The
Shulba Sutras or
Śulbasūtras (
SanskritSanskrit is a historical Indo-Aryan language, one of the liturgical languages of Hinduism and Buddhism, and one of the 22 official languages of India. It is also declared as a classical language by the government of India....
: "string, cord, rope") are
sutraSūtra , literally means a thread or line that holds things together, and more metaphorically refers to an aphorism , or a collection of such aphorisms in the form of a manual...
texts belonging to the
Śrauta' traditions are conservative ritualistic traditions of historical Vedic religion in Hinduism, based on the body of Śruti literature. They persist in a few places in India today although constituting a clear minority within Hinduism...
ritual and containing geometry related to fire-altar construction.
Purpose and origins
The Shulba Sutras are part of the larger corpus of texts called the Shrauta Sutras, considered to be appendices to the
VedasThe Vedas are a large body of texts originating in Ancient India. The texts are composed in Vedic Sanskrit and form the oldest layer of Sanskrit literature, and the oldest sacred texts of Hinduism....
. They are the only sources of knowledge of
Indian mathematicsIndian mathematics is the mathematics that emerged in South Asia from ancient times until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today ...
from the
Vedic periodThe Vedic Period is the period during which the Vedas, the oldest sacred texts of the Indo-Aryans, were being composed. Scholars place the Vedic period in the second and first millennia BCE continuing up to the 6th century BCE based on literary evidence.The associated culture, sometimes referred...
. Unique fire-altar shapes were associated with unique gifts from the Gods. For instance, "he who desires heaven is to construct a fire-altar in the form of a falcon"; "a fire-altar in the form of a tortoise is to be constructed by one desiring to win the world of Brahman" and "those who wish to destroy existing and future enemies should construct a fire-altar in the form of a rhombus".
The four major Shulba Sutras, which are mathematically the most significant, are those composed by
BaudhayanaBaudhāyana, was an Indian mathematician, whowas most likely also a priest. He is noted as the author of the earliest Sulba Sutra — appendices to the Vedas giving rules for the construction of altars — called the , which contained several important mathematical results. He is older...
,
ManavaManava is the author of the Indian geometric text of Sulba Sutras.The Manava Sulbasutra is not the oldest , nor is it one of the most important, there being at least three Sulbasutras which are considered more important...
,
ApastambaThe Dharmasutra of Āpastamba forms a part of the larger Kalpasūtra of Āpastamba. It contains thirty praśnas, which literally means ‘questions’ or books. The subjects of this Dharmasūtra are well organized and preserved in good condition...
and Katyayana, about whom very little is known. The texts are dated by comparing their grammar and vocabulary with the grammar and vocabulary of other Vedic texts. The texts have been dated from around 800 BCE to
200Year 200 was a leap year starting on Tuesday of the Julian calendar.-Roman Empire:* Septimus Severus visits Syria, Palestine and Arabia....
CE, with the oldest being the sutra that was written by Baudhayana around 800 BCE to 600 BCE.
There are competing theories about the origin of the geometry that is found in the Shulba sutras, and of geometry in general. According to the theory of the ritual origins of geometry, different shapes symbolized different religious ideas, and the need to manipulate these shapes lead to the creation of the pertinent mathematics. Another theory is that the mystical properties of numbers and geometry were considered spiritually powerful and consequently, led to their incorporation into religious texts.
Pythagorean theorem
The sutras contain discussion and non-axiomatic demonstrations of cases of the
Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
and Pythagorean triples. It is also implied and cases presented in the earlier work of
ApastambaThe Dharmasutra of Āpastamba forms a part of the larger Kalpasūtra of Āpastamba. It contains thirty praśnas, which literally means ‘questions’ or books. The subjects of this Dharmasūtra are well organized and preserved in good condition...
and
BaudhayanaBaudhāyana, was an Indian mathematician, whowas most likely also a priest. He is noted as the author of the earliest Sulba Sutra — appendices to the Vedas giving rules for the construction of altars — called the , which contained several important mathematical results. He is older...
, although there is no consensus on whether or not Apastamba's rule is derived from Mesopotamia. In Baudhayana, the rules are given as follows:
1.9. The diagonal of a square produces double the area [of the square].
[...]
1.12. The areas [of the squares] produced separately by the lengths of the breadth of a rectangle together equal the area [of the square] produced by the diagonal.
1.13. This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36.
The
Satapatha Brahmana and the
Taittiriya Samhita were probably also aware of the Pythagoras theorem. Seidenberg (1983) argued that either "Old Babylonia got the theorem of Pythagoras from India or that Old Babylonia and India got it from a third source". Seidenberg suggested that this source might be
SumerSumer was a civilization and historical region in southern Iraq . It is the earliest known civilization in the world and is known as the Cradle of Civilization...
ian and may predate 1700 BC.
Pythagorean triples
Pythagorean triples are found in
ApastambaThe Dharmasutra of Āpastamba forms a part of the larger Kalpasūtra of Āpastamba. It contains thirty praśnas, which literally means ‘questions’ or books. The subjects of this Dharmasūtra are well organized and preserved in good condition...
's rules for altar construction. They were used for the construction of right angles. The complete list is:
However, since these triples are easily derived from an old Babylonian rule, Mesopotamian influence is not unlikely.
Geometry
The Baudhayana Shulba sutra gives the construction of geometric shapes such as squares and rectangles. It also gives, sometimes approximate, geometric area-preserving transformations from one geometric shape to another. These include transforming a
squareIn geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
into a
rectangleIn Euclidean geometry, the term rectangle normally refers to a quadrilateral with four right angles. This is a simple rectangle. A simple rectangle with vertices ABCD would be denoted as ....
, an isosceles
trapeziumThe word trapezium has several meanings:* - a quadrilateral with a pair of parallel sides ....
, an isosceles
triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
, a
rhombusIn geometry, a rhombus or rhomb is a quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.In general, a polygon whose...
, and a
circleA circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the centre. The common distance of the points of a circle from its center is called its radius....
, and transforming a circle into a square. In these texts approximations, such as the transformation of a circle into a square, appear side by side with more accurate statements. As an example, the statement of circling the square is given in Baudhayana as:
2.9. If it is desired to transform a square into a circle, [a cord of length] half the diagonal [of the square] is stretched from the centre to the east [a part of it lying outside the eastern side of the square]; with one-third [of the part lying outside] added to the remainder [of the half diagonal], the [required] circle is drawn.
and the statement of squaring the circle is given as:
2.10. To transform a circle into a square, the diameter is divided into eight parts; one [such] part after being divided into twenty-nine parts is reduced by twenty-eight of them and further by the sixth [of the part left] less the eighth [of the sixth part].
2.11. Alternatively, divide [the diameter] into fifteen parts and reduce it by two of them; this gives the approximate side of the square [desired].
The constructions in 2.9 and 2.10 give a value of π as 3.088, while the construction in 2.11 gives π as 3.004.
Square roots
Altar construction also led to an estimation of the
square root of 2The square root of 2, also known as Pythagoras' constant,is the positive real number that, when multiplied by itself, gives the number 2....
as found in three of the sutras. In the Baudhayana sutra it appears as:
2.12. The measure is to be increased by its third and this [third] again by its own fourth less the thirty-fourth part [of that fourth]; this is [the value of] the diagonal of a square [whose side is the measure].
which leads to the value of the square root of two as being:
One conjecture about how such an approximation was obtained is that it was taken by the formula: with and
which is an approximation that follows a rule given by the twelfth century Muslim mathematician Al-Hassar. The result is correct to 5 decimal places.
This formula is also similar in structure to the formula found on a Mesopotamian tablet from the Old Babylonian period (1900-1600 BCE):
which expresses in the
sexagesimalSexagesimal is a numeral system with sixty as the base. It originated with the ancient Sumerians in the 2000s BCE, was transmitted to the Babylonians, and is still used—in modified form—for measuring time, angles, and geographic coordinates....
system, and which too is accurate up to 5 decimal places (after rounding).
Indeed an early method for calculating square roots can be found in some Sutras, the method involves the
recursiveRecursion, in mathematics and computer science, is a method of defining functions in which the function being defined is applied within its own definition. The term is also used more generally to describe a process of repeating objects in a self-similar way...
formula: for large values of x, which bases itself on the non-recursive identity for values of
r extremely small relative to
a.
Numerals
Before the period of the Sulbasutras was at an end, the
Brahmi numeralThe Brahmi numerals are an indigenous Indian numeral system attested from the 3
rd century BCE . They are the direct graphic ancestors of the modern Indic and Arabic numerals. However, they were conceptually distinct from these later systems, as they were not used as a positional system...
s had definitely begun to appear (c. 300BCE) and the similarity with modern day numerals is clear to see. More importantly even still was the development of the concept of decimal place value. Certain rules given by the famous
IndiaIndia, officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the south, the Arabian Sea on the west, and the Bay of Bengal...
n grammarian Panini (c. 500 BCE) add a zero suffix (a suffix with no phonemes in it) to a base to form words, and this can be said somehow to imply the concept of the mathematical
zero0 is both a number and the numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, zero is used as a placeholder in place value systems...
.
Incommensurables
It has sometimes been suggested the sutras contain knowledge of irrationality, but such claims are not well substantiated and unlikely to be true.
List of Shulba Sutras
The following Shulba Sutras exist in print or manuscript
- Apastamba
- Baudhayana
- Manava
- Katyayana
- Maitrayaniya (somewhat similar to Manava text)
- Varaha (in manuscript)
- Vadhula (in manuscript)
- Hiranyakeshin (similar to Apastamba Shulba Sutras)
Further reading
- Parameswaran Moorthiyedath, "Sulbasutra"
- Seidenberg, A. 1983. "The Geometry of the Vedic Rituals." In The Vedic Ritual of the Fire Altar. Ed. Frits Staal. Berkeley: Asian Humanities Press.
- Sen, S.N., and A.K. Bag. 1983. The Sulbasutras. New Delhi: Indian National Science Academy.