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Ancient Creation Stories told by the Numbers

by H. Peter Aleff

 

 

Footnotes :

 

 

1 Eli Maor: “e - The Story of a Number”, Princeton University Press, 1994, pages xiii and 23.


2 Calvin C. Clawson: “Mathematical Mysteries: The Beauty and Magic of Numbers”, Plenum Press, New York, 1996”, pages 99 to 103.


3 Eli Maor: “e - The Story of a Number”, cited above, page 24.


4 Wolfram von Soden: “Introduction to the Study of the Ancient Near East”, Wissenschaftliche Buchgesellschaft, Darmstadt, 1985, page 159. (in German, my translation)


5 André Pichot:  “La naissance de la science”, Gallimard, Paris, 1991, edition consulted  “Die Geburt der Wissenschaft ...” (“The Birth of Science -- from the Babylonians to the early Greeks”),  pages 65 to 73, page 89 for tens of thousands of mathematics- related tablets excavated but only a fraction of them so far deciphered, and page 90 middle for joyful playing with numbers.


6 Sir Thomas Heath: “A History of Greek Mathematics”, Clarendon Press, Oxford, 1921, edition consulted Dover, New York, 1981, Volume 1, page 41 bottom. See also Richard J. Gillings: “Mathematics in the Time of the Pharaohs”, 1972, edition consulted Dover Publications, New York, 1982, pages 45 to 70.


7 Richard K. Guy: “Unsolved Problems in Number Theory”, Springer Verlag, New York, 1994, see pages 158 to 166 which include more than three pages of a bibliography on Egyptian fractions that “shows only a fraction of what has been written”.


8 B. L. van der Waerden:  “Science Awakening I: Egyptian, Babylonian, and Greek Mathematics”, 1954, edition consulted Scholar’s Bookshelf, Princeton Junction, New Jersey, 1988,  page 42 top.


9 Aryeh Kaplan: “Sefer Yetzira: The Book of Creation”, Samuel Weiser, Inc., York Beach, Maine, 1993, pages 190 and 191. See also pages 265 and 275 where this stanza appears as 4:12 and 4:4 in other versions of the text.


10 B. L. van der Waerden: “Science Awakening I:...”, cited above, see page 16 and jacket text.


11 London, 1917, Cambridge University Press, 1961, abridged edition consulted 1975.


12 London, 1914, edition consulted Dover, New York, 1979.


13 W.W Rouse Ball: “A Short Account of the History of Mathematics”, first published in 1908, edition consulted Dover Publications, New York, 1960, page 367 top.


14 Eli Maor: “e - The Story of a Number”, cited above, page 127; see Figure 50 on page 128 for Bernouilli’s tombstone in the Münster cathedral of his home city Basel, Switzerland.


15 Eli Maor: “e - The Story of a Number”, cited above, “The Hanging Chain”, pages 140 to 146.



16 Eli Maor: “e - The Story of a Number”, cited above, “The Most Famous of all Formulas”, pages 153 to 161.

 



17 Eli Maor: “e - The Story of a Number”, cited above, “The Most Famous of all Formulas”, page 160.

See also George F. Simmons: "Calculus Gems: Brief Lives and Memorable Mathematics", McGraw-Hill, New York, 1992, page 162.

 

 

 

 

  

 

  

  Numerals and constants  

 

 tell the creations of numbers and world    

 
 

The constant e of growth and renewal

Like the other constants, the number e is not necessarily confined to the modern cage where current lore likes to place it.  We know it as the base of the natural logarithms and as a pillar at the gate to higher mathematics, and our histories of mathematics trace this constant back only to Renaissance times.  Many people assume therefore almost automatically that it must be a relatively recent discovery.

However, some of those who investigated the history of e leave the door open about its origins. For instance, the mathematician Eli Maor explains in his delightful book about this number:

“In the course of my research, one fact became immediately clear: the number e was known to mathematicians at least half a century before the invention of the calculus (it is already referred to in Edward Wright’s English translation of John Napier’s work on logarithms, published in 1618). How could this be? 

One possible explanation is that the number e first appeared in connection with the formula for compound interest. Someone -- we don’t know who or when -- must have noticed the curious fact that if a principal P is compounded n times a year for t years at an annual interest rate r, and if n is allowed to increase without bound, the amount of money S, as found from the formula 

S = P (1 + r / n)nt

seems to approach a certain limit. This limit, for P = 1 and r = 1, is about 2.718

This discovery -- most likely an experimental observation rather than the result of rigorous mathematical deduction -- must have startled mathematicians of the early seventeenth century, to whom the limit concept was not yet known. Thus, the very origins of the number e and the exponential function ex may well be found in a mundane problem: the way money grows with time. (...)

From time immemorial, money matters have been at the center of human concerns. No other aspect of life has a more mundane character than the urge to acquire wealth and achieve financial security. So it must have been with some surprise that an anonymous mathematician -- or perhaps a merchant or moneylender -- in the early seventeenth century noticed a curious connection between the way money grows and the behavior of a certain mathematical expression at infinity.

Central to any consideration of money is the concept of interest, or money paid on a loan. The practice of charging a fee for borrowing money goes back to the dawn of recorded history; indeed much of the earliest mathematical literature known to us deals with questions related to interest

For example, a clay tablet from Mesopotamia, dated to about 1,700 BCE and now in the Louvre, poses the following problem: How long will it take for a sum of money to double if invested at 20 percent interest rate compounded annually?”1

Replace the term “money” with “grain” or “livestock” or any other commodities that can be lent and borrowed, and the above scenario of some mathematically inclined lender exploring the options for the growth of his or her capital can fit equally well into any farming, herding, or trading economy even long before the beginning of writing.

To illustrate how elementary e is, and what important role it plays in economic life, another mathematician, Calvin C. Clawson, described a hypothetical scenario how even some neolithic peasants could have come upon this number when lending grain to their neighbors who had to pay back the loan plus some part of the harvested yield2.

As to the ability of the ancients to conceive the mathematics that lead to e, Maor continues about the Mesopotamians:

“...in a way, the Babylonians did use a logarithmic table of sorts. Among the surviving clay tablets, some list the first ten powers of the numbers 1/36, 1/16, 9, and 16 (...) -- all perfect squares.  Inasmuch as such a table lists the powers of a number rather than the exponent, it is really a table of antilogarithms, except that the Babylonians did not use a single, standard base for their powers. It seems that these tables were compiled to deal with a specific problem involving compound interest rather than for general use.”3

Several other tablets survive on which some Sumerians and Old Babylonians computed compound interest and put exponents through their paces. They left us tables of multiplications and their reciprocals as well as lists of powers and roots. 

From at least the early second millennium BCE on, Old Babylonian scholars went far beyond such utilitarian tools for calculation. The Assyriologist Wolfram von Soden describes that they created also

“... other lists that demonstrate a theoretical interest in the peculiarities of the number realm. For instance, someone followed the number 225 which is written 3;45 in the sexagesimal system up to its tenth power because this produced strikingly repetitive digit sequences

Another scholar continued the powers of two up to 230 and formed then also the reciprocal to this number which has ten digits in our notation and even more in sexagesimal terms. Such tablets could not have any practical interest whatsoever, but we do not know why people made such unusual lists. The situation was similar for geometry which had grown far beyond the requirements of its many practical applications.”4

Along the same lines, the historian of science André Pichot notes the Mesopotamians’ extraordinary ability for handling numbers.  He mentions a tablet from the time of the First Dynasty of Babylon (1900 to 1600 BCE) that gives the square root of two with an accuracy of one part in almost two million, and others that teach the extraction of cubic roots and preserve lists of exponents that he, too, compares with logarithm tables. 

He further reports that their early scribes used such lists to compute compound interest, in one case correctly for 30 years, and that they left us many other groups of numbers: astronomical data collections and arithmetic and geometric progressions, some that were based on purely mathematical speculation and curiosity about the world of numbers, some that remain hard to interpret, and a great many more that sit in museum storage rooms and have never been read.5

*

The calculation of e with the above formula of compound interest is quite lengthy.  However, it could also have been obtained quite easily with the well documented ancient method of unit fractions (1 / x)

The ancient Egyptians handled all their divisions with this method which was so well designed that the Greeks and Romans and their successors used it long into the Byzantine period, more than a thousand years after the last pharaohs were gone6

(Egyptian fractions are still a subject of great interest to mathematicians and continue to provide these with many new and challenging questions, quite a few of which are still unsolved.7

Also, some of the oldest Sumerian texts8, dating from about 2000 BCE, were tables of multiplications and of inverses (1 / x)

Such unit fractions can generate the ratio e with the sort of manipulations that formed the basis for much of ancient mathematics and at which many scribes were highly skilled. 

One such method is the simple formula 

e = 2 + 1/2! + 1/3! + 1/4! + ... 

in which the exclamation points mean “factorial”, that is, “multiply all the integers up to the one so marked with each other”. 

Written in the form of Egyptian fractions, this series becomes  2 + 1/ 2 + 1/ 6 + 1/ 24 + 1/ 120 + 1/ 720 + 1/ 5040 + ..., and just these few members yield already the close approximation of  2.7182539.

Factorials were an important concept in the early Jewish traditions that led to the Kabbalah. For instance, in stanza 4:16 of the Sefer Yetzira, which is one of the oldest texts from those traditions, the author proposed as a subject of exploration the number of permutations possible with the 22 letters of the Hebrew alephbet.  These letters were to the Kabbalists “stones quarried from the great Name of God”:

“Two stones build two houses;
Three stones build six houses;
Four stones build 24 houses;
Five stones build 120 houses;
Six stones build 720 houses;
Seven stones build 5,040 houses.
From here on go out and calculate
that which the mouth cannot speak
and the ear cannot hear.”9

The remaining permutations cannot be spoken or heard because their quantities grow so rapidly that no one could pronounce them all, even in a long lifetime of uninterrupted reciting: 22 factorial works out to over a sextillion (1021) different combinations of all the letters. 

However, this impossibility of pronouncing all the possible permutations would not have prevented the adepts from exploring their numbers, as that verse instructed them to do.  

It also seems likely that some of those who investigated these factorial numbers would have been curious enough to invert them, as in the ancient Mesopotamian tablets that listed inverses, or as in the Egyptian system of unit fractions that relied entirely on reciprocals.  The small step of adding up these inverses, the way the users of Egyptian fractions did for their results, would have shown them soon that the sum approaches a limit, and that this limit was the constant e.

*

The modern discovery of e is ascribed to curious minds who performed interest calculations in an environment of commercial activity that encouraged the exploration of the relevant mathematics. 

The conditions for this discovery were similarly favorable in the ancient Near East during several centuries- long historical periods of relative stability and active trade.  Such periods could equally well have existed in prehistoric times because people have been trading and hoarding wealth and counting and reckoning for far longer than they have been writing. 

Already the pre- dynastic Egyptians of the Delta region exchanged goods and ideas with their Mesopotamian counterparts, as attested, for instance, by imported cylinder seals found there.  Trading on any such scale requires attention to numbers, so the merchants or administrators back then had probably similar mathematical needs as their later colleagues.

Moreover, we saw above that numbers had supernatural properties. The sages and priests had therefore even stronger reasons than any accountant to explore the behavior of these all- pervading entities that were as timeless as the gods and seemed to offer insights into the workings of their world, just as they did and do into the workings of ours. 

And among these mysterious but computable numbers, some constants stand out, particularly e which displays much mathematically impressive behavior.

*

The properties of e have delighted and enthused many modern lovers of mathematics. After John Napier published in 1614 his tables of “ratio numbers” or “logarithms” which he had spent 20 years computing, people soon recognized that the logarithmic function has a universal base number, and that this number is e.  The more they worked with their new tool, the more they encountered this mysterious number. According to Maor, this constant

“remains central to almost every branch of mathematics, pure or applied. It shows up in a host of applications, ranging from physics and chemistry to biology, psychology, art, and music, (...) from the area under a hyperbola to the shape of a hanging chain, from the inner structure of a Nautilus shell to Bach’s equal-tempered scale and to the art of M. C. Escher.”10

The logarithmic spiral you saw earlier produced by the golden ratio is also derived from e because the equation of that spiral becomes in polar coordinates

r = eaT 

where r is the distance of a point on the spiral from its center, T is the angle through which the spiral has grown to that point, and a is a fixed quantity for a given spiral that describes how tightly that spiral winds itself at each turn. 

This simple equation thus links e to phi.  It also governs the principle of spiral growth in nature by steadily increasing the size of an organism while keeping its shape the same. 

This spiral is so universal, from the arms of galaxies to the way a fern leaf curls, and its mathematics are so spare and elegant, that many authors have praised its beauty. Examples are D’Arcy Thompson’s classic “On Growth and Form”11, and Theodore Andrea Cook’s equally wide-ranging “The Curves of Life”12.

The modern discoverer of this equation, Jakob Bernoulli (1654 to 1705), admired greatly how the “spira mirabilis” or “miraculous spiral” generated by that simple formula always reproduced its original shape, regardless to what mathematical transformations and inversions he subjected it that distorted all other curves beyond recognition. 

Jacob was so impressed with this ever self- same regeneration and resurrection of "his" spiral that he asked to have it engraved on his tombstone13 with the inscription eadem mutata resurgo, or “though changed, I shall arise the same”

His request reveals his hope for his own renewal in symbolic imitation of the phi and e in this spiral.  It also suggests that if any ancient mathematicians were aware of this number and had explored some of its properties, they could easily have formed similar associations between e and renewal. 

Such symbolic associations are based on unchanging properties of the number and on almost equally hard- wired habits of the human mind to use analogies and metaphors in trying to understand abstract ideas.  They are therefore not tied to superficial differences in cultures or religions.

Bernoulli’s request followed the tradition of Archimedes of Syracuse (287 to 212 BCE) who had asked that a sphere with a circumscribed cylinder be marked on his grave to recall one of his favorite discoveries. However, Bernoulli wound up even closer to this precedent- setter than he had intended.  As Maor discovered and tells the story:

“... the mason indeed cut a spiral on the grave, but it was an Archimedean instead of a logarithmic spiral (...) which no doubt would have made Jakob turn in his grave.”14

On the other hand, Jakob’s ghost would just as surely be tickled back to a healthy pink if he could see the splendid monument raised to the answer for an e-related question which he first proposed and in which he had asked for the curve of a hanging chain

The resulting equation for the catenary curve 

y = (ex + e–x) / 2

has impressed some admirers of mathematical elegance to the point that Eero Sarinen’s famous Gateway Arch of Saint Louis is a 630- foot high stainless steel embodiment of this equation15.

However, the builders seem to have held their plans upside down because the resulting shape is turned around, as if gravity pushed upwards.

*

This miraculous constant e is not only tied to the golden ratio phi, as explained above, but also to the circle constant pi.  These two are linked by means of an equation that many admirers, including Maor, have called "one of the most beautiful formulas in all of mathematics".16

We owe this equation to Leonard Euler (1707 to 1783), a pupil of Jacob Bernoulli's brother Johann and one of the most prolific mathematicians on record.  Euler explored many properties of e and gave it this current designation.  Maor reports the opinion that it could have been an abbreviation for "exponential", and he also suggests that the letters a to d were already used as mathematical symbols, so e was simply next. 

In any case, Euler discovered that e, elevated to the power of pi times i, the imaginary square root of minus one, equals minus one:

e pi x=  - 1

This wonderfully simple and elegant connection between these two transzendental constants by means of the basic imaginary number to produce the latter's real and integer square has led many people to see mystic meanings in this perplexing paradox of the number world, again according to Maor.17

Like the reaction of Jacob Bernoulli, the deep pondering of these people about the pixie- dust- like miracles of e suggests that if any ancient mathematicians had also discovered and explored some facets of this astounding constant, then they would have been even more likely to project major symbolical and religious meanings onto e.  

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