I am reading "Elementary Number Theory" by G. Burton. The book is technical, but the following historical comments are, perhaps, of general interest:
1.3. Early Number Theory
Before becomeing weighted down with details, we should say a few wordsabout
the origin of number theory.The theory of numbers is one of the oldest branches
of mathematics; an enthusiast, by stretching a point here and there, could extend
its roots back to a surprisingly remote date. Although it seems probable that the
Greeks were largely indebted to the Babylonians and ancient Egyptians for a core
of information about the properties of the natural numbers, the first rudiments of an
actual theory are generally credited to Pythagoras and his disciples.
Our knowledge of the life of Pythagoras is scanty, and little can he said with any
certainty. According to the best estimates, he was born between 580 and 562 B.C. on
the Aegean island of Samos. It seems that he studied not only in Egypt, but even may
have extended his journeys as far east as Babylonia. When Pythagoras reappeared
after years of wandering, he sought out a favorable place for a school and finally
settled upon Croton, a prosperous Greek settlement on the heel of the Italian boot.
The school concentrated on four mathemata, or subjects of study: arithmetica (arith-
metic, in the sense of number theory, rather than the art of calculating), harmonia
(music), geometria (geometry), and astrologia (astronomy). This fourfold division
of knowledge became known in the Middle Ages as the quadrivium, to which was
added the trivium of logic, grammar, and rhetoric. These seven liberal arts came to
be looked upon as the necessary course of study for an educated person.
Pythagoras divided those who attended his lectures into two groups: the Pro-
bationers (or listeners) and the Pythagoreans. After three years in the first class, a
listener could be initiated into the second class, to whom were confided the main dis-
coveries of the school. The Pythagoreans were a closely knit brotherhood, holding all
worldly goods in common and bound by an oath not to reveal the founder's secrets.
Legend has it that a talkative Pythagorean was drowned in a shipwreck as the gods'
punishment for publicly boasting that he had added the dodecahedron to the number
of regular solids enumerated by Pythagoras. For a time, the autocratic Pythagoreans
succeeded in dominating the local government in Croton, but a popular revolt in 501
B.C. led to the murder of many of its prominent members, and Pythagoras himself
was killed shortly thereafter. Although the political influence of the Pythagoreans
thus was destroyed, they continued to exist for at least two centuries more as a
philosophical and mathematical society. To the end, they remained a secret order,
publishing nothing and, with noble self-denial, ascribing all their discoveries to the
Master.
The Pythagoreans believed that the key to an explanation of the universe lay
in number and form, their general thesis being that "Everything is Number." (By
number, they meant, of course, a positive integer.) For a rational understanding of
nature, they considered it sufficient to analyze the properties of certain numbers.
Pythagoras himself, we are told "seems to have attached supreme importance to the
study of arithmetic, which he advanced and took out of the realm of commercial
utility."
The Pythagorean doctrine is a curious mixture of cosmic philosophy and number
mysticism, a sort of supernumerology that assigned to everything material or spiritual
a definite integer. Among their writings, we find that 1 represented reason, for reason
could produce only one consistent body of truth; 2 stood for man and 3 for woman;
society, the Pythagoreans classified the odd numbers, after the first two, as masculine
and divine.
Although these speculations about numbers as models of "things" appear friv-
olous today, it must be borne in mind that the intellectuals of the classical Greek
period were largely absorbed in philosophy and that these same men, because they
had such intellectual interests, were the very ones who were engaged in laying the
foundations for mathematics as a system of thought. To Pythagoras and his followers,
mathematics was largely a means to an end, the end being philosophy. Only with
the founding of the School of Alexandria do we enter a new phase in which the
cultivation of mathematics was pursued for its own sake.
We digress here to point out that mystical speculation about the properties of
numbers was not unique to the Pythagoreans. One of the most absurd yet wide-
spread forms that numerology took during the Middle Ages was a pseudoscience
known as gematria or arithmology. By assigning numerical values to the letters
of the alphabet in some order, each name or word was given its own individual
number. From the standpoint of gematria, two words were considered equivalent if
the numbers represented by their letters when added together gave the same sum.
All this probably originated with the early Greeks for whom the natural ordering
of the alphabet provided a perfect way of recording numbers; alpha standing for 1, beta
for 2, and so forth. For example, the word "amen" is "alpha_mu_eta_nu" in Greek; these letters
have the values 1, 40, 8, and 50, respectively, which total 99. In many old editions
of the Bible, the number 99 appears at the end of a prayer as a substitute for amen.
The most famous number was 666, the "number of the beast," mentioned in the
Book of Revelations. A favorite pastime among certain Catholic theologians during
the Reformation was devising alphabet schemes in which 666 was shown to stand
for the name of Martin Luther, thereby supporting their contention that he was the
Antichrist. Luther replied in kind: He concocted a system in which 666 became the
number assigned to the reigning Pope, Leo X.
It was at Alexandria, not Athens, that a science of numbers divorced from mystic
philosophy first began to develop. For nearly a thousand years, until its destruction
by the Arabs in 641 A.D., Alexandria stood at the cultural and commercial center of
the Hellenistic world. (After the fall of Alexandria, most of its scholars migrated to
Constantinople. During the next 800 years, while formal learning in the West all but
disappeared, this enclave at Constantinople preserved for us the mathematical works
of the various Greek schools.) The so-called Alexandrian Museum, a forerunner of
the modern university, brought together the leading poets and scholars of the day;
adjacent to it there was established an enormous library, reputed to hold over 700,000
volumes-hand-copied-at its height. Of all the distinguished names connected with
the Museum, that of Euclid (circa 350 B.C.), founder of the School of Mathematics,
is in a special class. Posterity has come to know him as the author of the Elements,
the oldest Greek treatise on mathematics to reach us in its entirety. The Elements is a
Scarcely any other book save the Bible has been more widely circulated or studied.
Over a thousand editions of it have appeared since the first printed version in 1482,
and before its printing, manuscript copies dominated much of the teaching of math-
ematics in Western Europe. Unfortunately, no copy of the work has been found that
actually dates from Euclid's own time; the modern editions are descendants of a
revision prepared by Theon of Alexandria, a commentator of the 4th century A.D.
