There are difficulties encountered when one studies the history of mathematics in ancient India. Some of this is due to gaps in original source material. There are also discrepancies among secondary and tertiary sources. Additionally, at one point in the West, the history of Indian mathematics was often overlooked in favor of a more Eurocentric view. More recently, there have been efforts to correct this neglect with the publication of popular books for mainstream audiences like The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph [Joseph], and more reliable specialized research by scholars such as Kim Plofker, author of Mathematics in India [Plofker2] and a chapter on Indian mathematics in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, edited by Victor J. Katz [Plofker1]. For more information on the debate about Eurocentrism in the study of Indian mathematics, see Clemency Montelle’s review [Montelle] of The Crest of the Peacock.

The earliest evidence of ancient Indian mathematics dates back to 3000 BCE with the Harappan culture. [Joseph, p. 217] The earliest texts, dated during the second millennium BCE, are the Vedas, written in a form of Sanskrit. The word “Veda” translates as “knowledge.” In addition to hymns, these texts contain procedures for conducting religious rituals. [Plofker2, p. 5 and Plofker1, p. 385]  Elements such as fire and water, as well as gods such as Agni (of fire) and Indra (of rain and thunderstorms) were worshiped in the religion of the Vedic period. The religious rituals involved the recitation of hymns and sacrifices. Priests performed these rituals for both nobles and wealthy commoners with the goal of enhancing fertility, wealth, and an afterlife shared with their ancestors. This style of worship is also seen today in the Hindu religion. There are a vast number of Vedic rituals, we mention only several. See the recent book by Shrikant Prasoon [Prasoon] for further discussion.

• The Agnistoma ritual involving the ceremonial beverage soma. The beverage soma was made by juicing the stalks of a certain plant, although the identity of this plant remains unclear, and is frequently speculated about. Soma imbued the drinker with immortality. For example, the gods Agni and Indra were said to drink soma.
• Rituals involving the construction of sacrificial fire altars. Examples include the Agnihotra, an oblation to Agni, as well as the Agnicayana.
• Healing rituals found in the Atharvaveda, a Veda on medicine.
• The Ashvamedha ritual for the prosperity of a nation.
• The Rajasuya, a ritual performed by a powerful king after a successful military campaign.

Figure 2.  Boys working on a model of the bird-shaped fire altar in an Agnicayana ritual in 2011 in Panjal, Kerala. (Photo courtesy of Professor Michio Yano.)

As mentioned above, the focus of this article is on the construction of sacrificial fire altars in ancient India by means of rope geometry. Cryptic instructions for building these fire altars using measuring cords were given in the Vedic texts called Śulba-sūtras, which translates as “Rules of the cord.” According to Plofker [Plofker2, p. 16-18], there were four main Śulba-sūtras which are known by their authors’ names:

• Baudhāyana-śulba-sūtra, from the Middle Vedic period, about 800-500 BCE,
• Mānava-śulba-sūtra, also from the Middle Vedic period, about 800-500 BCE,
• Āpastamba-śulba-sūtra, from a time after the previous two but before the following,
• Kātyāyana-śulba-sūtra, possibly from the mid-fourth century BCE.

The dating of these writings is not certain, but is based on comparison of the style and grammar used in these texts with other texts. For more information on the Śulba-sūtras, see Mathematics in India by Kim Plofker [Plofker2] starting on page 16, and also page 387 and following in Chapter 4 of The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, edited by Victor J. Katz [Plofker1].

Ritual was an extremely important part of the ancient Hindu religion. Fire altars were built for both rituals that occurred at regularly scheduled times throughout the year, as well as for special rituals. The special rituals would involve requests to gain benefits or favors such as wealth, and the shape of the fire altar depended on the favor being requested. The altars were temporary and destroyed once the ritual was completed. For example, the Agnicayana ritual called for a bird-shaped altar constructed out of 1005 bricks in homage to the god Agni. The ritual took 12 days to perform. The purpose of the ritual was to build an immortal body that would transcend suffering and death, both hallmarks of mortal existence [Converse]. The table below shows the various shapes of the fire altars, along with the favor being requested, and the source within the Śulba-sūtras for either information about and/or instructions for building that particular fire altar. The sources are given in the table by using the first letter of each of the four Śulba-sūtras followed by the appropriate verses. The English translation of the Śulba-sūtras by Sukumar N. Sen and Amulya Kumar Bag [Sen & Bag] was used to make the table.

According to Plofker [Plofker2, p. 17],

Many of the altar shapes involved simple symmetrical figures such as squares and rectangles, triangles, trapezia, rhomboids, and circles. Frequently, one such shape was required to be transformed into a different one of the same size. Hence the Śulba-sūtra rules often involve what we would call area–preserving transformations of plane figures, and thus include the earliest known Indian versions of certain geometric formulas and constants.

A 12-day fire altar sacrifice ritual was filmed by scholars in 1975 and made into a documentary called Altar of Fire. For more information on the documentary and to view a 9-minute preview that shows the altar bricks being made and a measuring stick being used, visit the Documentary Educational Resources website. The entire 58-minute documentary is available for purchase at the website.

The first step in any fire altar construction was to lay out the cardinal directions, especially the East-West line. The East-West line had special significance in the construction of the Vedic fire altar. Indeed, on the East-West line are two altars. The altar at the eastern end is square, and contains the Āhavanīya fire, symbolizing the celestial world, heaven. The one at the western end is circular and contains the Gārhapatya fire, symbolizing the terrestrial world. There is a third fire in the southern direction as well, the Dakṣiṇāgni, which symbolizes the air world. See [Kramrisch & Burnier] for further discussion. The GeoGebra applet below moves, one step at a time, through these instructions for laying out the cardinal directions from the Kātyāyana-śulba-sūtra. Click on “Go” to proceed to the next step.

Figure 3. This applet outlines the construction of the East-West line as described in the Kātyāyana-śulba-sūtra, which has special significance in the construction of Vedic fire altars. Note the translation of the original text in quotes. Click "Go" to advance to the next step.

Also, at the beginning of the Baudhāyana-śulba-sūtra, instructions are given for using a measuring cord to construct a square, providing another construction for laying out the cardinal directions. See Activities 2 and 3 on the Student Activities page of this article for indoor and outdoor classroom activities that model this ancient Indian way of constructing a square. The picture below shows the result of Activity 2. Note that there are other methods for constructing a square in the Śulba-sūtras in addition to the two just given, which are not included in this article. For example, one involves Pythagorean triples and is similar to the construction of the Great Altar which follows.

Figure 4. The result of constructing a square with a measuring cord, using an ancient Indian procedure explained in Activity 2, except for the very last step of connecting the four corners (dots) with line segments to form the square.

As mentioned above, a related construction is that of the so-called Great Altar, which is in the shape of an isosceles trapezoid with its altitude parallel to the East-West line, and its longer base facing West. Its bases have lengths 30 paces and 24 paces, and its altitude has length 36 paces. The Great Altar was used in rituals involving the Vedic ceremonial beverage soma [Plofker2, p. 25]. The converse of the Pythagorean Theorem is implicitly used in the construction, in which a mark is made 15 units from an end of a 54 unit rope. If we attach the ends of the 54 unit rope to stakes in the ground 36 units apart, and pull on the mark until the rope is taut, then the resulting triangle has side lengths \(36,\) \(15,\) and \(54-15=39\) units. But then the converse of the Pythagorean Theorem forces the triangle to be a right one. The GeoGebra applet below presents the construction described in the Śulba-sūtra of Āpastamba, and a similar construction also appears in that of Baudhāyana. The applet moves through these instructions one step at a time. Click on “Go” to proceed to the next step.

Figure 5. This applet outlines the construction of the Great Altar as described in the Śulba-sūtra of Āpastamba. Note the translation of the original text in quotes. Click "Go" to advance to the next step.

Āpastamba also described the classical construction of transforming an isosceles trapezoid into a rectangle of equal area, and thus calculated the area enclosed by the Great Altar. That is, cut off a right triangle with leg lengths 3 and 36 units from the northern edge of the trapezoid, and glue it to the southern edge, so as to obtain a rectangle. This yields a rectangle with side lengths 36 and 27 units, and hence area of 972 square units. Below is a translation of the original text from the Śulba-sūtra of Āpastamba, Section 5.7 [Plofker2, p. 26]:

The Great Altar is a thousand [square] paces [or double-paces] less twenty-eight. One should bring [a line] from the south[east] corner twelve units toward the south[west] corner. One should place the cut-off [triangle] upside-down on the other [side]. That is an oblong quadrilateral. In that way one should consider it established.

Also found in the Śulba-sūtras of Baudhāyana, Āpastamba, and Kātyāyanai are techniques for adding and subtracting squares; that is, given two squares, construct a third square whose area is the sum or difference of the two given squares. Note that we solve the classical problem of “doubling a square” by simply adding two squares of the same area. As before, these techniques rely on the Pythagorean Theorem, and are illustrated in the corresponding GeoGebra applet. Below is a translation of the original source, found in the Śulba-sūtra of Āpastamba, Section 2.4 [Plofker2, p. 21]:

Now the combination of two [square] quadrilaterals with individual [different measures]. Cut off a part of the larger with the side of the smaller. The cord [equal to] the diagonal of the part [makes an area which] combines both. That is stated.

Figure 6. This applet demonstrates the method in the Śulba-sūtra of Āpastamba for constructing a square with area equal to the sum of the areas of two given squares. Click "Go" to advance to the next step. The areas of the two given squares can be adjusted by sliding points \(D\) and \(H.\) The area of square \(JMDI\) can also be adjusted by sliding point \(I.\) The desired area is achieved when point \(I\) meets point \(K.\)

Consider two squares \(ACDB\) and \(EGHF.\) In the Geogebra applet in Figure 6, the reader may change the dimensions of the two squares, by adjusting the location of points \(D\) and \(H.\) Without loss of generality, we assume that \(EGHF\) has area less than or equal to that of \(ACDB.\)

First we mark points \(K\) and \(L\) on sides \(AB\) and \(CD\) so that \(KB\) and \(LD\) both have length equal to the side length of the smaller square \(EGHF.\) Also observe that the length of \(KL\) is the side length of \(ACDB.\)

Now the square with side length equal to that of \(KD\) is the desired square. Its area is \(KD^2\) and by the Pythagorean Theorem, \[{\rm{Area}}(ACDB)+{\rm{Area}}(EGHF)=KL^2 + LD^2=KD^2.\] In the applet, notice as we slide point \(I\) from \(B\) to \(K,\) the area of the constructed square increases, finally meeting the sum of the given areas when \(I\) meets \(K.\) Notice that when the areas of the two given squares are equal, the side of the desired square is the diagonal of a given square.

A similar technique is used to find the difference of two squares, also from Āpastamba's Śulba-sūtra, Section 2.5 [Plofker2, p. 21]:

Removing a [square] quadrilateral from a [square] quadrilateral: Cut off a part of the larger, as much as the side of the one to be removed. Bring the [long] side of the larger [part] diagonally against the other [long] side. Cut off that [other side] where it falls. With the cut-off [side is made a square equal to] the difference.

Figure 7. This applet demonstrates the method in the Śulba-sūtra of Āpastamba for constructing a square with area equal to the difference of the areas of two given squares. Click "Go" to advance to the next step. The areas of the two given squares can be adjusted by sliding points \(D\) and \(H.\) The area of square \(PNDJ\) can also be adjusted by sliding point \(I.\) The desired area is achieved when point \(I\) meets point \(M.\)

We again consider two squares \(ACDB\) and \(EGHF,\) again assuming that \(EGHF\) has area less than or equal to that of \(ACDB.\) As before, we mark points \(K\) and \(L\) on sides \(AB\) and \(CD,\) so that \(KB\) and \(LD\) have length equal to the side length of \(EGHF.\)

Now consider a circular arc, with center \(D\) and radius equal to the length of \(DB.\) Extend the circular arc until it intersects \(KL,\) say at point \(M.\) The desired square has side length equal to that of \(ML.\) By the Pythagorean Theorem applied to right triangle \(MLD,\)

\[ML^2=DM^2-LD^2={\rm{Area}}(ACDB)+{\rm{Area}}(EGHF).\]

For the last equality, notice the length of \(DM\) is equal to that of side \(DB\) of \(ACDB,\) as they are both radii of the same arc, and that \(LD\) has side length equal to the side length of \(EGHF.\) In the corresponding applet, the reader may move point \(I\) along the constructed arc to its desired position at \(M,\) and observe the change in the area of the constructed square.

A method for constructing a square with area equal to that of a given rectangle is given in both the Śulba-sūtras of Baudhāyana and Āpastamba. This will use the previous technique of finding a square whose area is the difference of the areas of two given squares. Below is a translation of the original text from Āpastamba's Śulba-sūtra, verse 2.7 [Plofker2, p. 22]:

Wishing [to make] an oblong quadrilateral an equi-quadrilateral [square]: Cutting off a [square part of the rectangle] with [its] width, [and] halving the remainder, put [the halves] on two [adjacent] sides [of the square part]. Fill in the missing [piece] with an extra [square]. Its removal [has already been] stated.

Figure 8. This applet demonstrates the technique found in Āpastamba's Śulba-sūtra for transforming a rectangle into a square with equal area. Click "Go" to advance to the next step. The area of the given rectangle may be adjusted by sliding points \(B\) and \(D.\) The area of square \(RTSJ\) may also be changed by sliding point \(I.\) The desired area is achieved when point \(I\) meets side \(CB.\)

Consider a rectangle \(ADCB.\) Let us assume \(AB\) is the smaller side. In the GeoGebra applet in Figure 8, the reader may adjust the dimensions of the rectangle by moving points \(B\) and \(D.\)

First, mark points \(H\) and \(K\) on sides \(AD\) and \(CD,\) respectively, so that \(AHKB\) forms a square. Then find points \(E\) and \(M\) so that remaining rectangle \(HDCK\) is divided into two congruent rectangles \(HEMK\) and \(EDCM.\)

Next, construct \(BKGJ\) to the right of, and adjacent to \(AHKB,\) congruent to the rectangles \(HEMK\) and \(EDCM.\)  Now observe that the sum of the areas of \(AHKB,\) \(HEMK,\) and \(BKGJ\) is equal to the area of the original rectangle \(ADCB.\)

Then construct square \(KMFG.\) Now the area of square \(AEFJ,\) less the area of \(KMFG,\) is equal to the area of the original rectangle \(ADCB.\) Thus our goal becomes to find the difference of squares \(AEFJ\) and \(KMFG.\)

We now proceed with the established technique for this. Find where the circular arc with center \(J\) and radius \(FJ\) intersects the segment \(BM.\) In the GeoGebra applet in Figure 8, slide point \(I\) along the arc. As point \(I\) approaches segment \(BM,\) the area of \(RTSJ\) approaches the area of \(ADCB.\) Indeed, when point \(I\) lies on segment \(BM,\) we apply the Pythagorean Theorem to triangle \(JIB\) and obtain

\[{\rm{Area}}(RTSJ)=IB^2=JI^2-JB^2=(JI-JB)(JI+JB).\]

But by congruence, \(JI=FJ=EA,\) and \(JB=EH\) and therefore,

\[JI-JB=EA-EH=HA=AB\]

and \[JI+JB = EA+EH = EA + DE = AD,\]

from which we infer \({\rm{Area}}(RTSJ)=(AB)(AD),\) as desired.

It is interesting to consider when the constructed square \(RTSJ\) will have side length equal to that of \(EH.\) For when this occurs, the side \(RT\) of \(RTSJ\) will lie precisely on side \(CB\) of \(ADCB.\) Setting \(X=AB\) and \(Y=AD,\) note that \[EH = \frac{Y-X}{2},\] and the equation \({\rm{Area}}(RTSJ)=(AB)(AD)\) can be rewritten as \[\left(\frac{Y-X}{2}\right)^2=XY\] or \[X^2-2XY+Y^2=4XY\] or \[X^2-6XY+Y^2=0.\]

Viewing the last equation as a quadratic in \(X\) and solving for \(X/Y\) we obtain

\[X=\frac{6Y-\sqrt{36Y^2-4Y^2}}{2}= (3 – 2\sqrt{2})Y,\] or

\[\frac{X}{Y}=\frac{AB}{AD}= 3 – 2\sqrt{2}.\]

Consequently, the constructed square \(RTSJ\) has side \(RT\) lying on side \(CB\) of \(ADCB\) \(-\) equivalently, \(RTSJ\) has side length equal to that of \(EH = {(AD-AB)}/{2}\) \(-\) precisely when \({AB}/{AD} = 3-2\sqrt{2},\) which the reader may verify by changing the dimensions of the original rectangle \(ADCB\) in the GeoGebra applet in Figure 8.

The Śulba-sūtras of Āpastamba and Kātyāyana contain an ingenious approximation of the square root of \(2\) by a rational number, \[\sqrt{2}\approx 1+\frac{1}{3}+\frac{1}{3\cdot 4}-\frac{1}{3\cdot4\cdot34}=1+\frac{1}{3}+\frac{1}{12}-\frac{1}{408}=1.41422\dots.\] Indeed, the relative error in this approximation is less than 0.0003%! The original text in the Śulba-sūtras of Kātyāyana (Section 2.9) may be restated as [Joseph, p. 334]:

Increase the measure by its third and this third by its own fourth, less the thirty-fourth part of that fourth. This is the value with a special quantity in excess.

Note that this passage demonstrates knowledge that the approximation is a slight overestimate of the square root of \(2.\) However, no explanation of this approximation appears in the Śulba-sūtras. We will present the rather plausible explanation found in [Joseph, p. 334-336], originally due to Bibhutibhusan Datta in 1932 [Datta].

Begin with two unit squares, \(SPQR,\) and \(ADCB,\) as shown in the GeoGebra applet in Figure 9 below. The square with area \(2\) will of course have side length equal to the square root of \(2.\) The square \(SPQR\) is divided into two rectangles of dimension \(1\) by \(1/3,\) one square of dimension \(1/3\) by \(1/3,\) and \(8\) rectangles of dimension \(1/{12}\) by \(1/3.\) These rectangles are then arranged around \(ADCB,\) so that they fit into square \(GAEF,\) which has side lengths \[1+\frac{1}{3}+\frac{1}{12} = \frac{17}{12}.\] Notice that these rectangles do not fill up \(GAEF\) completely; that is, the red shaded rectangle in the upper right hand corner is not filled with a piece of \(ADCB.\) Therefore, the area of \(GAEF\) is precisely \(2\) plus the area of the red shaded rectangle, which is \(1/{144},\) being a square of side length \(1/{12}.\)

Figure 9. This applet outlines a preliminary approximation of the square root of \(2\) as the rational number \(17/12.\) Click "Go" to demonstrate that a square with side length \(17/12\) has area slightly greater than \(2.\) The overestimate of the area, shaded in red in the final step, is \(1/144.\) Therefore, \(17/12\) is an overestimate of the square root of \(2.\)

Indeed, square \(GAEF\) is the basis for our approximation, for its area is slightly greater than \(2,\) and, therefore, its side length of \[1+\frac{1}{3}+\frac{1}{12} = \frac{17}{12}\] is slightly greater than the square root of \(2.\) Our goal is to modify \(GAEF,\) so that its area is as close to \(2\) as possible. To this end, imagine cutting off two rectangular strips, of equal thickness, from the left and bottom edges of \(GAEF.\) This forms the “L” shaped region \(GAELNM,\) shown in the GeoGebra applet in Figure 10 below. We would like the area of \(GAELNM\) to be \(1/144,\) for then the area of square \(MNLF\) would be precisely \(2,\) and thus its side lengths would be precisely the square root of \(2.\)

Figure 10. This applet demonstrates the procedure described in the article for calculating just by how much \(17/12\) overestimates the square root of \(2.\) The key idea is to shave off an "L" shaped region of area \(1/144\) from the square with side length \(17/12.\)

Let \(X\) denote the “thickness” of region \(GAELNM.\) In the GeoGebra applet in Figure 11 (zoomed in around the lower left corner of \(GAEF),\) this is the length of segment \(HI.\)

Figure 11. After zooming in to the lower left corner of square, this applet demonstrates that the desired thickness of the "L" shaped region is approximately \(1/408.\) As a result, the square root of \(2\) is approximately \(17/12-1/408.\)

The area of region \(GAELNM,\) in terms of \(X,\) is \[2\left(\frac{17X}{12}\right)-X^2.\]

To see this, observe that \(GAELNM\) is the union of two rectangular strips of dimensions \({17}/{12}\) by \(X,\) with intersection a square of side length \(X.\) We therefore add the areas of the rectangular strips and subtract the area of the overlap \(X^2.\) Setting this equal to \(1/{144},\) ignoring the small term of \(X^2,\) and solving for \(X\) we obtain:

\[2\left(\frac{17X}{12}\right)-X^2\approx 2\left(\frac{17X}{12}\right)=\frac{1}{144},\] so that

\[X = \frac{1}{408} = \frac{1}{3\cdot4\cdot34}=0.0245\]

Therefore, the side length of a square of area \(2\) – that is, the square root of \(2\) – is approximately \[1 + \frac{1}{3} + \frac{1}{3\cdot 4} - \frac{1}{3\cdot4\cdot34}.\]

This is an overestimate of the square root of \(2,\) because we ignored the small, but positive, term \(X^2,\) when solving for \(X.\)

We caution the reader that knowledge of fractions, and operations with fractions, which we have used above, were most likely not known to the ancient Indians. In 2006, a reconstruction of the approximation of \(\sqrt{2}\) using only manipulation of measuring rope was discovered, which the reader may find in the article published that year by Satyanad Kichenassamy [Kichenassamy].

Activity 1. Linear Measure the Ancient Indian Way

This is an activity for elementary and middle school students that involves measuring using three units of length from ancient India:

Download Student Worksheet: Linear Measure the Ancient Indian Way.

Figure 12. Measurement of the height of a participant in an Agnicayana fire altar sacrificial ritual in 2011 in Panjal, Kerala. (Photo courtesy of Professor Michio Yano)

Activity 2. Constructing a Square an Ancient Indian Way (Indoors Version)

This activity can be used with middle school or high school students. Using cardboard, string, and pushpins, the students model one of the several ways for constructing a square used in ancient India in the building of a fire altar. Common Core Standard G.CO.12 recommends making formal geometric constructions with a variety of tools and methods.

Download Instruction Sheet for Instructors: Constructing a Square an Ancient Indian Way.

Activity 3. Constructing a Square an Ancient Indian Way (Outdoors Version)

The above activity can be completed outside in a more realistic fashion, using pegs or stakes instead of pushpins and a rope or cord instead of the string. If there is no open ground available, the activity can be completed on a parking lot using sidewalk chalk, with students holding the pegs in place instead of pounding the pegs into the ground.

Conclusion

In ancient India, geometry was used extensively in constructing fire altars without the use of modern measuring devices. By using ropes, they were able to form right angles and various shapes and to transform one shape into another with the same area. They also were able to devise a good approximation to the square root of 2. Students today can use these same methods to explore and engage with geometrical concepts through GeoGebra applets and hands-on activities, while at the same time gaining an appreciation of some of the mathematical contributions of the ancient Indian civilization.

Acknowledgments

The authors wish to thank the editor and referees for their helpful suggestions. We greatly appreciate the thorough review which was invaluable in improving the paper. We also wish to thank Michio Yano, Professor Emeritus, Kyoto Sangyo University, for allowing us to use the photographs in Figures 1, 2, and 12.

About the Authors

Cynthia J. Huffman is a University Professor in the Department of Mathematics at Pittsburg State University. She has always been interested in history of mathematics but her interest was especially sparked by participation in several of the MAA Study Tours. Her research areas include computational commutative algebra and history of mathematics. Dr. Huffman is a handbell soloist and has a black belt in Chinese Kenpo karate.

Scott V. Thuong is an Assistant Professor in the Department of Mathematics at Pittsburg State University. His research areas include topology, geometry, and the history of mathematics. In his spare time, Dr. Thuong enjoys a good game of badminton.

References and Resources

Common Core State Standards Initiative: http://www.corestandards.org/

[Converse] Converse, Hyla Stuntz. The Agnicayana Rite: Indigenous Origin?, History of Religions 14 (2), 1974, pp. 81-95.

[Datta] Datta, Bibhutibhusan. The Science of the Sulbas: A Study in Early Hindu Geometry. Calcutta University Press, 1932.

Gardner, Robert and J.F. Staal. Altar of Fire. Documentary. The Film Study Center at Harvard University, 1976. Available for purchase at http://www.der.org/films/altar-of-fire.html.

GeoGebra: http://www.geogebra.org

[Joseph] Joseph, George Gheverghese. The Crest of the Peacock: Non-European Roots of Mathematics, 3^rd edition. Princeton University Press, 2011.

[Kichenassamy] Kichenassamy, Satyanad, Baudhāyana’s rule for the quadrature of the circle, Historia Mathematica 33, 2006, p. 149-183.

[Kramrisch & Burnier] Kramrisch, Stella, and Raymond Burnier. The Hindu Temple, Motilal Banarsidass Publ., Vol. 1, 1976, p. 22-23.

[Montelle] Montelle, Clemency. Book Review of The Crest of the Peacock: Non-European Roots of Mathematics, Notices of the AMS, December 2013, Vol. 60, No. 11, pp. 1459-1463: http://www.ams.org/notices/201311/rnoti-p1459.pdf

[Plofker1] Plofker, Kim. Mathematics in India. Chapter 4 in Katz, Victor, editor. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, 2007, pp. 385-514.

[Plofker2] Plofker, Kim. Mathematics in India. Princeton University Press, 2009.

[Prasoon] Prasoon, Shrikant. Indian Scriptures. Pustak Mahal, 2010, Ch. 2.

[Sen & Bag] Sen, S.N., and A.K. Bag. The Śulbasūtras. Indian National Science Academy, 1983.