Mathematics in Early Civilizations In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation builds a new story to an old structure.

 Mathematics in Early Civilizations

In most sciences one generation tears down what another has built and what one has established

another undoes. In Mathematics alone each generation builds a new story to an old structure.

The Rhind Papyrus

Egyptian Mathematical Papyri

With the possible exception of astronomy,

mathematics is the oldest and most continuously

pursued of the exact sciences. Its origins lie

shrouded in the mists of antiquity. We are often

told that in mathematics all roads lead back to

Greece. But the Greeks themselves had other ideas about where mathematics began. A

favored one is represented by Aristotle, who in his Metaphysics wrote: “The mathemat-

ical sciences originated in the neighborhood of Egypt, because there the priestly class

was allowed leisure.” This is partly true, for the most spectacular advances in mathe-

matics have occurred contemporaneously with the existence of a leisure class devoted to

the pursuit of knowledge. A more prosaic view is that mathematics arose from practical

needs. The Egyptians required ordinary arithmetic in the daily transactions of commerce

and state government to x taxes, to calculate the interest on loans, to compute wages,

and to construct a workable calendar. Simple geometric rules were applied to determine

boundaries of elds and the contents of granaries. As Herodotus called Egypt the gift of

the Nile, we could call geometry a second gift. For with the annual ooding of the Nile

Valley, it became necessary for purposes of taxation to determine how much land had

been gained or lost. This was the view of the Greek commentator Proclus (A.D. 410–485),

whose Commentary on the First Book of Euclid’s Elements is our invaluable source of

information on pre-Euclidean geometry:

According to most accounts geometry was rst discovered among the Egyptians and origi-

nated in the measuring of their lands. This was necessary for them because the Nile over ows

and obliterates the boundaries between their properties.

Although the initial emphasis was on utilitarian mathematics, the subject began eventu-

ally to be studied for its own sake. Algebra evolved ultimately from the techniques of

calculation, and theoretical geometry began with land measurement.

Most historians date the beginning of the recovery of the ancient past in Egypt from

Napoleon Bonaparte’s ill-fated invasion of 1798. In April of that year, Napoleon set sail

from Toulon with an army of 38,000 soldiers crammed into 328 ships. He was intent

on seizing Egypt and thereby threatening the land routes to the rich British possessions

in India. Although England’s Admiral Nelson destroyed much of the French eet a

month after the army debarked near Alexandria, the campaign dragged on another 12

months before Napoleon abandoned the cause and hurried back to France. Yet what had

been a French military disaster was a scienti c triumph. Napoleon had carried with his

expeditionary force a commission on the sciences and arts, a carefully chosen body of

167 scholars—including the mathematicians Gaspard Monge and Jean-Baptiste Fourier—

charged with making a comprehensive inquiry into every aspect of the life of Egypt

in ancient and modern times. The grand plan had been to enrich the world’s store of

knowledge while softening the impact of France’s military adventures by calling attention

to the superiority of her culture.

The savants of the commission were captured by the British but generously allowed

to return to France with their notes and drawings. In due course, they produced a truly

monumental work with the title D´escription de l’Egypte. This work ran to 9 folio volumes

of text and 12 volumes of plates, published over 25 years. The text itself was divided into

four parts concerned respectively with ancient Egyptian civilization, monuments, modern

Egypt, and natural history. Never before or since has an account of a foreign land been

made so completely, so accurately, so rapidly, and under such dif cult conditions.

The D´escription de l’Egypte, with its sumptuous and magni cently illustrated folios,

thrust the riches of ancient Egypt on a society accustomed to the antiquities of Greece

and Rome. The sudden revelation of a ourishing civilization, older than any known

so far, aroused immense interest in European cultural and scholarly circles. What made

the fascination even greater was that the historical records of this early society were

in a script that no one had been able to translate into a modern language. The same

military campaign of Napoleon provided the literary clue to the Egyptian past, for one

of his engineers uncovered the Rosetta Stone and realized its possible importance for

deciphering hieroglyphics.

Most of our knowledge of early mathematics in Egypt comes from two sizable papyri,

each named after its former owner—the Rhind Papyrus and the Golenischev. The latter

is sometimes called the Moscow Papyrus, since it reposes in the Museum of Fine Arts in

Moscow. The Rhind Papyrus was purchased in Luxor, Egypt, in 1858 by the Scotsman

A. Henry Rhind and was subsequently willed to the British Museum. When the health

of this young lawyer broke down, he visited the milder climate of Egypt and became an

archaeologist, specializing in the excavation of Theban tombs. It was in Thebes, in the

ruins of a small building near the Ramesseum, that the papyrus was said to have been

found.

The Rhind Papyrus was written in hieratic script (a cursive form of hieroglyphics

better adapted to the use of pen and ink) about 1650 B.C. by a scribe named Ahmes, who

assured us that it was the likeness of an earlier work dating to the Twelfth Dynasty, 1849–

1801 B.C. Although the papyrus was originally a single scroll nearly 18 feet long and 13

inches high, it came to the British Museum in two pieces, with a central portion missing.

Perhaps the papyrus had been broken apart while being unrolled by someone who lacked

the skill for handling such delicate documents, or perhaps there were two nders and each

claimed a portion. In any case, it appeared that a key section of the papyrus was forever

lost to us, until one of those chance events that sometimes occur in archeology took place.

About four years after Rhind had made his famous purchase, an American Egyptologist,

Edwin Smith, was sold what he thought was a medical papyrus. This papyrus proved

to be a deception, for it was made by pasting fragments of other papyri on a dummy

scroll. At Smith’s death (in 1906), his collection of Egyptian antiquaries was presented

to the New York Historical Society, and in 1922, the pieces in the fraudulent scroll

were identi ed as belonging to the Rhind Papyrus. The decipherment of the papyrus was

completed when the missing fragments were brought to the British Museum and put in

their appropriate places. Rhind also purchased a short leather manuscript, the Egyptian

Mathematical Leather Scroll, at the same time as his papyrus; but owing to its very brittle

condition, it remained unexamined for more than 60 years.

A Key to Deciphering: The Rosetta Stone

It was possible to begin the translation of the Rhind Papyrus almost immediately

because of the knowledge gained from the Rosetta Stone. Finding this slab of polished

black basalt was the most signi cant event of Napoleon’s expedition. It was uncovered by

of cers of Napoleon’s army near the Rosetta branch of the Nile in 1799, when they were

digging the foundations of a fort. The Rosetta Stone is made up of three panels, each

inscribed in a different type of writing: Greek down the bottom third, demotic script

of Egyptian (a form developed from hieratic) in the middle, and ancient hieroglyphic

in the broken upper third. The way to read Greek had never been lost; the way to

read hieroglyphics and demotic had never been found. It was inferred from the Greek

inscription that the other two panels carried the same message, so that here was a trilingual

text from which the hieroglyphic alphabet could be deciphered.

The importance of the Rosetta Stone was realized at once by the French, especially

by Napoleon, who ordered ink rubbings of it taken and distributed among the scholars

of Europe. Public interest was so intense that when Napoleon was forced to relinquish

Egypt in 1801, one of the articles of the treaty of capitulation required the surrender of

the stone to the British. Like all the rest of the captured artifacts, the Rosetta Stone came

to rest in the British Museum, where four plaster casts were made for the universities

of Oxford, Cambridge, Edinburgh, and Dublin, and its decipherment by comparative

analysis began. The problem turned out to be more dif cult than imagined, requiring 23

years and the intensive study of many scholars for its solution.

The nal chapter of the mystery of the Rosetta Stone, like the rst, was written by a

Frenchman, Jean Franc¸ois Champollion (1790–1832). The greatest of all names associated

with the study of Egypt, Champollion had had from his childhood a premonition of the

part he would play in the revival of ancient Egyptian culture. Story has it that at the age

of 11, he met the mathematician Jean-Baptiste Fourier, who showed him some papyri

and stone tablets bearing hieroglyphics. Although assured that no one could read them,

the boy made the determined reply, “I will do it when I am older.” From then on, almost

everything Champollion did was related to Egyptology; at the age of 13 he was reading

three Eastern languages, and when he was 17, he was appointed to the faculty of the

University of Grenoble. By 1822 he had compiled a hieroglyphic vocabulary and given

a complete reading of the upper panel of the Rosetta Stone.

Through many years hieroglyphics had evolved from a system of pictures of com-

plete words to one that included both alphabetic signs and phonetic symbols. In the hi-

eroglyphic inscription of the Rosetta Stone, oval frames called “cartouches” (the French

word for “cartridge”) were drawn around certain characters. Because these were the

only signs showing special emphasis, Champollion reasoned that symbols enclosed by

the cartouches represented the name of the ruler Ptolemy, mentioned in the Greek text.

Champollion also secured a copy of inscriptions on an obelisk, and its base pedestal,

from Philae. The base had a Greek dedication honoring Ptolemy and his wife Cleopatra

(not the famous but ill-fated Cleopatra). On the obelisk itself, which was carved in hi-

eroglyphics, are two cartouches close together, so it seemed probable that these outlined

the Egyptian equivalents of their proper names. Moreover, one of them contained the

same hieroglyphic characters that lled the cartouches found on the Rosetta Stone. This

cross-check was enough to allow Champollion to make a preliminary decipherment. From

the royal names he established a correlation between individual hieroglyphics and Greek

letters. In that instant in which hieroglyphics dropped its shroud of insoluble mystery,

Champollion, worn by the years of ceaseless effort, was rumored to cry, “I’ve got it!”

and fell into a dead faint.

As a tting climax to a life’s study, Champollion wrote his Grammaire Egyptienne

en Encriture Hieroglyphique, published posthumously in 1843. In it, he formulated a

system of grammar and general decipherment that is the foundation on which all later

Egyptologists have worked. The Rosetta Stone had provided the key to understanding

one of the great civilizations of the past.