Mathematics in Greece and Rome

Ø 1) Guess what these proper names (the name of a country, a city, a part of the world, culture, a sea) mean: Alexandria, Greece, the Orient, the Near East, Rome, Babylon, Hellenism, Athens, Syracuse?

Ø 2) Does Hellenism refer to Rome or Greece?

Ø 3) Which names of these great mathematicians would you connect with Greece and which with Rome: Euclid, Archimedes, Ptolemy and Diophantus?

(1) The early Greek study of mathematics had one main goal: the understanding of man’s place in the universe according to a rational scheme. Mathematics helped to find order in chaos, to arrange ideas in logical chains, to find fundamental principles. It was the most rational of all sciences, and although there is little doubt that the Greek merchants became acquainted with Oriental mathematics along their trade routes, they soon discovered that the Orientals had left most of the rationalization undone.

(2) When Alexander the Great died at Babylon in 323 B.C. the whole Near East had fallen to the Greeks. The period of Hellenism began. Greek mathematics, thus transplanted to new surroundings, kept many of its traditional aspects, but experienced also the influence of the problems in administration and astronomy which the Orient had to solve. It is also remarkable that the greatest flowering of this Hellenistic mathematics occurred in Egypt under the Ptolemies. Egypt was now in a central position in the Mediterranean world. Alexandria, the new capital, was built on the sea coast and became the intellectual and economic center of the Hellenistic world. Besides Alexandria there were other centers of mathematical learning, especially Athens and Syracuse. Athens became an educational center, while Syracuse produced Archimedes, the greatest of Greek mathematicians.

(3) Among the first scholars associated with Alexandria was Euclid, one of the most influential mathematicians of all times. Euclid, about whose life nothing is known with any certainty, flourished probably during the time of the first Ptolemy (306 - 283 B.C.). His most famous and most advanced texts are the thirteen books of “The Elements.” “The Elements” form, next to the Bible, probably the most reproduced and studied book in the history of the Western World. More than a thousand editions appeared since the invention of printing, and before that time manuscript copies dominated much of the teaching of geometry. Most of our school geometry is taken, often literally, from eight or nine of the thirteen books; and the Euclidean tradition still weighs heavily on our elementary instruction. For the professional mathematician these books have always had an inescapable fascination and their logical structure has influenced scientific thinking perhaps more than any other text in the world.

(4) The greatest mathematician of the Hellenistic period was Archimedes (287 – 212 B.C.) who lived in Syracuse as adviser to King Hiero. The most important contributions which Archimedes made to mathematics were his books, such as “Measurement of the Circle, ” “On the sphere and Cylinder, ” “Quadrature of the Parabola, ” “On Spirals, ” “On Conoids and Spheroids, ” “On Floating Bodies”. In all these works Archimedes combined a surprising originality of thought with a mastery of computational technique and rigor of demonstration. In his computational proficiency Archimedes differed from most of the productive Greek mathematicians.

(5) The third and last period of antique society is that of the Roman domination. Syracuse fell to Rome in 212, Carthage in 146, Greece in 146, Mesopotamia in 64, and Egypt in 30 B.C. The entire Roman-dominated Orient, including Greece, was reduced to the status of a colony ruled by Roman administrators.

(6) As long as the Roman Empire showed some stability, Eastern science continued to flourish as a curious blend of Hellenistic and Oriental elements. Alexandria remained the center of antique mathematics. Computational arithmetic and algebra of an Egyptian-Babylonian type were cultivated side by side with abstract geometrical demonstrations. We have only to think of Ptolemy, Heron, and Diophantus to become convinced of this fact.

(7) One of the earliest Alexandrian mathematicians of the Roman period was Nicomachus of Gerasa (c. A.D. 100) whose “Arithmetic Introduction” is the most complete exposition of Pythagorean arithmetic, still existing.

(8) One of the greatest documents of this second Alexandrian period was Ptolemy’s “Great Collection, ” better known under the Arabicized title of “Almagest” (A.D. 150). The “Amalgest” was an astronomical opus of supreme mastership and originality, even though many of the ideas may have come from Babylonian astronomers. Also it contained a trigonometry, with a table of chords belonging to different angles ascending by halves of an angle, equivalent to a sine table. The Oriental touch is even stronger in the “Arithmetica” of Diophantus (A.D. 250). We do not know who Diophantus was - he may have been a Hellenized Babylonian. His book is one of the most fascinating treaties preserved from Greco-Roman antiquity.

Ø 4) Which of the sentences may be included into this text?

a) Counting by fingers, that is, counting by fives and tens, came at a certain stage of social development.

b) Neolithic man also developed a keen feeling for geometrical patterns.

c) We possess reliable editions of Euclid and Archimedes.

d) The main result of the Greek victory was the expansion and hegemony of Athens.

e) The immediate consequence of Alexander’s campaign was the acceleration of the advance of Greek civilization over large sections of the Oriental world.

e) Euclid’s treatment is based on a logical deduction of theorems from a set of definitions, postulates, and axioms.

f) The most important contributions of Archimedes to mathematics were in the domain of what we now call the “integral calculus.”

Ø 5) Name the paragraphs which give answers to these questions:

a) What kind of work is the “Amalgest” by Ptolemy?

b) What was the main goal of the early Greek study of mathematics?

c) What books on mathematics did Archimedes write?

d) What centers of mathematical learning in ancient Greece could you mention?

e) What happened to the Orient when Rome conquered it?

6.2 THE ORIGIN OF THE WORD “MONEY”

Ø 1) Read the title and the words from the text (moneta, goddess Juno, Juno Moneta, Rome, temple, the mint) and guess what this text is about.

The English word “money” is believed to come from the Italian word “moneta” which has an interesting history. Today the word means “coin, ” but in ancient Rome, and perhaps even earlier in Greece, the word meant “advisor, ” one who warns, or one who makes people remember.

There are several accounts of how the meaning of the word changed based on a similar story about the goddess Juno. She presided over many aspects of life. One of these aspects was an advisor of the Roman people, so one of her names was Juno Moneta.

A flock of geese in Juno’s sanctuary on the Capitoline Hill squawked the alarm that saved Rome from an invasion of the Gauls in 390 B.C. A temple was built in honor of Juno Moneta at the site because her sacred geese had “warned” of the attack.

The first Roman mint was built near Juno Moneta’s temple in 289 B.C. Originally it produced bronze and later silver coins. Many of these coins were struck with the head of Juno Moneta on the face. We don’t know if this was done in tribute to Juno Moneta or just to identify the mint, but “moneta” became the word for both coin and mint, and eventually for the word “money.”

Ø 2) Say if the statements are true, false or there is no evidence in the text:

a) The word “moneta” comes from the Russian language.

b) The meaning of the word “moneta” changed in the course of time.

c) The goddess Juno was in charge of monetary matters.

d) The goddess Juno lived in the IVth century B.C.

e) The first Roman mint was built to commemorate the goddess Juno.

f) The first Roman mint produced gold coins.

g) Many coins had the head of Juno Moneta on its face.