recoveredscience.com              We offer surprises about     

and numerals and their ancient religious uses     in our e-book

Ancient Creation Stories told by the Numbers

by H. Peter Aleff

 

 

  

Footnotes :

 

 

[1]  Gay Robins and Charles Shute: “The Rhind Mathematical Papyrus: an ancient Egyptian text”, British Museum Publications, London , 1987, see “Squaring the Circle”, pages 44 to 46.

 

[2]  Otto Neugebauer: “The Exact Sciences in Antiquity”,  Brown University Press, 1957, edition consulted Dover , New York , 1969, pages 46 and 47.

 

[3]  André Pichot: “The Birth of Science ...” = “La naissance de la science”, Gallimard, Paris , 1991, edition consulted “Die Geburt der Wissenschaft: Von den Babyloniern zu den frühen Griechen”, Wissenschaftliche Buchgesellschaft, Darmstadt , 1995, page 78 top.

 

[4]  Petr Beckmann: “A History of Pi”, first published by The Golem Press, 1971, edition consulted Barnes and Noble Books, New York , 1993, pages 63 to 66.

 

[5]  Mesopotamian dates are firm only back to 911 BCE.  The chronology for earlier periods is based on king lists with gaps, overlaps, scribal errors, and inconsistencies, and proposed years for Hammurabi’s accession to the throne range from 1848 to 1702 BCE.  The dates in this book follow the mainstream “middle chronology” proposed in 1940 by Sidney Smith and used by Georges Roux in “Ancient Iraq”, first published 1964, updated third edition consulted Penguin Books, London , 1992,  pages 25, 26, and 436.

 

 

[6] B.L. van der Waerden: “Science Awakening I -- Egyptian, Babylonian, and Greek Mathematics”, 1954, edition consulted The Scholar’s Bookshelf, Princeton, NJ, 1975, page 76, see also pages 78 and 79 on the “Plimpton 322” tablet from the same corpus, and its picture on Plate 8 opposite page 61.

 

[7]  André Pichot: “The Birth of Science ...”, cited above, see pages 197 and 80 to 85 on “The Theorem of Pythagoras”.

 

[8]  B.L. van der Waerden: “Science Awakening ...” cited above, see pages. 65 and 69 to 73, also Pichot pages. 65 to 67.

 

[9]  Petr Beckmann: “A History of Pi”, cited above, Chapter 10 “The Digit Hunters”, see page 102.

 

 

 

 

  

 

  

Numerals and constants  

 

 tell the creations of numbers and world

 
 


Ancient values for Pi

Among the eight written mathematical documents that survive from ancient Egypt , only two refer to circles, the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus, and only the Rhind does so clearly.  

Five of the 84 example problems in the Rhind deal with the volumes of cylindrical granaries and use a simple rule of thumb for computing the area of the circular base.  That rule was easy to use for beginners and apparently close enough for the intended application: take the diameter of the circle, subtract its ninth part, and square the result. 

Although those problems do not mention any constant nor relate the diameter of a circle to its circumference, translated into modern terms the results from that method amount to a value for pi of  3 1/6  = 3.1667.  This approximation is 0.8% too high compared with the 3.14159... in our pocket calculators, but it is said to be the best that survives in writing from the pre- Hellenic world[1].

Indeed, the corresponding values on several ancient Mesopotamian tablets are even worse.  Otto Neugebauer, long one of the leading researchers in the area of premodern science, described in his classic 1957 book The Exact Sciences in Antiquity:

“... only a very crude approximation for the area of a circle is known so far [from Mesopotamia ], corresponding to the use of 3 for p.  Several problems concerning circular segments and similar figures are not yet fully understood and it seems to me quite possible that better approximations of p were known and used in cases where the rough approximation would lead to obviously wrong results. (...) 

After completion of the manuscript, new discoveries were made. (...) [One still unpublished tablet seems to imply] the approximation p = 3 1/8  [= 3.125], thus confirming finally my expectation that the comparison of the circumference of the regular hexagon with the circumscribed circle must have led to a better approximation of p than 3.”[2]

The then still unpublished tablet Neugebauer mentioned does not seem to have made it into print quickly because André Pichot, another prominent scholar in this field, still wrote in 1991:

“The Mesopotamian formulae [for circle area and circumference] used p = 3 which is a rather poor approximation compared with the very accurate value for the square root of 2 [previously presented as correct to one part in almost two million].  (...) 

The value of p = 3 appears on several tablets which also give the circumference of the circle as six times the radius.  This derives without doubt from equating the circumference around a circle of radius r with the circumference of a hexagon having a side length of r which explains also the division of the circle into 360 degrees.  In addition, there is at least one tablet with values for various circle arcs, chords, secants, etc., but it is hard to interpret.”[3]

The earliest improvement over the value implied in the Rhind Papyrus is currently credited to the Greek Archimedes of Syracuse who lived from about 287 to 212 BCE.  He fenced in the range of pi between 3 10/71 and 3 1/7, or 0.024% below and 0.040% above the correct result.

The method Archimedes used to achieve this result is simple.  He reasoned that the circumference of a regular convex polygon inscribed into a circle is smaller than that of the surrounding circle, and that the sides of any such polygon drawn around the circle must add up to a greater circumference.  He could also see that polygons with more sides hugged the circle closer than those with fewer, and that their circumferences came therefore closer to that of the circle between them. 

Archimedes started with hexagons in and around his circle.  Hexagons are easy to dissect into triangles for computing the lengths of their sides because these sides are all equal, and they yield the crude circumference- to- diameter ratio of  3 we saw above on the Babylonian tablets.  Then he doubled the number of polygon sides and calculated the dimensions of the new triangles which now had half the previous center angles, and he repeated this process until he arrived at the polygon pair with 96 sides which gave the above result [4]. 

This procedure is called the “method of exhaustion” because it exhausts the difference between the circumferences of the polygons and that of the circle with its ever closer approximations.  The only mathematical tools required for it are the so-called Theorem of Pythagoras, about the sum of the squares over the sides of a right- angled triangle, and the ability to extract square roots. 

Both these tools were available in the ancient Near East long before Pythagoras.  Already the Old Babylonians  of king Hammurabi’s time (1792 to 1750 BCE)[5] were familiar with the contents of that theorem.  They may not have explicitly formulated it in modern terms or spelled out its derivation on any of the few surviving tablets, but they used it routinely to solve questions of the type:

“A beam of length 30 stands against a wall. The upper end has slipped down a distance 6.  How far did the lower end move?”[6] 

The writers of these questions answered them more than a thousand years before Pythagoras, and the Kahun Papyrus from Middle Kingdom Egypt hints that their contemporaries along the Nile were aware of the relationships in that mis- attributed theorem, too[7]. 

As to the computing of roots, various ancients regularly solved quadratic and even cubic equations[8].  For instance, a scribe from the First Dynasty of Babylon worked out the square root of  two to within 0.000047 per cent of the correct value.  Similarly, the Berlin Papyrus from Egypt’s late Middle Kingdom (about 1800 BCE) contains a second degree equation; its author clearly knew how to find square roots although, like Archimedes, he did not describe his method for doing so.

Doubling the number of polygon sides again and again allows anyone with the same basic knowledge to compute the value of pi to any desired accuracy, depending only on that person’s patience and on the amount of time s/he cares to set aside for the repetitive calculations. 

Archimedes probably had better things to do than to tediously hunt for additional digits of the circle constant that were of no practical or theoretical value whatsoever.  Refining pi any further had clearly no “Eureka” value.

Some of Archimedes’ successors, however, continued where he had left off.  One of them, Ludolph van Ceulen (1539 to 1610), professor of mathematics and military science at the Dutch University of Leyden , spent several decades to nail down the first 35 decimal places of pi and so attached his first name to that ratio for a while in German- speaking countries. 

Similarly, the Japanese mathematician Takebe (» 1722) started with an octagon and subdivided his polygons until they had 1,024 sides.  This produced p to 41 decimals.  Such relentless efforts may have given a new meaning to the term “method of exhaustion”, but they did not develop any new mathematics for these stunts. They all still used the same old approach as Archimedes[9]. 

Even many thousands of such computing steps would have presented no obstacle to ancient people who consistently piled up many millions of baked bricks or hewn stone blocks for symbolic reasons.  If they had any motivation to do so, some determined Egyptian mathematician could easily have computed pi with great precision, greater than in the formula for granary contents from the Rhind Papyrus learner’s manual, or than in Archimedes’ result.  

Some well placed number researchers might even have delegated that tedious chore to their otherwise under- occupied assistant priests to give them something useful to do.  These, in turn, would have found their task probably more entertaining, despite the endless repetitions, than watching the TV programs available back then.

In any case, the booty quantities on Narmer's mace approximate three simple combinations of  pi with phi.  The quantities of 120,000 prisoners and 400,000 oxen do not tell us how good the values for these constants known to the mace designer were because these quantities are rounded.  However, the total of 1,942,000 items for pi divided by phi shows that the product of his combined errors for both constants was only about 0.02 per cent, or in the same range as Archimedes’ error for pi alone.  

Continue reading about early mathematics, and why these constants on Narmer's mace are less of an anachronism than some academics declare them to be, or return directly to that mace and examine the astronomical constants embedded in its ratios. 
 

 

 

Return to navigation bar  ¦  Back to top   ¦  About us
Our Privacy Policy  ¦   Useful Links  ¦   Rebranding

Contact us at recoveredscience.com
2097 Cottonwood Drive, Vineland, NJ 08361  USA

All not otherwise credited material on this site is
©1982 to 2015 H. Peter Aleff. All rights reserved.