Some constructions in the M\=anava \'Sulvas\=utra

S. G. Dani

Some constructions in the M\=anava \'Sulvas\=utra

T0 review · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Manava Sulvasutra records a circumference ratio of 3.2 and, under a new reading, a trisector-based circle-squaring rule accurate to about 0.5 percent.

desk verdict A careful, honest proposal that the Manava Sulvasutra contains a trisector-based quadrature with ~0.5% error and an explicit awareness that pi exceeds 3; plausible but depends on an inferred reading and a garbled formula. read the letter →

arxiv 1908.00440 v1 pith:EQVLRFFG submitted 2019-08-01 math.HO

S. G. Dani This is my paper · ORCID

classification math.HO

keywords sulvasconstructionssomeutraanavadiscussotherutras

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

The Sulvasutras are ancient Indian texts with rules for building altars. The Manava version is usually treated as the least polished of the four major ones. This paper argues that it contains two overlooked mathematical ideas. One verse says that the circumference of a circle is three diameters plus one fifth of a diameter, or 3.2 instead of the older round value 3. The paper reads the accompanying phrase "not a hair-breadth remains" as a sign that the author knew the old value was too small, not as a claim of exact accuracy. It also suggests the choice of a fifth was natural because dividing by five is easy in a decimal number system.

The second and main claim is about a verse usually read as a rule for making a circle with the same area as a square. Earlier translators either doubted it was about quadrature or gave constructions that did not fit the text well. The paper proposes that the square is divided into nine parts, the trisecting lines are extended until they meet the circle through the square's corners, points are marked one fifth of the way from the square's side to that circle, and a circle is drawn through the eight marked points. For a square of side 1, this circle has area about 0.9947, an error near 0.5 percent, better than the older rule's 1.7 percent.

The mathematical check is simple. The historical conclusion is conditional: the Sanskrit is terse, and the interpretation depends on reading one unusual word in a particular way and on accepting an emendation. If Sanskrit scholars confirm the reading, the Manava Sulvasutra holds a uniquely accurate construction not found in the other Sulvasutras.

Extended reading notes

Core claim

The load-bearing assertion is that the Manava Sulvasutra verse 10.3.2.15, read as "divide the square into nine parts... on the parts jutting out mark the points at one-fifth (from the square) and draw the circle through them," describes a genuine and previously unrecognized construction for a circle of the same area as a square, with an error of about 0.5 percent rather than the 1.7 percent of the Baudhayana rule. If the paper is correct, this verse is the second original mathematical idea in the Manava Sulvasutra, along with the statement that the circumference is three diameters plus one fifth of a diameter.

Load-bearing premise

The new reading assumes that the terse second half of the verse refers to the trisectors of the square extended to the circle through the square's vertices, and that the intended radius is set by points one fifth of the way from the side of the square to that circle. Neither the trisectors nor the circumcircle is named in the Sanskrit; both are inferred from context, symmetry, and the analogy with the Baudhayana construction, and the author admits the text may need emendation. If a Sanskritist shows that "utsedha" and the rest of the verse do not license these points, the paper's main novel claim fails even though the arithmetic is correct.

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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

Dear colleague,

The paper is worth your time. Its new claim is that Manava Sulvasutra 10.3.2.15 describes a trisector-based quadrature that achieves about 0.5% error, better than Baudhayana's 1.7%. The author also makes a strong case that the earlier verse 10.2.3.13 concerns circumference, not a square of equal area, so the Manava explicitly recognizes that pi exceeds 3.

What is genuinely good: the geometry is simple enough to check. I re-derived the construction from the verbal description and got the stated area. The paper treats the earlier translations fairly and shows why they are problematic. The argument that the first part of the verse (divide into nine parts) must be relevant is persuasive, and the construction is a natural generalization of Baudhayana's bisector rule.

Where I hesitate: the new reading is inferred, not established. The verse literally names no trisectors, no circumcircle, and no one-fifth point. The author says a Sanskritist may confirm it 'possibly after some emendation'—an honest caveat, but it means the central result is conditional. If an alternative reading of 'utsedha' or 'purisena' is licensed, the construction vanishes. Also, the displayed formula in Section 3 is inconsistent with the stated area. From the text I get r^2 = 1/36 + (1/2 + (√17−3)/30)^2, which gives area 0.994...; the printed expression gives something else. Probably a typo, but it should be corrected.

The paper is not overclaiming; it flags its own speculative step. For historians of ancient Indian mathematics, this is a genuinely useful contribution, and the 0.5% error result is worth knowing. I'd send it to peer review with a referee who can judge the Sanskrit. The mathematical core is solid; the philological claim needs expert testimony.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted numerical parameters and no new entities. Its central claim depends instead on interpretive axioms about Sanskrit word meanings, textual reliability, and the inferred geometry behind the verse; these are listed above.

assumptions (4)

domain assumption The standard lexical meanings of 'vishkambha', 'parikshepa', and 'utsedha' are correct for these Sulvasutra verses.
The circumference reading turns on 'parikshepa' meaning circumference rather than area, and the quadrature reading turns on interpreting 'utsedha' as a jutting-out segment on the trisectors.
domain assumption The Manava Sulvasutra text at 10.3.2.15 is corrupt or ambiguous and may be emended on grammatical and contextual grounds.
The author states the interpretation may need some emendation and argues against the literal translations in [7], [11], and [12], so the reading rests on a text that is not taken at face value.
ad hoc to paper The intended circle is the circumcircle through the square's vertices, with the eight marked points placed symmetrically.
No word in the verse names the circumcircle or the symmetric eight-point set; the paper introduces them from overall formulations and symmetry considerations to make the construction work.
standard math Modern Euclidean formulas for circumference and area apply when evaluating the accuracy of ancient constructions.
The claimed errors of 1.7 and 0.5 percent are computed with modern formulas for circle area and radius.

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Cite this review

Pith. "Pith review of Some constructions in the M\=anava \'Sulvas\=utra." pith.science (2026). https://pith.science/paper/EQVLRFFG

@misc{pith2026190800440,
  author       = {Pith},
  title        = {Pith review of: Some constructions in the M\=anava \'Sulvas\=utra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EQVLRFFG}},
  note         = {Machine review of arXiv:1908.00440}
}

read the original abstract

The M\=anava \'Sulvas\=utra, while less sophisticated than the other \'sulvas\=utras, is seen to contain some mathematical ideas and constructions not found in the other \'sulvas\=utras. Here we discuss some of these constructions and discuss their significance in the overall context of the \'sulvas\=utra literature.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Vaman Shivram Apte, The Students’ Sanskrit English Dictionary, Motilal Banarasidass, Delhi 10

work page
[2]

Dani, Geometry in the Sulvasutras, in Studies in History of Mathemat- ics, Proceedings of Chennai Seminar, Ed

S.G. Dani, Geometry in the Sulvasutras, in Studies in History of Mathemat- ics, Proceedings of Chennai Seminar, Ed. C.S. Seshadri, Hindustan Bo ok Agency, New Delhi, 2010

work page 2010
[3]

Bibhutibhusan Datta, Ancient Hindu Geometry: The Science of th e ´Sulba, Calcutta Univ. Press. 1932; reprint: Cosmo Publications, New Delhi, 1993

work page 1932
[4]

Gupta, New Indian values of π from the M¯ anava´Sulvas¯ utra, Centaurus 31 (1988), 114 - 125

R.C. Gupta, New Indian values of π from the M¯ anava´Sulvas¯ utra, Centaurus 31 (1988), 114 - 125

work page 1988
[5]

Gupta, Vedic circle-square conversions: new texts and rule s, Ganita Bharati 26 (2004), 27 -39

R.C. Gupta, Vedic circle-square conversions: new texts and rule s, Ganita Bharati 26 (2004), 27 -39

work page 2004
[6]

Takao Hayashi, A new Indian rule for the squaring of a circle: M¯ an ava ´Sulvas¯ utra3.2.9-10, Ganita Bharati 12 (1990), 75-82

work page 1990
[7]

Raghunath P. Kulkarni, Char Shulvas¯ utre(in Marathi), Maharashtra Rajya Sahitya Sanskrit Mandal, Mumbai, 1978; Hindi translation Char Shulbs¯ utra, Maharshi Sandipani Rashtriya Vedavidya Pratishthana, Ujjain, 2 000

work page 1978
[8]

Monnier Williams, Motilal Banarasidass, Delhi

work page

Show all 12 references

[9]

Kim Plofker, Mathematics in India : 500 BCE - 1800 CE, Princeton Un i- versity Press, 2008

2008
[10]

Saraswati Amma, Geometry in Ancient and Medieval India, M otilal Banarsidas, Delhi, 1979

T.A. Saraswati Amma, Geometry in Ancient and Medieval India, M otilal Banarsidas, Delhi, 1979

1979
[11]

Sen and A.K

S.N. Sen and A.K. Bag, The ´Sulbas¯ utras, Indian National Science Academy, New Delhi 1983

1983
[12]

Jeanette van Gelder, The M¯ anava ´Srautas¯ utra, English translation, New Delhi, 1963. S.G. Dani UM-DAE Centre for Excellence in Basic Sciences Campus of University of Mumbai, Kalina, Santrcruz Mumbai 400 098 India E-mail: shrigodani@gmail.com 11

1963

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Vaman Shivram Apte, The Students’ Sanskrit English Dictionary, Motilal Banarasidass, Delhi 10

work page

[2] [2]

Dani, Geometry in the Sulvasutras, in Studies in History of Mathemat- ics, Proceedings of Chennai Seminar, Ed

S.G. Dani, Geometry in the Sulvasutras, in Studies in History of Mathemat- ics, Proceedings of Chennai Seminar, Ed. C.S. Seshadri, Hindustan Bo ok Agency, New Delhi, 2010

work page 2010

[3] [3]

Bibhutibhusan Datta, Ancient Hindu Geometry: The Science of th e ´Sulba, Calcutta Univ. Press. 1932; reprint: Cosmo Publications, New Delhi, 1993

work page 1932

[4] [4]

Gupta, New Indian values of π from the M¯ anava´Sulvas¯ utra, Centaurus 31 (1988), 114 - 125

R.C. Gupta, New Indian values of π from the M¯ anava´Sulvas¯ utra, Centaurus 31 (1988), 114 - 125

work page 1988

[5] [5]

Gupta, Vedic circle-square conversions: new texts and rule s, Ganita Bharati 26 (2004), 27 -39

R.C. Gupta, Vedic circle-square conversions: new texts and rule s, Ganita Bharati 26 (2004), 27 -39

work page 2004

[6] [6]

Takao Hayashi, A new Indian rule for the squaring of a circle: M¯ an ava ´Sulvas¯ utra3.2.9-10, Ganita Bharati 12 (1990), 75-82

work page 1990

[7] [7]

Raghunath P. Kulkarni, Char Shulvas¯ utre(in Marathi), Maharashtra Rajya Sahitya Sanskrit Mandal, Mumbai, 1978; Hindi translation Char Shulbs¯ utra, Maharshi Sandipani Rashtriya Vedavidya Pratishthana, Ujjain, 2 000

work page 1978

[8] [8]

Monnier Williams, Motilal Banarasidass, Delhi

work page