# The oldest example of applied geometry in the world

## An Old-Babylonian tablet reveals sophisticated understanding of mathematics

##### Published: 16 September 2021 (GMT+10)

A cuneiform clay tablet dated at 3,700-years from Babylonia, has recently been described as “one of the oldest examples of applied geometry from the ancient world.”^{1} The tablet is inscribed on one side with what appears to be a map of a field, complete with surveyor’s measurements that obey trigonometric relationships. On the reverse side is a tabulation, containing the results of the surveyor’s area calculations. The evidence demonstrates that ancient people’s understanding of mathematics “during this era was more sophisticated than previously assumed.”^{2}

The tablet, designated Si. 427 (resembling a biscuit, see photos), was unearthed in 1894 by Father Jean-Vincent Scheil’s French archaeological expedition at Tell Abu Habba, ancient Sippar-Yahrurum, of ancient Babylonia—southwest of modern Baghdad. It was stored in the Musée impérial de Constantinople, now the İstanbul Arkeoloji Müzeleri (Istanbul Archaeology museums).^{3} The tablet’s significance lay undiscovered for over a century, until Daniel Mansfield, a mathematician from the University of New South Wales, Australia (UNSW) recognized its importance.

The discovery of the tablet’s purpose challenges the way the history of mathematics is understood. Specifically, that the application of trigonometric theory, commonly attributed to the Greek mathematician Pythagoras (c. 570–c. 495 BC), was preceded by the Babylonians by more than a thousand years.

Si. 427 is a remarkable tablet, dating to the Old Babylonian [OB] period 1900–1600 BC. This dating is based on comparisons of certain vocabulary found on other OB tablets. It reveals geometric and legal details about an agricultural plot of land belonging to a prominent land-owner called Sîn-bēl-apli, who is known from other OB cuneiform tablets from Sippar.^{4} The survey plan was drawn-up to establish the field’s area and new boundaries after a portion of the land was sold-off. It is thought the land lay in-between a river and a road. The top half was marshy, and there was also a threshing floor and a tower on the land, which must have acted as fixed points for the survey. Si. 427 is the only known example of a cadastral document (surveyor’s map) from this period.

## The tablet text

The translation of the table reveals that the field was surveyed twice: first with shapes that have horizontal alignments, followed by a second survey using shapes with vertical alignments.^{5} This likely gave a way to check the results and give the most accurate sum of the field’s areas.

The obverse top of Si. 427 gives an area: “of a marshy field together with the tower and threshing-floor” whose OB area is measured at “1 bur GANA 45 sar” (approx. 16 acres).

The obverse bottom gives: “a total area of the field” of “1 bur 1 eŝe 4 iku 33 sar” which is approximately 25 acres. The OB units of measure are given in sexagesimal form (see below).

The reverse top gives a table of measurements made by the surveyor of the field.

The reverse bottom states: “total (surface area) of the field, together with various marshes of Sîn-bēl-apli apart from [OB measurement of the] field acquired through purchase” of “7 iku” (approx. 6 acres).^{6}

## Why survey was essential in Babylonia

Surveying must have been essential in re-establishing land boundaries after the Tigris and Euphrates rivers regularly flooded, washing away boundary markers and land. Also, farmers needed to know what the area of their land was, so they could calculate the seed required to sow for the best crop-yields. Such information was also essential for the Babylonian government, who required taxes from the farmers, based on the area of their land. Babylonians constructed roads and canals, which necessarily required surveying. Another factor is that during the OB period, land that typically belonged to the king’s estates and temples was starting to be sold-off and became available to citizens to purchase. Therefore, land needed to be accurately measured, mapped, divided, and new boundaries carefully established so as to avoid disputes. Therefore, the profession of the surveyor was a highly respected one, involving much training in mathematics, law, and land survey method.

The Si. 427 tablet demonstrates the Babylonian surveyors were highly skilled at producing mathematically accurate boundaries. The information engraved in the clay reveals the methods of the land surveyor as he performed his duties.

Other cuneiform inscriptions discovered from the OB period allow historians to piece together the development of land survey, which also included resolving private boundary disputes. This is indicated by a cuneiform text, about a young surveyor, Enki-manšum, who brags about his scribal training:

“When I go to divide a plot, I can divide it; when I go out to apportion a field, I can apportion the pieces, so that when wronged men have a quarrel I soothe their hearts.”^{7}

## Fundamentals of Babylonian survey

Si. 427 represents a map drawn up by a surveyor to calculate the area and establish boundaries of the plot of land he was mapping. Mansfield explains:

“Much like we would today, you’ve got private individuals trying to figure out where their land boundaries are, and the surveyor comes out but instead of using a piece of GPS equipment, they use Pythagorean triples.”^{8}

It is likely that the surveyor first set out along the longest boundary, what we refer to as a ‘Pythagorean triple’ (PT, see below) which is a kind of right(-angled) triangle. The surveyor of Si. 427 also made use of a variety of diagonal PTs likely chosen to match the field’s shape.^{9} Then he would have extended the sides of the PTs to form perpendicular lines, essential for accurately determining the areas of the enclosed areas. Succeeding perpendicular lines would likely have been set out by establishing new PTs, or by measuring out using a measuring rod and rope^{10} equal distances along parallel lines. Si. 427 is most likely the recording of the actual surveying process. This is the most likely scenario, which may be proved by the discovery of more tablets from this period.

The surveyor likely recorded his measurements on the tablet of clay in his hand with a wedged wooden stylus as he went about his business. To calculate the area of the land, he first wanted to describe it in terms of shapes, whose areas could easily be calculated (e.g., squares, rectangles, right triangles, and right trapezoids). The rectangles would have been described in terms of two PT right triangles. The surveyor measured off distances from fixed points, such as the corner of the field, tower, or threshing-floor. Once he started measuring out distances, he recorded everything he set out on the clay. No doubt he hammered wooded pegs into the ground as fixed points when needed, much like modern surveyors do. His aim would be to divide the land as efficiently as he could into standard shapes, whose areas could easily be calculated. What was found was that the shapes (PTs) he chose were standardised and chosen to give accurate, whole numbers.

## Pythagorean Triples

A fool-proof and achievable way for surveyors or draftsmen to construct perpendicular boundary lines was to make use of the so-called “Pythagorean triple” (PT), (which really are Babylonian triples, used more than 1000 years before Pythagoras). Mansfield states of these:

“Once you understand what Pythagorean triples are, your society has reached a particular level of mathematical sophistication.”^{11}

PTs are specific types of triangles that ensure their sides are true right angles to each other (perpendicular). Consider the right-angled triangle below, with sides labelled base ‘b’, height ‘h’. The long diagonal side ‘d’ is equal to the square of the other two sides added together, which can be expressed as: b^{2}+ h^{2} = d^{2}.

When it comes to calculating the sides of these triples, there were certain sizes that were easier to handle in terms of producing whole numbers for ease of calculations. Specifically, (3, 4, 5), (5, 12, 13), and (8, 15, 17) triangles. It was these specific PTs that the surveyor preferred to use in terms of convenience to produce a systematic grid with true parallel lines so as to accurately solve the area of the piece of land he was surveying. How he knew what PTs to use is discussed later (see Plimpton 322).

An important factor in the survey process was the scaling up of PTs. This was because scaling preserves ratios and proportions of sides (they are called *similar triangles*). Such triangles could be marked out on the ground and extended by rope and measuring rod to the desired length using the PT calculations.

Such was the essential task of the ancient Babylonian surveyor, which enabled the sides of the rectangle to be stretched out by sight to form perpendicular boundary lines. Such an ability demonstrated that Babylonian surveyors had a thorough working understanding of the geometry of rectangles and right triangles and used it to solve practical problems, like the dividing of a field for property rights.

According to Mansfield, one simple way to make an accurate right angle is to make a rectangle with sides 3 and 4, and diagonal 5 units. These special numbers form the 3-4-5 Pythagorean triple and a rectangle with these measurements has mathematically perfect right angles, which would be important to ancient surveyors. Mansfield explains:

“The ancient surveyors who made Si. 427 did something even better: they used a variety of different Pythagorean triples, both as rectangles and right triangles, to construct accurate right angles. This raises a very particular issue—their unique base 60 number system means that only some Pythagorean shapes can be used. It seems that the author of Plimpton 322 [see later] went through all these Pythagorean shapes to find these useful ones. This deep and highly numerical understanding of the practical use of rectangles earns the name ‘proto-trigonometry’ but it is completely different to our modern trigonometry involving sin, cos, and tan.”^{12}

When it came to the properties of triangles, the ancient Babylonians didn’t think in terms of angles, or sin, cos, tan (these are approximate values). Rather, they thought in terms of rectangles, which could be divided into right triangles, whose sides could be accurately calculated in terms of whole numbers.

## Babylonian mathematics

To understand the map engraved on Si. 427 and the surveyor’s method, an appreciation of Babylonian mathematics is helpful. Babylonian mathematics was founded upon a base-60 numbering system (sexagesimal), which is arguably the most sophisticated and powerful of all number systems ever devised. We still use it today to tell the time, calculate angles, and for geographic coordinate systems. Our familiar base 10 system probably arose from people using all their fingers to count. The Babylonian numbering system was inextricably linked to their understanding of geometry. Their number system is completely different to our own, which was developed from Hindu-Arabic symbols using base ten. Our system makes use of 9 special symbols ‘1–9’, and a ‘0’ to indicate zero.

Alternatively, Babylonian scribes could write any number with just two cuneiform symbols: 1) a vertical dash (𒁹 made in clay) to represent units (e.g. 3 dashes equals 3, and the fourth and seventh become stacked (𒑖 to save space). 2) A left-right wedge symbol (𒌋) represents 10s, e.g., 3 wedges equals 30, (and becomes stacked (𒑩 ) to represent the numbers 40 and 50). So, for example, 47 would be written with 4 wedges on the left and 7 dashes on the right to make a ‘digit’—‘47’. Babylonian digits went from 1 to 59 (see table below). There was no separate digit for zero, rather, a space would be left in the clay.

Such a system is called a “place value system” (referring to the position of each digit in a number), in this case a sexagesimal one. However, there is a base 10 system element (like ours) to the Babylonian system— because the 59 digits, were built from a ‘unit’ symbol and a ‘ten’ symbol.

When it comes to the number ‘60’ that would be represented as 𒁹(+ space). Significantly, apart from the space, this number appears identical to the Babylonian ‘1’. This may seem problematic to our eyes, but apparently not for the Babylonian scribes. In our modern base-10 notation, Babylonian 60 can be expressed as 1 x 60^{1} + 0.

What distinguishes the Babylonian base-60 number system from the base-10 system we are familiar with is the number of ‘factors’ available in the Babylonian system. A ‘factor’ is a number that divides into another number exactly without leaving a remainder. In our base 10 number system remainders are expressed as fractional numbers which are placed at the right of the decimal point, (referred to as a ‘floating point’ system). In our modern number system non-regular fractions give repeating answers (e.g. 3.333… etc.). The Babylonians preferred exact results, so would avoid calculating numbers (divisions involved in calculating the lengths of sides of triangles, or rectangles) that would not give exact results.

The Babylonian system used the factors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 for base 60, while our western system only has factors 1, 2, 5, and 10 for base 10. Although the base 60 system is not ubiquitously used as it once was in Babylonia, it has some key advantages over our base 10 system—because the number 60 has many more divisors compared to any smaller positive whole number. So when it came to calculating the sides of triangles (by division) this became essential for the surveyor, offering a powerful method of calculation and very accurate and exact results.

The Babylonians didn’t use multiplication tables like we are familiar with either. Rather, they made use of tables of squares (up to a huge 59 squared). These were available to scribes, (or surveyors) on clay tablets.

## Plimpton 322 theoretical tabulation

The trigonometric information used by the surveyor of Si. 427 comes from another tablet, discovered from the OB period. This particular tablet is designated Plimpton 322 (P. 322) which is housed in the G.A. Plimpton Collection at Columbia University and has been previously discussed in *Creation magazine*.^{13} It is from the city of Larsa, dated to c. 1,800 BC, which places it in the reign of Hammurabi. It has been translated and identified as representing the oldest yet discovered example of a ratio-based trigonometric table, which is astonishingly accurate, and deals with exceptionally large numbers. However, until the recent discovery of Si. 427 and its purpose, it wasn’t recognized what kind of problem P. 322 helped to solve. The new evidence gained from Si. 427 reveals the true purpose of P. 322. It is now understood to represent a comprehensive chart of Pythagorean triples, which relate to triangles and rectangles. It also contains information that enabled Babylonian surveyors to use the information in real-world surveys.

It is now theorized that the surveyor who made tablet Si. 427 must have used physical tools based upon this information tabulated in P. 322, as well as the standard rope and measuring reed. This was probably a set of Pythagorean triple triangle templates, which he likely kept in his workshop along with his other tools.

The information given in P. 322 provided the information the surveyor required to perform such calculations. The new discovery of Si. 427 reveals the function of both tablets—P. 322 is the theoretical solution—to the real-world challenges that the Babylonian surveyor encountered from Si. 427.

Both tablets are tremendously important mathematically and historically for understanding aspects of daily life in ancient Babylonia. Si. 427 represents the practical application and P. 322, the theoretical understanding of geometry. The mathematics involved was impressively advanced for the time.

## Surveying and Scripture

The ancient Babylonian knowledge is entirely consistent with what we know from the Bible, which is not silent about the subject of surveying and measurement. Firstly, we recognize that humans, created in the Image of God, have always been intelligent and resourceful, and have not arisen from sub-human, primitive ancestors as claimed by human evolutionary theory. From Scripture we know that:

- Within one generation from the creation of Adam, Cain built the first ‘city’ (Genesis 4:17).
- Tubal-Cain, by the seventh generation, achieved the forging of instruments of bronze and iron (Genesis 4:22).
- By the tenth generation, God told Noah to prepare an ark for the salvation of his household and the animals that went on-board (Genesis 7:1). The ark was a huge vessel measuring 300 × 50 × 30 cubits (approx. 140 × 23 × 13.5 metres or 459 × 75 × 44 feet) (Genesis 6:15). Such an undertaking must have required skills in draughtsmanship and survey as well as sophisticated construction techniques.
- Then there is the interesting verse in Genesis 10:25 which refers to the “dividing of the earth” in “Peleg’s day”, which has received various interpretations,
^{14}but which, in context, is referring to linguistic division, of people groups after the Babel confusion of languages. However, this division could also have included a survey of land territories that the various people groups would claim after Babel (Genesis 10:25). - The tower of Babel, (Genesis 11:4–5) was the beginning of post-Flood civilisation and empire under the leadership of Nimrod. He was one of Noah’s great grandsons, through his son Ham. Babel and its tower would have required huge amounts of planning, forethought, draughtsmanship, and survey, which must have been essential, required skills.

It is here, in the cradle of civilisation, in the OB period (1900–1600 BC) that solid evidence of survey and drafting skills can be seen. It is no surprise, when seen in terms of biblical history. Such skills and abilities are always “more sophisticated than previously assumed” for ancient peoples—when evolutionary presuppositions are held as being true.

- Survey and closely associated skills of measuring and recording are enshrined in the Mosaic Law, for instance:

“You shall not remove your neighbour’s landmark, which the men of old have set, in your inheritance which you will inherit in the land that the Lord your God is giving you to possess” (Deuteronomy 19:1).

The point being, someone is not going to know, or be able to prove a landmark or border has been moved unless it has first been measured in relation to a fixed point and recorded for posterity, which are fundamental to surveying.

- Honest and accurate weights and measures are considered essential to a good, moral, and functioning society as stipulated in Deuteronomy 25:13–16. An unchanging, standard measure is essential for civilisation to function. Standardised and unchanging measures are the basis for the modern International System of Units (SI).
- Pasture land was commanded to be surveyed and measured so as to be given to the Levites as part of their inheritance when the Israelites had possessed Canaan (Numbers 35:4–5).

All these examples, required skills in measuring and recording, using various techniques and tools designed especially for the purpose.

## Conclusion

The ancient clay tablets discussed in this article Si. 427 and Plimpton 322 are examples of sophisticated mathematics and survey that go back to the very beginnings of civilization. The time frame of these artifacts are post-Babel. However, they take us to within arms-length of the original engineers, architects, and surveyors who helped construct the tower of Babel and the first empire, under the leadership of Nimrod. He in turn would have learned all he knew from his father Ham, who in turn learned from his father Noah, who himself must have possessed great architectural and engineering skills developed in the pre-Flood world.

People, created in God’s Image have always been intelligent and resourceful, the ancient Babylonian artifacts are testimony to the accuracy of biblical history as outlined in Genesis 1–11 which speaks against the supposed primitive ancestry of humans, from long ago.

### References and notes

- Mansfield, D.F
*., How ancient Babylonian land surveyors developed a unique form of trigonometry — 1,000 years before the Greeks*, theconversation.com. Return to text. - UNSW,
*Daniel Mansfield uncovers world’s oldest known example of applied geometry*, 5 Aug 2021; maths.unsw.edu.au. Return to text. - Mansfield, D.F., Perpendicular Lines and Diagonal Triples in Old Babylonian Surveying,
*J. Cuneiform Studies*,**72**:87–99, (p. 91), 2021. Return to text. - Mansfield, Ref. 3, p. 92. Return to text.
- Mansfield, Ref. 3, p. 98. Return to text.
- Mansfield, Ref. 3, pp. 92, 96. Return to text.
- ETCSL 5.4.1, lines 30–32. Robson, E.,
*Mathematics in Ancient Iraq, a Social History*, Princeton University Press, Princeton, p. 122, 2008. Return to text. - Lu, D.,
*Australian mathematician discovers applied geometry engraved on 3,700-year-old tablet*, 4 August, 2021; theguardian.com. Return to text. - Mansfield, Ref. 3, p. 98. Return to text.
- Mansfield, Ref. 3, p. 88. Return to text.
- Lu, Ref. 8. Return to text.
- News Staff,
*3,700-year-old Babylonian clay tablet is earliest known example of applied geometry*, 5 Aug, 2021; sci-news.com. Return to text. - Anon, Mathematics as old as we thought!
*Creation***40**(1):10, 2018. Return to text. - Sibley believes Peleg’s division refers to post-Glacial melting causing natural land divisions, Sibley, A. Post-glacial flooding of coastal margins within the biblical timeframe of Peleg,
*J. Creation***18**(3):96–103, 2004. Return to text.

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