New Angles on Ancient Babylonian Geometry (Part 1)

In recent years, insights from two Sydney University mathematicians, Prof Norman Wildberger and Dr Daniel Mansfield, have shed fascinating new light on aspects of ancient Babylonian geometry. In particular, fresh analysis of two tablets, known as Si427 and Plimpton 322, has revealed the role and importance Pythagorean triangles played in their cosmology. As these and other tablets date to as early as 1900BC, this is more than a millennia before Pythagoras himself and the achievements of Greek mathematics.

Si427 (shown above) was discovered in 1894 by a French archaeological expedition at Sippar in central Iraq. This tablet clearly shows how Pythagorean triangles were employed to lay out rectangles and other shapes with accuracy.

“Si.427 dates from the Old Babylonian period (1900-1600 BCE),” said Dr. Daniel Mansfield, a mathematician in the School of Mathematics and Statistics at the University of New South Wales.

“It’s the only known example of a cadastral document from this period, which is a plan used by surveyors define land boundaries.”

“In this case, it tells us legal and geometric details about a field that’s split after some of it was sold off.”

“This is a significant object because the surveyor uses what are now known as Pythagorean triples to make accurate right angles.”

“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,” Dr. Mansfield said.

By using Pythagorean triangles, made up of whole number measures, right angles can be established, calibrated and maintained. This gives rise readily to a system which employs grids of a regular measure. Angles can be specified by joining grid points of particular separation. Indeed, this is the way that the Babylonians conceived of angles. They did not conceive them as we do, as divisions of a circle, or degrees.

Instead, for the Babylonians, the only angles they were concerned with were those formed by the diagonal of a rectangle of sides of whole number measure.

The most prized diagonals in such a system also have a hypotenuse of whole number measure. These are the Pythagorean triangles. Si427 demonstrates that use of Pythagorean triangles aligned to the north-south grid in land measure, but it was not only employed for legal questions of contracts and land ownership.

This same methodology of surveying Pythagorean triangles in landscape has been identified by researchers including Robin Heath, and Howard Crowhurst. Heath discovered the so-called Lunation Triangle, a very impressive (5:12:13) Pythagorean triangle laid out at huge scale, in units of Megalithic yards, aligned north-south, between Stonehenge, Lundy Island and the Preselli Hills in Wales, from whence the stones came. This same triangle, at smaller scale, is also found within the ground plan of Stonehenge itself.

Howard Crowhurst has uncovered many examples of the practice in his research at Carnac, in Brittany, and elsewhere in France and indeed all over the ancient world. The motif appears to be universal: the use of different Pythagorean triangles, aligned to the north-south grid, to create desired angles.

It is very encouraging to see that there has been some fruitful exchanges of ideas and even collaboration between Professor Wildberger and Howard Crowhurst. The two researchers work complements each other exquisitely. On the one hand there is evidence from these tablets from the second millennia BC of the use of Pythagorean triangles in questions relating to land surveying, while on the other hand there is evidence from the traces in the ancient landscape itself, in the layout of the stones, that just these techniques were in use.

In Crowhurst's work, there is an additional element introduced. In some instances these alignments set up at angles defined by Pythagorean triangles are also aligned to particular rising positions of the sun, and certain stars. There seems to be a principal at work here: for the ancient engineers, it was desirable to construct alignments to bearings corresponding to key astronomical features on the horizon; however, these angles could not merely be created by simply laying down a line pointing in the desired direction. They must be formed by valid Pythagorean triangles, laid out in whole numbers of some measure, and aligned to the north-south grid.

An excellent example of this practice may be found at Crucuno, near Carnac in northern France. This is a rectangle made of standing stones, aligned precisely to the north-south grid. It was surveyed by Alexander Thom, and has been discussed by Heath and Crowhurst. The rectangle of standing stones has sides of 30 and 40 Megalithic Yards (MY). This gives a diagonal of 50 MY.

At this latitude, the sun rises on the summer solstice at a bearing of 53°. This corresponds to one of the angles of the (3,4,5) triangle. As the rectangle is aligned north-south, the diagonal is directed at this bearing of 53°, the (3,4,5) angle.

The hypotenuse of the triangle, or diagonal of the rectangle, is therefore aligned to the summer solstice sunrise at that latitude. I included a Google Earth image of it in The Map and the Manuscript as shown below.

The use of grids as basis for geometrical construction also underlies ancient Egyptian art. This topic is explored in great detail by Professor Gay Robins in her 1994 book Proportion and Style in Ancient Egyptian Art. She shows that grids were used throughout the history of dynastic Egypt to provide a canon of proportion for the depiction of the human body and indeed the entire panoply of gods, other living creatures and objects. The grid system provided accuracy and consistency in representation. It also offered the opportunity to create particular angles by joining grid intersections. Thus for example, joining two points separated by squares (8,11) apart creates a 36° angle (accuracy <0.03°).

So we see the same or at least related systems of grid geometry underling art, surveying, and the laying out of stones in landscape, already present in the second millennia BC.

The second tablet, known as Plimpton 322 (shown below), has presented some challenging problems for scholars for nearly a century. It dates from around 1800BC, and shows a table of numbers in Babylonian sexagesimal, or base-60, notation.

What is it? Wildberger and Mansfield have shown that it is a table of values analogous to a modern trigonometric table, with some important differences.

Instead of a list of angles classified by whole numbers of degrees, or 1/360th division of the circle, the Babylonians constructed this table solely from angles of certain carefully selected Pythagorean triangles. As a consequence of the choices made, all of the trigonometric results are expressed exactly, with no approximations. That is to say, the lengths of the sides of the triangles/rectangles given in the list for the set of Pythagorean triangles selected, all resolve by calculation in base 60 to exact values, rather than recurring figures as occurs in modern tabulations. Here is the abstract from their paper, with links to the full paper and further resources online below.

Abstract: We trace the origins of trigonometry to the Old Babylonian era, between the 19th and 16th centuries B.C.E. This is well over a millennium before Hipparchus is said to have fathered the subject with his ‘table of chords’. The main piece of evidence comes from the most famous of Old Babylonian tablets: Plimpton 322, which we interpret in the context of the Old Babylonian approach to triangles and their preference for numerical accuracy. By examining the evidence with this mindset, and comparing Plimpton 322 with Madhava's table of sines, we demonstrate that Plimpton 322 is a powerful, exact ratio-based trigonometric table.

“Nobody expected that the Babylonians were using Pythagorean triples in this way. It is more akin to pure mathematics, inspired by the practical problems of the time,” Dr. Mansfield said.

There's something else about these Pythagorean triples. Most people today have heard of (3,4,5), and probably others: (5:12:13), (7:24:25), (8,15, 17), (9,40,41) and others. The examples on Plimpton 322 include these, but they also include some that are on another level completely. Converted to our decimal notation they include such incredible specimens as (12,709; 13,500, 18,541) and (4,061; 8,100; 9,061). How did they manage to work out that such sets of three numbers were Pythagorean triples?

More to follow in Part 2.

Resources for further reading and viewing:

Wildberger and Mansfield Papers

Plimpton 322 is Babylonian exact sexagesimal trigonometry. Daniel F. Mansfield, N.J. Wildberger

Plimpton 322: A Study of Rectangles. Daniel F. Mansfield

Wildberger and Mansfield Videos

Si.427 - one of the oldest and most complete examples of applied geometry from the ancient world. Dan Mansfield

Old Babylonian Mathematics and Plimpton 322: A new perspective (introduction)

Old Babylonian mathematics and Plimpton 322: Geometry in ancient Mesopotamia and Egypt

Old Babylonian mathematics and Plimpton 322: A new understanding of the OB tablet Plimpton 322

Old Babylonian Mathematics and Plimpton 322: How did the OB scribe construct P322?

Plimpton 322 and Old Babylonian Trigonometry History of Maths 34 N J Wildberger