Mathematics A Missing Factor in Assessing the Antiquity of Vedanga Jyotisa

Jaidev Dasgupta

REVIEW 4 major objections 5 minor 14 references

Mathematics A Missing Factor in Assessing the Antiquity of Vedanga Jyotisa

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The extant version of the Vedanga Jyotisa must have been composed in the early centuries of the Common Era, because its fractional arithmetic and rule-of-three calculations were not mathematically feasible before then.

desk verdict A short, genuinely useful challenge to Vedanga Jyotisa's date, but its load-bearing premise that fractions require written symbols is asserted and contradicted by the paper's own example. read the letter →

arxiv 2506.02216 v1 pith:7RCNOUN3 submitted 2025-06-02 math.HO

Jaidev Dasgupta This is my paper

classification math.HO MSC 01A32

keywords VedangaJyotisadatingofancienttextsfractionalarithmeticsymbolicnumeralsplace-valuesystemhistoryIndianastronomyrulethreeBrahmiscript

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that previous attempts to date the Vedanga Jyotisa (VJ) overlooked the feasibility of its mathematics. The text's calculations, such as the result that the moon crosses $1809/124$ asterisms per fortnight and the use of the rule of three, require symbolic fraction arithmetic, while India had no written numerals or place-value system until centuries after the proposed dates. The author concludes that the extant version of VJ was composed in the early centuries of the Common Era, after such arithmetic became possible. If correct, the oldest Indian astronomical text is several centuries younger than the 1400-400 BCE range most scholars assign, and the history of early Indian calendar science needs revising.

What carries the argument

The load-bearing mechanism is the fractional arithmetic embedded in the five-year yuga: $1809/124 = 14\frac{73}{124}$ asterisms per fortnight, a quantity the calendar repeatedly accumulates, together with the rule of three, which the paper reads as an algebraic proportion $x/b = c/d$. The author's argument is that these operations cannot be represented or carried out without symbolic numerals, a place-value system, and zero. That historical feasibility constraint does the dating work: it fixes the earliest possible date for VJ's extant text at the point when such arithmetic became available.

What would settle it

A concrete reconstruction showing that a person without written numerals can reliably compute and accumulate $1809/124$ asterisms per fortnight across a five-year cycle using tally marks, ropes, or memory alone would falsify the paper's central premise; so would epigraphic or manuscript evidence of fractional astronomical computation in India before the third century BCE.

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Extended reading notes

Core claim

The central claim is that the extant Vedanga Jyotisa cannot be older than the period in which symbolic arithmetic was available. In a five-year yuga the moon traverses $67 \times 27 = 1809$ asterisms through 124 fortnights, giving a per-fortnight rate of $1809/124$, or $14\frac{73}{124}$, and this fractional quantity is repeatedly accumulated; the text also uses the rule of three, an algebraic proportion procedure. Because Brahmi script appeared only in the third century BCE and decimal place-value notation with zero only in the fifth to seventh century CE, the author places the text's current form in the early centuries of this era, after Pingala (c. 200 BCE) and before Aryabhata's fraction rules at the end of the fifth century CE. Earlier dates obtained from precession measure the winter-solstice conjunction at Dhanishtha, not the moment of composition. The paper's conclusion therefore re-dates VJ's creation to the early centuries CE and reads the Dhanishtha conjunction as an inherited oral reference point.

Load-bearing premise

The argument stands or falls on the premise that fractional arithmetic cannot be represented or performed without written symbolic numerals; if such calculations could be done orally or with concrete aids, the early-centuries-CE conclusion does not follow.

Editorial extensions

If this is right

The precession-based dates of Sastry and Dikshit identify the winter-solstice conjunction at Dhanishtha, not the composition of VJ, so the text can be centuries younger than those dates.
VJ's current form was likely created by adding fractional calculations to an older five-year calendar of the kind described in the Arthashastra, which explains why Arthashastra lacks the Dhanishtha conjunction.
Composition must fall after Pingala, c. 200 BCE, and before Aryabhata's systematic fraction rules around the end of the fifth century CE, narrowing the plausible window to the early centuries CE.
Later texts that use VJ's astronomy would need to be checked against this later date, and the development of Indian mathematical astronomy would be compressed into a shorter span than usually assumed.
Since the calendar assumes a fixed sky, the Dhanishtha solstice was a repeatable reference point, not a fresh observation made at the time of writing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same arithmetic-feasibility test could be applied to other early astronomical texts with fractional schemes; if written numerals are truly necessary for such computation, several ancient dates across cultures may need similar revision.
The paper's central premise about the necessity of written symbols could be tested by trying to reconstruct VJ's fortnightly moon-position table with only tally sticks or memorized ratios; a successful oral method would weaken the conclusion.
Because the paper asserts rather than demonstrates this necessity, a historical study of oral arithmetic traditions, or of computation with physical tokens, would either harden or overturn the chronological floor it proposes.
If future epigraphy or manuscript finds show fractional astronomical computation in India before 300 BCE, the paper's early-centuries-CE date would have to be abandoned, while absence of such evidence would strengthen it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dasgupta asks a good question that the dating literature has mostly bypassed: could the mathematics in Vedanga Jyotisa actually have been done at 1400–400 BCE? The example—1809/124 asterisms per fortnight and the repeated addition of a mixed fraction—makes the issue concrete, and the paper is right that precession-based dates only fix a conjunction, not a text's composition. Applying a mathematical feasibility criterion as the decisive test is new relative to Sastry, Dikshit, Ohashi, Pingree, and Plofker, all of whom rely on language, content, or precession. These are real merits.

The soft spot is the middle of the argument. Everything turns on the claim that 'without symbolic numerals, representing and performing these mathematical operations on fractions are incomprehensible' (Section 3). That premise is asserted with support from Dasgupta (2023) and Devlin (2000), but it is not demonstrated, and the paper itself offers a counterexample. In Section 4, the author notes that a 1/6th tax 'could be paid simply by separating the sixth measure for tax after every five measures by weight ... without any need for calculation.' That is a concrete, non-symbolic implementation of a fractional operation. If grain can be partitioned that way, then fractions are not incomprehensible without numerals; they are just done with physical measures. The Sulvasutra rope-and-peg geometry likewise realizes ratios without symbolic arithmetic. So the necessity claim fails internally, and the firm conclusion that 'the extant version of VJ has to be from the early centuries of this era' does not follow.

The additional arguments are weaker. Arthashastra's silence about VJ proves little, and the suggestion of a simpler pre-existing calendar to which fractions were later added is an invented entity, not a documented one. The paper is honest about some of this, but the 'has to be' remains too strong.

This is not a careless paper. It is clear, fair to the main sources, and the historical question it raises is legitimate. But the central premise is unestablished, so the dating revision is not convincing. I would send it to a referee who knows the Indian material; a good referee could ask for either direct evidence for the cognitive necessity claim or a softened conclusion. I would not cite it as a reliable date, but it could be worth a mention in a survey of dating debates. A reading group could use it to discuss how an interesting idea can be oversold.

Referee Report

4 major / 5 minor

Summary. This short note argues that previous dates for the Vedanga Jyotisa (VJ), ranging from 1400 BCE to 400 BCE, are too early because they overlook the mathematical content of the text. The author claims that VJ contains fractional arithmetic and a rule-of-three calculation that require symbolic numerals, which in India became available only after the advent of Brahmi script in the 3rd century BCE and, more fully, with place-value notation and zero in the 5th to 7th century CE. The paper concludes that the extant version of VJ must be from the early centuries of the Common Era, and it offers supporting arguments from the absence of VJ in the Arthashastra and from the supposed imitation of Pingala.

Significance. If the argument were correct, it would force a substantial reconsideration of the chronology of early Indian astronomy, moving VJ from the late Vedic period into the first centuries CE and challenging a long-standing scholarly consensus. The paper also performs a useful service by insisting that the mathematical technical content of ancient texts should be a factor in dating arguments. However, the central inference is not established: the claim that fractional arithmetic requires written symbolic numerals is asserted rather than demonstrated, and it is internally contradicted by the paper's own description of a concrete, non-symbolic procedure for a 1/6th tax. The additional arguments from silence and imitation are weak and do not support the strong early-centuries-CE conclusion.

major comments (4)

[p. 4, paragraph beginning 'Other ancient texts'] The claim on p. 5 that 'without symbolic numerals, representing and performing these mathematical operations on fractions are incomprehensible' is contradicted by the paper's own discussion of the 1/6th tax on p. 4, where it states that the tax 'could be paid simply by separating the sixth measure for tax after every five measures by weight ... without any need for calculation.' This is exactly a concrete, non-symbolic implementation of a fractional operation. The paper does not explain why the specific fractional calculations in VJ, such as the repeated addition of 14 73/124 asterisms per fortnight, could not in principle be realized by analogous concrete procedures, for instance using tally marks, counting rods, or physical partitioning. Because this necessity premise is load-bearing for the conclusion, the internal contradiction is a fatal flaw in the paper's central argument.
[p. 3, paragraph beginning 'As has been discussed in detail elsewhere'] The premise that 'Mathematics using symbols could not have been possible until after the advent of Brahmi script in the 3rd century BCE' is asserted rather than demonstrated. The support offered is a self-citation (Dasgupta 2023) and a popular book (Devlin 2000), neither of which is shown to establish the cognitive-historical claim that fractional arithmetic requires written symbols. The paper does not engage with evidence from other ancient cultures for non-symbolic or concrete arithmetic, such as counting boards, token systems, or oral calculation methods, nor does it address the possibility that the VJ calculations could be performed with a mixed-integer and remainder approach without symbolic manipulation. This unproven premise carries the entire dating argument.
[p. 6, paragraph beginning 'Also, it must be noted that Arthashastra does not mention VJ.'] The argument from Arthashastra's silence is an argument from absence that is not methodologically secure. The Arthashastra's own date is uncertain (300 BCE to 300 CE), and silence about a particular calendrical text does not license the conclusion that VJ was created later by adding fractional calculations to a simpler calendar. The 'very likely' scenario described is speculative and lacks direct textual or comparative evidence. At best this argument provides a weak terminus post quem, not the strong conclusion that VJ is from the early centuries CE.
[p. 6, paragraph beginning 'This timing also agrees with Pingree's observation'] The inference from VJ imitating Pingala's Chandah-sutra to a composition date after Pingala (c. 200 BCE) is questionable even if the imitation is granted. Both texts could derive from a common earlier tradition, and Pingala's own date is disputed. Moreover, even if VJ postdates Pingala, that would only place VJ after 200 BCE; it is consistent with earlier estimates such as Pingree's 400 BCE as well as with the early centuries CE, and it does not by itself force the narrow early-centuries-CE conclusion. The paper's 'has to be' is much stronger than the evidence supports.

minor comments (5)

[p. 3, fractional calculation example] The sentence 'Obviously, there is a fractional calculation involved here' would benefit from explaining why the repeated addition of 14 73/124 asterisms per fortnight could not be handled by a non-symbolic procedure, such as counting whole asterisms and accumulating fractional remainders with tallies or physical markers.
[p. 4] The paper uses 'Brahmi script' and 'place-value system' nearly interchangeably in the discussion of numerals; these are distinct developments and should be treated separately, since the prior existence of script does not imply place-value notation and vice versa.
[p. 5, sentence citing Datta and Singh] The statement that 'Datta and Singh (2004) suggest that writing and reduction of fractions to lowest terms appeared ... around 200 CE' is a strong historical claim drawn from a secondary source; it should include a specific quotation or page reference so that readers can verify the attribution.
[Throughout] The manuscript has no numbered sections, which makes it difficult to refer to specific arguments; adding numbered sections would improve clarity and usability.
[Abstract] The phrase 'the latter was dated much before the period when its mathematics was actually feasible' is ambiguous about which 'latter' is meant; consider rewriting for clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the dating argument rests on an under-supported and internally contradicted feasibility premise, but it does not reduce to its own inputs by construction.

full rationale

The paper's derivation chain is: VJ contains fractional arithmetic (e.g., 1809/124 = 14 73/124 asterisms per fortnight); such arithmetic allegedly requires symbolic numerals; Brahmi script and place-value notation appear only in the 3rd century BCE and later; therefore the extant version of VJ must date from the early centuries CE. None of these steps is equivalent to an input by construction: no parameter is fitted, no quantity is defined in terms of the quantity being predicted, and the conclusion is not the same proposition as any premise. The one self-citation, Dasgupta 2023, supplies the characterization of Vedic mathematics as concrete and non-symbolic, and that premise is load-bearing in the sense that the argument needs it to rule out non-symbolic calculation. However, the chronological anchors (Brahmi epigraphy, Datta and Singh on fraction reduction, Katz and Joseph on place-value and zero) are independently cited, and the author's own example of paying a 1/6th tax by physically separating every sixth measure undermines, rather than circularly supports, the necessity premise. This is a correctness and evidence problem, not a circularity problem. The central claim retains independent content and does not reduce to a self-citation or to the paper's own equations.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper's dating conclusion rests on one load-bearing premise: that fractional arithmetic requires written symbolic numerals. This premise is asserted, not demonstrated. The rest of the ledger consists of weaker historical inferences about Arthashastra's silence and imitation of Pingala, plus reliance on secondary sources for the chronology of Indian fraction arithmetic.

assumptions (5)

ad hoc to paper Mathematics using symbols could not have been possible until after the advent of Brahmi script in the 3rd century BCE.
The paper asserts this as a precondition for fractional calculation and cites Dasgupta 2023 and general history, but provides no evidence that non-symbolic cultures cannot perform fractional arithmetic. This premise is load-bearing for the dating conclusion.
ad hoc to paper Without symbolic numerals, representing and performing operations on fractions are incomprehensible.
This is the specific cognitive claim used to move from absence of script to impossibility of VJ's computations. It is asserted with only a popular book (Devlin) and contradicts known pre-symbolic arithmetic practices in other cultures.
domain assumption VJ imitates Pingala's Chandahsutra, so VJ must postdate Pingala around 200 BCE.
The direction and date of imitation are assumed from Pingree's observation; the paper uses it to set a lower bound but does not defend the inference or Pinagala's date.
domain assumption Arthashastra's silence about VJ means VJ was created later.
Absence from one text is treated as evidence of later composition, though Arthashastra is not an exhaustive astronomical record.
domain assumption Datta and Singh's date of around 200 CE for writing and reduction of fractions is reliable and serves as an anchor for the early centuries CE.
The paper uses this secondary-source date as a terminus for VJ without discussing the evidence behind it or the debates around the chronology of Indian fraction arithmetic.

invented entities (1)

A simpler pre-existing calendar (in Arthashastra or oral tradition) from which VJ was created by adding fractional calculations
purpose: Explains how VJ can contain a fixed Dhanishtha solstice conjunction yet be composed later; it postulates an earlier source for the astronomical content.
No independent evidence is provided for this hypothetical earlier calendar; it is inferred from the absence of the conjunction in Arthashastra and from the assumption that VJ's astronomy is older than its math.

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Cite this review

Pith. "Pith review of Mathematics A Missing Factor in Assessing the Antiquity of Vedanga Jyotisa." pith.science (2026). https://pith.science/paper/7RCNOUN3

@misc{pith2026250602216,
  author       = {Pith},
  title        = {Pith review of: Mathematics A Missing Factor in Assessing the Antiquity of Vedanga Jyotisa},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7RCNOUN3}},
  note         = {Machine review of arXiv:2506.02216}
}

read the original abstract

While reading ancient texts one has to be cognizant of the assumptions made about the past. One has to ask: Are these assumptions valid? Are we projecting the present views into the past? A case in point is the dating of Vedanga Jyotisa. This note reports that due to mathematics being overlooked in the various attempts of assessing the time of composition of the text, the latter was dated much before the period when its mathematics was actually feasible.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Bag, A.K., Binomial theorem in Ancient India, Ind. Jl. Hist. Sci. 1(1), 1966

work page 1966
[2]

Bárhaspatyah (Lála Chhote Lál), The Obscure Text of the Jyotisha Vedānga, Indian Press, Allahabad, 1907

work page 1907
[3]

Dasgupta, J., From Concrete to Abstract in Indian Mathematics, Gaṇita Bhāratī 45, (1), 2023 7

work page 2023
[4]

and Singh, A

Datta, B. and Singh, A. N. , History of Hindu Mathematics V ol. 1. Bhartiya Kala Prakashan: Delhi, 2004

work page 2004
[5]

Devlin, K., The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, Basic Books, Widenfeld & Nicolson, Great Britain, 2000

work page 2000
[6]

Dikshit, S.B., Bhartiya Jyotish Sastra, Pt. 1, Govt. of India, Delhi, 1969

work page 1969
[7]

Joseph, G.G., Indian Mathematics: Engaging with the world from ancient to modern times, World Scientific Publishing Europe Ltd., 2016

work page 2016
[8]

Katz, V .J., A History of Mathematics: An Introduction, Addison-Wesley, 2009

work page 2009

Show all 14 references

[9]

Ramasubramanian, Sule and Vahia), Science and Heritage Initiative (SandHI), IIT, Bombay, 2016

Ohashi, Y ., The Mathematical and Observational Astronomy in Traditional India in History of Indian Astronomy- A Handbook (eds. Ramasubramanian, Sule and Vahia), Science and Heritage Initiative (SandHI), IIT, Bombay, 2016

2016
[10]

VI (4) (ed

Pingree, D., Jyotiḥśāstra: Astral and Mathematical Literature in A History of Indian Literature, V ol. VI (4) (ed. J. Gonda), Wiesbaden: Harrassowitz, 1981

1981
[11]

Press, 2009

Plofker, K., Mathematics in India, Princeton Univ. Press, 2009

2009
[12]

(Ed., Tr.), The Arthashastra, Penguin Books, 1992

Rangarajan, L.N. (Ed., Tr.), The Arthashastra, Penguin Books, 1992

1992
[13]

& tr.), Vedāṅga Jyotiṣa of Lagadha, INSA, New Delhi, 1984

Sastry, T.S.K., (ed. & tr.), Vedāṅga Jyotiṣa of Lagadha, INSA, New Delhi, 1984

1984
[14]

Singh, N., Linguistics and Oral Tradition in the Period between the Decline of Harappan Culture and the Rise of Magadhan Culture in History of Science and Technology in Ancient India – The Beginnings by D. P. Chattopadhyaya, FIRMA KLM Pvt. Ltd., Calcutta, 1986, note 23 on p. 420

1986

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Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Bag, A.K., Binomial theorem in Ancient India, Ind. Jl. Hist. Sci. 1(1), 1966

work page 1966

[2] [2]

Bárhaspatyah (Lála Chhote Lál), The Obscure Text of the Jyotisha Vedānga, Indian Press, Allahabad, 1907

work page 1907

[3] [3]

Dasgupta, J., From Concrete to Abstract in Indian Mathematics, Gaṇita Bhāratī 45, (1), 2023 7

work page 2023

[4] [4]

and Singh, A

Datta, B. and Singh, A. N. , History of Hindu Mathematics V ol. 1. Bhartiya Kala Prakashan: Delhi, 2004

work page 2004

[5] [5]

Devlin, K., The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, Basic Books, Widenfeld & Nicolson, Great Britain, 2000

work page 2000

[6] [6]

Dikshit, S.B., Bhartiya Jyotish Sastra, Pt. 1, Govt. of India, Delhi, 1969

work page 1969

[7] [7]

Joseph, G.G., Indian Mathematics: Engaging with the world from ancient to modern times, World Scientific Publishing Europe Ltd., 2016

work page 2016

[8] [8]

Katz, V .J., A History of Mathematics: An Introduction, Addison-Wesley, 2009

work page 2009