ideal-typical kingdom heavily reliant on computation, particularly with respect to state finances. Nevertheless, computational practices themselves are little discussed, and no general study of them in the Arthaśāstra yet exists. This chapter is a primarily philological effort to frame an initial inquiry into such practices in the text through a study of the terms through which computation is expressed or implied. After introducing the Arthaśāstra, I examine: 1. various means of assigning value as laid out in the text (including an overview of mensuration in the Arthaśāstra); 2. some of the most prevalent numerical operations and procedures; 3. the use of these in a few examples of state activities; and 4. how computation was conceived among other evaluative activities.
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 269804.
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Notes
- 1.
The date of the Arthaśāstra has been the subject of disagreement since its rediscovery in the early twentieth century. I have proposed (McClish 2009, 2019) an extended compositional history of the text in which an original compilation was substantively redacted to bring the text more or less to its extant form. The original recension to the text was probably composed in the first century BCE or first century CE, and it was heavily redacted around the third century CE. Various perspectives on the date of the text can be found in the relevant works provided in the bibliography.
- 2.
Passages from the Arthaśāstra given here follow Kangle’s critical edition (1969). I preserve his sandhi, but do not include the punctuation marks that he adds to the text. The most recent and most useful translation is that of Olivelle (2013), who provides abundant notes to the often-obscure passages as well as helpful appendices. I give Olivelle’s translations unless otherwise stated. Where I make minor terminological amendments, I give his original in a footnote. The uniform changes I make to Olivelle’s translations are: 1. rendering all measurement units in lower case italics, i.e., ‘pala’ instead of ‘Pala’; 2. rendering numbers with words rather than numerals; 3. using ‘per cent’ rather than ‘percent’; and 4. following standard British punctuation. When defining terms in text, I often give his translation alongside other possibilities.
- 3.
The extant text is attributed to an individual named Kauṭilya. I have argued (McClish 2019) that this is probably, instead, the name of a redactor who substantially updated the text some centuries after its initial composition. This textual history is not brought to bear on the present study.
- 4.
The relationship between śāstra and prayoga has been treated in many publications, but a good starting point is Pollock (1985).
- 5.
For an overview of sources for the study of the ancient state in South Asia, see Scharfe (1989).
- 6.
References to the Arthaśāstra follow the format: book.chapter.sentence, so this refers to book two, chapters six through nine.
- 7.
- 8.
The reader is referred to the treatment of Sanskrit number words and their construction with other nouns in the standard grammars: e.g., Speijer (1886: 221–227), Whitney (1888: 177–185), Wackernagel and Debrunner (1930: 329–430). On the presentation of numbers in the Mahābhārata, see Hopkins (1902). Some variation is evident in the presentation of numbers in the Arthaśāstra, but it is not of particular importance to the present study.
- 9.
Olivelle (2013: 112): ‘accounting’.
- 10.
Olivelle (2013: 139): ‘sold by the number of pieces’.
- 11.
The term pramāṇa is also used generally to mean something like ‘size’, ‘measure’, ‘amount’ or ‘extent’ (1.16.9; 1.16.27; 1.19.6; 2.2.14; 2.7.31, etc.).
- 12.
This is the translation favored by Shama Sastry (1967: 61), Kangle (1972: 79) and Olivelle (2013: 110). Cf., however, the commentary Cāṇakyaṭīkā of Bhikṣuprabhamati (Harihara Sastri 2011: 122), which explains māna as a capacity measure ‘such as a prastha’ and unmāna as a weight measure, ‘such as a tulā’.
- 13.
These are the meanings assigned by Kangle (1972: 88) and Olivelle (2013: 116). Shama Sastry (1967: 68) understands māna as referring to ‘weights and measures’ and māpana as ‘counting articles’. Bhikṣuprabhamati (Harihara Sastri 2011: 130) takes an entirely different approach: mānaviṣamaḥ mahatā prasthādinā āyam upasaṅgṛhya alpenopanayati | māpanaviṣamaḥ hastakauśalāt tenaiva mānena prabhūtam ākṛṣya vyayakāle nyūnaṃ māpayati | ‘Discrepancy with regard to māna: after receiving income with a large <measuring instrument > such as a prastha, he disburses with a small one [cf. 4.2.13]. Discrepancy with regard to māpana: because of his deft hand, with the same measuring instrument he draws out a large amount and causes less to be measured at the time of making payments [cf. 4.2.20]’. See also 2.15.63.
- 14.
mānapratimānonmānāvamānabhāṇḍa. Cf. Kangle (1972: 81): ‘the weight, the measure, the height, the depth, and the container’; and Shama Sastry (1967: 62): ‘the counterweights (pratimāna) used in weighing them, their member, their weight, and their cubical measure’. Bhikṣuprabhamati (Harihara Sastri 2011: 124), however, probably offers the best reading: mānabhāṇḍaṃ prasthādi kāṣṭhamayam | pratimānabhāṇḍam ayomayādi | unmānabhāṇḍaṃ tulā | apamānabhāṇḍam api tatrādiparicchedārtham | rajjavaḥ śalākāḥ, ‘a māna instrument, made of wood, such as prastha <a measure of capacity>; a pratimāna instrument, such as one <i.e., a weight for use with a balance> made of iron ; an unmāna instrument: a balance; and also an apamāna instrument used to make spatial determinations and so forth: ropes; rods’. Note Bhikṣuprabhamati (Harihara Sastri 2011: 124) reads apamāna where Kangle (1969: 43) gives the reading avamāna.
- 15.
For an overview of measurements in the ancient period, see Srinivasan (1979), who has extended discussions of linear measures (6–34), area measures (35–68), capacity measures (69–92), weights (93–117) and time (118–161). Sen (1967: 84–95) provides a good summary of weights and measures in the Arthaśāstra and Olivelle (2013: 455–459) gives a list of weights and measures in the text, including a glossary and equivalences with contemporary metrology.
- 16.
We find also the term gaurava, which can also refer, among other things, to the ‘weight’ of an object (1.21.7; 2.12.1; 2.12.7), although it seems to do so in a less quantified sense. The term prayāma is used to mean ‘additional weight’ (2.7.2; 2.19.24); see Kölver (1995).
- 17.
- 18.
Phaseolus radiatus. On the cultural history of the māṣa bean, see Mehram (1970), which, however, does not discuss its importance to metrology.
- 19.
Abrus precatorius.
- 20.
Sinapis alba.
- 21.
Cassia tora or Senna tora.
- 22.
Five guñja berries are given as equivalent to a māṣaka of gold (2.19.2).
- 23.
Twenty śimba beans are given as equivalent to a dharaṇa of silver (2.19.6).
- 24.
One must be attentive to the distinction in the text between the gold māṣaka weight, silver māṣaka weight and a copper coin called the māṣaka (also spelled māṣika, as at 2.21.24). As Sircar (1968: 94) notes, the copper māṣaka was so-called because it was understood to be worth 1/16th of the silver kārṣāpaṇa coin (cf. 2.12.24).
- 25.
By giving both the suvarṇa and karṣa as equivalent measures (2.19.3) and defining a four karṣas as a pala (2.19.4), the text has linked the māṣaka/suvarṇa weights, discussed in the context of equal arm balances, with those denominated in palas, discussed in the context of steelyard balances. The suvarṇa stands in relation to the māṣaka by a ratio of 1:16 (2.19.3), while the karṣa equates to the pala in a ratio of 4:1 (2.19.4). We have, seemingly, a concatenation of different systems.
- 26.
There is a problem, however, in trying to make this equivalence work with the metrological systems just discussed, as we seemingly have either a pala or dharaṇa here that is a different weight from the pala and dharaṇa discussed prior (2.19.4; 2.19.6). Calculating the weight of the pala based on 2.19.2–4, we arrive at the following: if a guñja berry weighs 118.6 mg (=1.83 grains; Sircar 1968: 74), then a karṣa weighs 9.49 g and a pala weighs about 38 g (2.19.2–4). The dharaṇa mentioned here at 2.19.20 would, therefore, weigh about 3.8 g. But, this is not the value we find for the dharaṇa if we calculate its weight based on the ratios given in the earlier discussion of the silver dharaṇa (2.19.5–6). The text does not provide the ratio between white mustard seeds and guñja berries, but contemporary authorities agree that the ratio is 18:1 (Sircar 1968: 73). Hence, a silver māṣaka (=88 white mustard seeds) is equal to 4.9 guñja berries (=581 mg). According to these instructions, a silver dharaṇa is sixteen māṣakas and weighs, therefore, about 9.3 g. Thus, a pala of ten of these silver dharaṇas would weight 93 g, more than twice as much as a pala of four gold karṣas (=38 g). Sircar (1968: 89) sees the problem: either two different kinds of pala are meant or two different kinds of dharaṇa. Following Prabhamati (Harihara Sastri 2011: 170), he posits that the text is describing here at 2.19.20 a different, heavier pala (a so-called ‘dharaṇa-pala’; Sircar 1968: 89). Kangle (1972: 134), for his part, simply reduces the value of the silver dharaṇa, from roughly 9.3 to 3.8 g, by positing that the ratio between white mustard seeds and guñja berries must be 44:1, which ratio is contradicted by contemporary sources. I follow Olivelle (2013: 144, 457) in differentiating between the silver dharaṇa and the dharaṇa mentioned here, which is, by definition, 1/10 of a pala. I disagree, however, that a different pala is intended. This interpretation is supported by the following passage (2.19.21–23), whose purpose is to describe a series of steelyard balances whose maximum measured values are five, ten and fifteen per cent less than the standard balance (see below). Each balance is still divided into one hundred units called pala, which are further subdivided into ten units called dharaṇa/dharaṇika. Hence, as the maximum value measured by the balances decreases, the value of each pala decreases proportionately: tāsām ardhadharaṇāvaraṃ palam, ‘Of these <balances>, a pala is progressively less by half a dharaṇa’ (2.19.23). We can see that the dharaṇa/dharaṇika is a unit used to express the difference between palas as they diminish five, ten and fifteen per cent (i.e., one-half, one and one and a half dharaṇas/dharaṇikas) from the standard pala, respectively. It is much easier to assume that this dharaṇa/dharaṇika is simply a term for 1/10th of a standard pala, as required to express the relationship between these balances, than that a new kind of pala is being introduced, for which this no other evidence.
- 27.
- 28.
Sircar (1968: 78). kākaṇī as weight: 2.13.16; 2.13.48; 2.13.50; 2.14.8; 2.14.9. Not to be mistaken with the coins called kākaṇī and half-kākaṇī (2.12.24; 2.19.42; 4.1.40, 42) or the kākaṇī shell used for gambling (3.20.8–10; 4.10.9).
- 29.
Another bridge given between weight and capacity or volume in the text is firewood: at 2.19.26 the text reads kāṣṭhapañcaviṃśatipalaṃ taṇḍulaprasthasādhanam | eṣa pradeśo bahvalpayoḥ, ‘twenty-five palas of firewood <in weight> cooks one prastha of rice <in volume>. This is exemplary with regard to higher and lower quantities’. As Olivelle (2013: 552) notes, however, this appears to be a gloss that has ‘found its way into the text’. On so-called ‘canonical measures of weight’ for grains, which connect weight with other types of measurement, see Bosak (2014).
- 30.
Vessels for measuring capacity were made of wood and designed so that one-quarter of the total capacity measured by the vessel was comprised of the ‘heap’ (śikhā) that mounded up above the mouth of the vessel (2.19.34). The exceptions to this were certain vessels for measuring such things as liquids, which ‘possess an inner heap’ (antaḥśikha), apparently meaning that the entire capacity was measured within the vessel itself. The description of the antaḥśikha vessel in the text is unclear. See Olivelle (2013: 552–553).
- 31.
- 32.
On gaṇita see Datta (1929).
- 33.
Olivelle (2013: 148): ‘number’.
- 34.
Olivelle (2013: 112): ‘accounting’.
- 35.
Dasgupta (1957): 360, specifically with reference to the Nyāya tradition. Cf. Guha (2012: 47), who interprets tarka within classical Nyāya as ‘a cognitive act that validates a content (of a doubt or a cognition or a speech-act) by demonstrating its logical fitness or invalidates a content by demonstrating its logical unfitness’.
- 36.
On tarka in the Arthaśāstra, see Scharfe (1993: 263).
- 37.
For the purposes of this article, procedures are defined as numerical processes that involve more than one operation.
- 38.
I do not discuss here the presentation of numbers implying or expressing addition, such as dvicatvāriṃśat (lit., ‘two-forty’, i.e., 42; 2.20.15) or sāśītiprakaraṇaśata (lit., ‘with-eighty-prakaraṇa-hundred’, i.e., 180 prakaraṇas; 1.1.18).
- 39.
See 2.21.20 and 3.19.27.
- 40.
Here, the final –a of sa has euphonically combined with the initial a- of ardha to generate a long ā: sārdha. We find also use of the prefix sa in sapādacatuṣ- (‘fifty and a quarter’; 2.27.11). Cf. 1.1.18.
- 41.
The fluid application of these expressions become even more apparent when we realize that the words ca and adhika are also used in other sources to express single number values, as in nava ca navatiś ca (‘nine and ninety’) and aṣṭādhikanavati (‘ninety increased by eight’). Cf. Whitney (1888: 179).
- 42.
- 43.
adhika as ‘more by’: 2.3.33; 2.15.35; 2.20.24; 2.25.20; 2.25.26; 2.29.35(2); 3.4.24.
- 44.
vṛddhi as ‘increase of’: 2.15.26; 2.15.28; 2.15.36; 2.30.12. The first three of these occurrences describe changes to grains as they are worked in different ways.
- 45.
Three rajjus is thirty daṇḍas, so an increase of two daṇḍas amounts to one-fifteenth of the original value (2.20.23–24). In the second case (3.4.24), a number of values are generated off of the original value of one (year). The wives of men in the four social classes mentioned have to wait one, two, three and four years respectively. One year is added to that when there are children, generating two, three, four and five years. The addend (one year) does not exceed the augend (one year).
- 46.
For other examples using this term, see 2.3.16, 17; 2.7.11; 2.13.16.
- 47.
See also 3.4.24; 3.18.4; 3.18.7. As mentioned above, instances of iterative addition (except 2.7.27; see below) delimit a range within which the iteration will take place. Sometimes that range is provided by context, as in the examples we have just seen. In other cases, a range of values is explicitly articulated. The upper and lower values are given in the ablative case, with the particle ā denoting the upper limit. In the following examples ā has euphonically combined with the initial sound of aṣṭa, to yield āṣta-: prākāram ubhayato meṇḍhakam adhyardhadaṇḍaṃ kṛtvā pratolīṣaṭtulāntaraṃ dvāraṃ niveśayet pañcadaṇḍād ekottaram āṣṭadaṇḍād iti caturaśraṃ ṣaḍbhāgam āyāmād adhikam aṣṭabhāgaṃ vā | pañcadaśahastād ekottaram āṣṭādaśahastād iti talotsedhaḥ | (2.3.16–17) ‘Having made the wall circular by one and a half daṇḍas on both sides, he should erect a gate large enough for six beams of a postern gate. It should be a quadrangular in shape, with a minimum of five daṇḍas and increasing by one daṇḍa up to a maximum of eight daṇḍas, and one-sixth or one-eighth more than its length. The height of the floor is a minimum of fifteen hastas and increasing by one up to a maximum of eighteen hastas’.
- 48.
In one case the term abhikrānta appears to be synonymous with guṇa in this sense: tryabhikrāntapakṣa, ‘when its wings are increased [Kangle (1972: 451): ‘augmented’] threefold’ (10.6.18).
- 49.
guṇa expressing a relationship between measured quantities: 2.15.32–35; 2.15.37; 2.15.49; 2.15.53; 2.15.56; 2.19.17; etc.
- 50.
See Olivelle (2013: 522–523) for possible interpretations.
- 51.
Other instances can be found at 2.7.12–14; 2.7.23; 2.7.36; 2.7.37.
- 52.
kṣaya used in the sense of ‘loss’: 2.7.2; 2.8.3; 2.8.4; 2.14.8; 2.14.9; 2.15.24; 2.18.20; 3.12.30, etc. ; hrāsa: 2.6.28. Cf. 2.19.10; 2.23.6; 2.30.43.
- 53.
pari + hā: 2.7.10, 2.7.19, 2.9.14.
- 54.
Most often, atirikta is paired with another term that means ‘with a deficiency of’ or ‘deficient by’, such as hīna or ūna. Together, as in the compound hīnātirikta, they refer to ‘too little or too much’ or, in clearly quantitative contexts, ‘less or more by’. We find this expression used several times in a passage (4.2.2–12) that lays out fines for measurement instruments that are not true: parimāṇīdroṇayor ardhapalahīnātiriktam adoṣaḥ | (4.2.3) ‘In the case of a parimāṇī or a droṇa, to be half a pala less or more is not an offense’. These instructions, however, do not require computation, but only the taking of a physical measurement (see below). Computation is, however, indicated in an example where a judge who incorrectly fines people must pay eight times the amount the actual fine was ‘less or more’ than the appropriate fine (4.9.18).
- 55.
2.21.10; 2.24.16. Compare 9.2.2.
- 56.
-ūna: 2.3.4; 2.6.12; 2.15.25; 2.15.29; 2.15.30; 2.15.31; 2.15.45; 2.19.36; 2.22.12; 2.26.2; 2.28.16; 2.29.44; 3.14.29; 10.5.22. –nyūna: 2.20.49. –hīna: 1.9.2; 1.16.3; 1.16.4; 2.14.4; 2.25.20; 2.30.23; 4.1.5; 4.1.37; 4.2.3; 4.2.4; 4.2.6; 4.2.7; 4.2.9; 4.2.10.
- 57.
In this sense, these terms are counterparts to adhika and vṛddhi (‘more by’), discussed above, as well as, in a different sense, atirikta (‘too much by’). Of the three terms here, hīna has the broadest range and application. It is used frequently to describe qualitative deficiencies as well.
- 58.
Here the final –a of ardhabhāga is euphonically combining with the initial ū- of ūna to generate ardhabhāgona. Cf. tribhāgona (tribhāgono in euphonic context).
- 59.
Different, however, is the compound caturbhāgāvara at 2.19.30, which cannot mean ‘less by one-quarter’, but must mean something like ‘progressively less in the amount of one-quarter’ (i.e., less by three-quarters). One should also be careful not to confuse the usage described here with instances where compound-final –avara is used to indicate the minimum amount in a range of numbers, such as 2.1.2: kulaśatāvara, ‘a minimum of one hundred families’.
- 60.
See also 2.20.42; 2.30.14–17; 2.30.20; 2.31.12.
- 61.
Because the verse does not give the maximum fine (i.e., eight paṇas) from which the two paṇa subtraction is to be progressively made, it seems that the composer is not thinking of a decrease applied as one works forward through the list, but is actually thinking in terms of an increase in the fine, starting at two paṇas and accumulating as one works toward the beginning of the list.
- 62.
Note, however, dvātriṃśadbhāga at 2.13.51, which appears to mean ‘thirty-two parts’ rather than ‘one-thirty-second’ (cf. 2.13.54). We have also instances of cardinal numbers in compound with a final –bandha (2.7.21; 2.7.38; 2.8.6; 2.8.11; 3.1.20; 3.1.21; 3.11.33[2]; 3.12.6; 3.13.33; 3.13.34; 3.16.18) and –amśa (3.1.22; 3.5.8). Some of these instances seem to be construed as fractions like compounds ending in –bhāga. But, these usages are complicated by what appear to be particular and restricted meanings of these terms in legal discourse. Amśa can refer specifically to a ‘share’ of the patrimonial estate, so that at 3.5.8, dvyaṃśa refers to ‘two shares’, while at 3.1.22, aṣṭāṃśa appears to mean only ‘one-eighth’. Bandha, on the other hand, refers in these cases to different ‘amounts’ relevant to legal disputes, such as a bond, an amount under dispute, the value of some property and so forth (3.11.17, Olivelle 2013: 517–518). Hence, daśabandha (3.1.21) appears not to mean ‘one-tenth’ generally, but specifically ‘one-tenth of the amount <under dispute>’. Further study is required, although it is at least evident that we have a usage here particular to certain discourses, such as law, that deviates somewhat from other expressions of fractions. Fractions can also be rendered using compound-final -bhāgaka or –bhāgika, such as with ṣoḍaśabhāgika (‘one-sixteenth’; 3.10.22). Likewise, fractions can be formed simply by adding the -ka suffix to a numeral, as in ṣaṭka (‘one-sixth’; 2.15.38). In the example below (2.15.43), the compound-final –bhāga is merely implied: sūpaṣoḍaśa, lit., ‘stew-sixteen’, but meaning, ‘one-sixteenth of the amount of stew’. In one instance (9.3.4), fractions are expressed not with the suffix –bhāga, but instead as a compound giving the denominator first and the term ekīya (‘single’) second . Hence, we have the compounds sahasraikīya (lit., ‘one in a thousand’; i.e., ‘one-thousandth part’) and śataikīya (lit., ‘one in a hundred’; i.e., ‘one-hundredth part’). Both expressions intend to communicate very small portions and are, in this sense, less mathematically specific than rhetorically potent. See also caturthapañcabhāgika, which combines an ordinal, caturtha (‘fourth’), and a cardinal, pañca (‘five’). Kangle (1972: 150) and Olivelle (2013: 153) render as ‘one-fourth or one-fifth’ (2.24.16). The use of ordinals to denote fractions is seen at 3.7.19 in the term tṛtīyāmśa, ‘one-third share’ with respect to the division of inherited property (cf. 3.6.20).
- 63.
Note also ardhabhāga, which means, somewhat redundantly, ‘a half-part’. It appears to be used at 2.13.55, however, to denote ‘half a part’, which ‘part’ can be identified by context as one-third of the whole; hence, ardhabhāga here means ‘one-sixth’. See Kangle (1972: 115), Olivelle (2013: 539). Cf. aṣṭamabhāga (2.31.5), meaning a part that is an eighth, in this case a reference to one part of the half-day understood as divided into eight parts.
- 64.
Between the last two types mentioned, instances of the latter, i.e., where the numerator is not in compound with the denominator, are more common. Cf. 2.20.39. Such variations may reflect differences in the Arthaśāstra’s own sources. Note also the formulation caturguṇapañcabandha, ‘four times one-fifth the amount’ at 3.12.6. See below.
- 65.
Olivelle (2013: 137) has ‘increase one and half times’, a gain of one hundred fifty per cent. This is unlikely as the list here gives grains in order of their increase and following items double, triple, quadruple and increase fivefold, gains of one hundred, two hundred, three hundred, and four hundred per cent, respectively. So, Kangle (1972: 125): ‘become one and a half times (in volume)’.
- 66.
Kangle (1972: 124n.) gives the value of eleven-eighteenths. The calculation depends on the amount from which the value for one-ninth is to be determined. If we follow the procedure exhibited in the rest of the passage, it would not mean one-ninth of the whole, but only of one-half, meaning one-eighteenth. One-half plus one-eighteenth yields ten-eighteenths or five-ninths.
- 67.
For examples of medieval mathematical tables, see Sarma (1997).
- 68.
Other possibilities exist, but the provision of these ratios in the text suggests that they were understood to be applicable to computational practices.
- 69.
See footnote 67.
- 70.
At any rate, the grosser inequities in the rubric would only have emerged as the values increased, and we can expect that in practice most inspected measuring instruments would only have deviated a little from the standard measurement unit (large deviations being harder to hide), so that the lack of consistency in the rubric was itself probably of diminished practical relevance.
- 71.
What this passage may also tell us about processes of rounding is yet unclear and requires further investigation.
- 72.
See Datta (1927).
- 73.
We find percentages used elsewhere in the text, always in context of values conceptualized in monetary denominations. See 2.12.30; 2.25.39; 3.16.24; 3.17.15; 3.20.10; 4.2.28.
- 74.
The best treatment of this topic is Schlingloff (2013).
- 75.
Rather than ‘one-quarter’ (Olivelle 2013: 103). The term is pādona. See above.
- 76.
Such language is found more generally in instructions for construction (such as for horse and elephant stalls), but, like the instructions for the moats, they do not require computation to be put into practice.
- 77.
- 78.
Hayashi (2003: 365) suggests that saṃkhyāna here possibly refers to the ‘relatively elementary skill of computation beginning with the counting of numbers’.
- 79.
There is also a kind of salaried official in the text called a saṃkhyāyaka (2.1.7; 2.9.28; 2.9.30; 5.3.14). We are nowhere told precisely what he does, but the context of his service indicates that the saṃkhyāyaka was an accountant. We read that supervisors (adhyakṣas), overseers of various government functions, were assisted by ‘accountants <saṃkhyāyakas>, scribes, examiners of coins, receivers of balances, and higher ranking supervisors’ (2.9.28). Given the prominence of financial considerations in the operation of state bureaus, we can presume that the saṃkhyāyaka primarily served in just such a capacity. Cf., however, Hayashi (2003: 365).
- 80.
Grimes (1996: 41).
- 81.
Anumāna: 2.7.29; 2.14.50; 2.14.51; 2.15.59; 4.2.12. On the use of anumāna and the related term anumeya (‘to be inferred’) in the Arthaśāstra, see Scharfe (1993: 263–264).
- 82.
At 2.7.29, it is clear that anumāna refers to a means for detecting malfeasance, probably by investigating the records of accounts themselves. Computational operations, therefore, are strongly implied, although to the extent that anumāna refers more broadly to a means for determining innocence or guilt, it might also include reasoned judgments aided by the results of computation.
- 83.
On connections with astronomical literature, see Kangle (1972: 139–140).
- 84.
Of technical mathematical terms, such as the names given by Pṛthūdakasvāmin to the twenty parikarmans (‘mathematical operations’) (**ree 1981: 57), some few, such as pratyutpanna, varga and ghana, are found in the Arthaśāstra where, however, they do not refer to mathematical operations in any way. The same is generally true for technical terms referring to vyavahāra (‘mathematical determinations’). A few of these appear in the Arthaśāstra, although usually with different meanings. Among these, the term kṣetra (‘land’) is, of course, common in the Arthaśāstra, but never refers to ‘plane geometry’. The same can be said for rāśi (‘heap’, as of grain), which only refers in the Arthaśāstra to the actual heap of grain itself and not to an aspect of solid geometry. The term khāta, ‘excavation’, is relatively common in the Arthaśāstra, but is used only in reference to actual excavations or excavated material and not to solid geometry itself. We do have mention of a measuring unit called khātapauruṣa, a linear measure used for excavations (2.20.17), and this indicates that measurements and, perhaps therefore, computations involved in the physical activity of excavation formed something of an independent system. A similar measurement unit (the krākacikakiṣku, a ‘sawyer’s kiṣku’; 2.20.15) exists for ‘sawing’ (krākacika), another aspect of solid geometry. The measuring unit called chāyāpauruṣa, the ‘shadow pauruṣa’, is used in determining time on a sundial (2.20.10). Technical terms for the other ‘determinations’ are not found. It would appear, then, that the Arthaśāstra recognizes the mathematically archetypal activities of excavation, sawing, and using a sundial already as independent areas, even if it shows no explicit knowledge of the later mathematical determinations that they come to express.
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McClish, M. (2020). Computation in the Arthaśāstra. In: Michel, C., Chemla, K. (eds) Mathematics, Administrative and Economic Activities in Ancient Worlds. Why the Sciences of the Ancient World Matter, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-48389-0_3
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