10 · ŚULBASŪTRA · GEOMETRY

The altar cord

Transforming a square without changing its area

Cords and stakes made it possible to lay out altars and transform geometric shapes while preserving their area.

Museum documentation · free public access, no account required.

Specific historical recreation of Transforming a square without changing its area, North India; diverse school traditions, Approximate composition and transmission c. 800–500 BCE, with revisions — **DECLARED VISUAL RECONSTRUCTION · EXISTING EDITORIAL RESOURCE · TRACEKEEPERS**
Generated reconstruction · Declared visual hypothesis
Declared visual reconstruction for museographic context; it is not a photograph of the original nor documentary evidence.

History and context

The Śulbasūtras describe how to lay out altars, orient lines and transform figures while conserving areas. Taut ropes and stakes allowed the construction of squares, diagonals and complex forms with bricks of controlled measure.

The texts include notable approximations and geometric relationships. Calling them a modern theorem discovered by a single person distorts their form: they are operational rules transmitted in ritual schools and dated broadly.

The rules of the Śulbasūtras allow laying out angles, diagonals and area transformations for brick altars. Rope and stakes gave precision to forms that had to satisfy simultaneously geometric and ritual requirements.

Speaking of the 'Pythagorean theorem' may orient the visitor, but a race to decide who arrived first impoverishes distinct contexts. Nor should all Śulbasūtras be mixed without date as the work of a single author.

Evidence and interpretation

Documented fact: manuals such as those attributed to Baudhāyana and Āpastamba contain geometric procedures for altars. Interpretation and caution: their exact dates, priority over other traditions and direct relation to specific archaeological altars are not settled.

Catalogue facts

Narrative date: Approximate composition and transmission c. 800–500 BCE, with revisions
Location: North India; diverse school traditions
Material: Rope · stakes · clay bricks · oral text later written down
Context: Practical geometry for ritual construction
Curatorial reading: The documented fact and its interpretation appear separately in the evidence block.
Date: Approximate composition and transmission c. 800–500 BCE, with revisions
Location: North India; diverse school traditions
Support: Rope · stakes · clay bricks · oral text later written down
Documentary coverage: 2 museum, heritage, primary corpus or research sources for this stop.

A SHARED TOPIC, DIFFERENT MOMENTS

Objects connected through time

Follow a topic across the collections. A shared tag is a way to compare objects, not proof that one culture influenced another.

The order uses the midpoint of each approximate date range; ranges can overlap. Spacing does not represent elapsed time. Swipe, scroll or use the arrows. With the sequence focused, use ← / →, Home or End.

THE CURRENT OBJECT

Its date is not sufficiently established to place it on the timeline. We keep it here, separate from the dated objects.

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Approximate composition and transmission c. 800–500 BCE, with revisions

The altar cord

Vedic and later Indian worlds

Date of composition and textual transmission, not of a conserved rope or altar.

Current record

Measuring and weighing4 dated objects

1 / 4In the centre: Susa accounting bulla

You can scroll the first group without JavaScript. The other topics are also available below.

Mathematics

Current object, without a comparable date: The altar cord.

Plimpton 322 · Paleo-Babylonian period · around 1800 BCE · Sumer
Si.427 tablet · Towards 1900–1600 BCE · paleobabylonian; uncertain chronological attribution in CDLI · Sumer

Plant fibres

Current object, without a comparable date: The altar cord.

Architecture

Current object, without a comparable date: The altar cord.

Before Pre-Inca Cultures · Kotosh-Mito Tradition, c. 2200–1500 BCE; the enclosure is usually situated around 1800 BCE. · Pre-Inca cultures
Ishtar Gate · 7th–6th centuries BCE · work of Nebuchadnezzar II · Sumer
Heads on the wall · c. 800–300 BCE · Pre-Inca cultures
The god on the roof · Towards 510–500 BCE · Etruscans and Rome
Glazed brick guards · 522–486 BCE · Achaemenid Persia
Twenty-three delegations · 6th–5th centuries BCE, mainly under Darius and Xerxes · Achaemenid Persia
Torreáguila fountain spout · 101–125 CE · Etruscans and Rome
Zimbabwe birds · main period of Great Zimbabwe, c. 12th–15th centuries · Bantu-speaking worlds

Without a comparable date (not part of the chronological order):

A White Wall on the Danube · 6th century BCE · Celtic worlds
Cemetery · ca. 2600–1900 BCE · Indus civilisation
Dia’s strata · 9th century BCE–2nd millennium CE · Mande Worlds
Four gardens and water · 6th century BCE · Achaemenid Persia
Jenne-jeno contexts · ca. 250 BCE–1400 CE · Mande Worlds
Kumbi Saleh · medieval occupation; prominent phases between the 11th and 13th centuries · Mande Worlds
Livestock and mounds · c. 2050–1500 BCE · Nubia, Kerma and Kush
Lothal · ca. 2400–1600 BCE · Indus civilisation
Rice on stilts · Especially 1st–3rd centuries CE · Yayoi Japan
Rome learns in terracotta · End of the sixth century BCE · Etruscans and Rome
The quarry remained visible · Mainly 6th–5th centuries BCE · Gojoseon and early Korea
Tichitt walls · II–I millennia BCE · Mande Worlds

Worship and offerings

Current object, without a comparable date: The altar cord.

Uruk Vase · ca. 3300–3000 BCE · Sumer
Worshiper of Tell Asmar · Towards 2900–2600 BCE · Sumer
Statue of Ebih-Il · Towards 2400 BCE · Sumer
Before Phoenicia · 20th–18th centuries BCE · Phoenicians
Jue · c. sixteenth century BCE · Ancient China
Rubber before the stadium · c. 1600–1200 BCE for the oldest specimens · Maya worlds
Wooden faces · c. 1600–1000 BCE, in several phases · Maya worlds
Liding · 12th–11th centuries BCE · Ancient China
Astarte from El Carambolo · 8th century BCE · Phoenicians
Astarte from El Carambolo · 8th century BCE · Iberians
The invisible mosaic · c. 900–600 BCE · Maya worlds
Offering 4 · c. 800–600 BCE · Maya worlds
Uni was also Astarte · Towards 510 BCE · Etruscans and Rome
The god on the roof · Towards 510–500 BCE · Etruscans and Rome
Thoth in the form of a baboon · 664–30 BCE · Ancient Egypt
Poroina Mare Rhyton · Second half of the 4th century BCE · Thracians
Eight voices in one hand · 3rd–2nd centuries BCE · Gojoseon and early Korea
Thirty-nine buried bells · Buried towards the 1st century BCE–1st century CE · Yayoi Japan
Betatun altar · 1st century BCE–1st century CE · Iberians
Kongo crucifix · 16th–17th centuries · Bantu-speaking worlds

Without a comparable date (not part of the chronological order):

A tradition that invents · collected in 1993 or 1995 · Batwa / Twa histories
Four offerers · Iberian period · individual dating according to catalogue · Iberians
Lararium of Vilauba · Roman period · Etruscans and Rome
Livestock and mounds · c. 2050–1500 BCE · Nubia, Kerma and Kush
Rome learns in terracotta · End of the sixth century BCE · Etruscans and Rome
Soma stones · Rigvedic strata; later ritual tradition · Vedic and later Indian worlds
The Thracian Horseman · Roman era · diffusion especially in the 2nd–3rd centuries CE · Thracians
Translating faith · 16th–18th centuries · Bantu-speaking worlds

A closer look

About 1 min to read · 1 section. Further reading and sources · free access. Editorial review recorded: September 5, 2026.

A diagonal calculated by means of fractions

The rules attributed to Āpastamba and Kātyāyana include an approximation of the square root of two expressed by sums and a fractional correction: 1 + 1/3 + 1/12 − 1/408. The result, 577/408, approximates closely the ratio of diagonal to side of a square. The St Andrews synthesis underscores that the text preserves the rule, but not the explanation of how it was obtained. Its accuracy can be verified today; the reconstruction of the original reasoning remains a different historical question from the calculation.

Research · University of St Andrews, MacTutor · Indian Sulbasutras, J. J. O’Connor and E. F. RobertsonAccess recorded: September 5, 2026.

Location and territory

REGIONAL AREA

North India; diverse school traditionsFrom Punjab and the north-west towards the Ganges plain, with regional rhythms. The footprint indicates the indicative centre of the sphere, not a border or an exact point.

Sources and references

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The altar cord

History and context

Evidence and interpretation

Catalogue facts

Objects connected through time

The altar cord

Susa accounting bulla

Nippur Cubit

Weights

Maya's cubit

A closer look

A diagonal calculated by means of fractions

Location and territory

Sources and references

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