From the history of early mathematics

6.1.1 THE BEGINNINGS

Ø 1) Before reading the text answer the questions:

a) What do you think were the first numerical terms?

b) What could stimulate the development of numerical terms?

Ø 2) Read the text and answer the questions.

a) What led to the numeration with five and ten as a base?

b) Were any other bases used in numeration?

c) How were numerical records kept?

d) When did the symbols for 5, 10 and 20 appear?

e) What resulted from the appearance of the symbols 5, 10, 20?

Numerical terms, expressing some of the most abstract ideas, came slowly into use. Their first occurrence made a distinction only between one, two, and many. The development of the crafts and of commerce stimulated this crystallization of the number concept. Numbers were arranged and bundled into larger units, usually by the use of the fingers of the hand or of both hands, a natural procedure in trading. This led to the numeration first with five, later with ten as a base, completed by addition and sometimes by subtraction, so that twelve was conceived as 10+2, or 9 as 10-1. Sometimes 20, the number of fingers and toes was selected as a base. Numerical records were kept by means of bundling, strokes on a stick, knots on a string, pebbles or shells arranged in heaps of five. From this method to the introduction of special symbols for 5, 10, 20 etc., was only a step, and we find such symbols in use at the beginning of written history, at the so-called dawn of civilization. Once it was reached, numbers could be expressed with reference to a base, with the aid of which large numbers could be formed; thus a primitive type of arithmetic originated. Fourteen was expressed as 10 + 4, sometimes as 15 – 1. Multiplication began where 20 was expressed not as 10 + 10, but as 2x10. Division began where 10 was expressed as “half of a body,” although conscious formation of fractions remained extremely rare.

Ø 3) Now that you’ve read the text, can you answer the questions in task 1?

6.1.2 THE ANCIENT ORIENTAL MATHEMATICS

Ø 1) Answer the questions:

a) There are four parts of the world: the West, the East, the South, and the North. “The

Orient” also means a part of the world. What part of the world is synonymous with “the Orient”?

b) What countries of the ancient Orient do you know? Choose from the following: Babylonia, India, Egypt, Germany, Mesopotamia, Russia, Sumeria, Persia.

Ø 2) Read the text and find:

a) the date of writing the Papyrus of Rhind and the Moscow Papyrus,

b) the date of King Hammurabi’s reign in Babylon,

c) the reason for the origin of mathematics,

d) the characteristics of mathematics in the Papyrus of Rhind and Moscow Papyrus,

e) the difference between Egyptian and Mesopotamian mathematics,

f) the reasons for the use of 60 rather than 10 as a unit for time and circle division.

During the fifth, fourth and third millennium B.C. newer and more advanced forms of society evolved from well-established Neolithic communities along the banks of great rivers in Africa and Asia, in subtropics or nearly subtropics regions. These rivers were the Niles, the Tigris and the Euphrates, the Indus and later the Ganges, the Huang Ho and later the Yang-tse. These territories became centers of civilization.

Oriental mathematics originated as a practical science in order to facilitate computation of the calendar, administration of the harvest, organization of the public works, and collection of taxes. The initial emphasis was on practical arithmetic and measurement. Arithmetic evolved into algebra, and measurement developed into the beginnings of a theoretical geometry.

The knowledge of Oriental mathematics is very sketchy. The mathematics of Babylonia and Egypt may be considered the most representative of the ancient orient mathematics because there exists a certain consistency in the factual character of the Babylonian and Egyptian texts throughout the centuries. Most of our knowledge of Egyptian mathematics is derived from two mathematical papyri: the Papyrus of Rhind, containing 85 problems and written about 1650 B.C.; and the Moscow Papyrus, perhaps two centuries older, containing 25 problems.

The mathematics in these papyri is based on a decimal system of numeration with special signs for each higher decimal unit – a system with which we are familiar through the Roman system which follows the same principle: MDCCCLXXVIII = 1878.

The most remarkable aspect of Egyptian arithmetic was its calculus of fractions. All fractions were reduced to sums of so-called unit fractions. The Papyrus Rhind has a table giving the equivalents in unit fractions for all odd “n” from 5 to 101. This work with unit fractions had been practiced for thousands of years, not only during the Greek period, but even during the Middle Ages. It should be noted that all texts point to an Egyptian mathematics of rather primitive standards.

Mesopotamian mathematics reached a far higher level than Egyptian mathematics ever obtained. Already the oldest texts, dating from the latest Sumerian period (the third dynasty of Ur, 2100 B.C.) show keen computational ability. These texts contain multiplication tables in which a well-developed sexagesimal system of numeration was added to an original decimal system. However, this was not their most characteristic feature. Whereas the Egyptians indicated each higher unit by a new symbol, the Sumerians used the same symbol but indicated its value by its position. Such a system had enormous advantages for computation, as we can see when we try to perform a multiplication in our own system and in a system with Roman numerals. This whole system seems to have developed as a direct result of the technique of administration, as is indicated in thousands of texts dating from the same period dealing with the delivery of cattle, grain, etc., and with arithmetical work based on these transactions. Eventually a special symbol for zero appeared, but much later, in the Persian era.

Both the sexagesimal system and the place value system remained in the permanent possession of mankind. Our present division of the hour into 60 minutes and 3600 seconds dates back to the Sumerians, as does our division of the circle into 360 degrees, each degree into 60 minutes and each minute into 60 seconds. There is a reason to believe that this choice of 60 rather than 10 as a unit occurred in an attempt to unify systems of measure, although the fact that 60 have many divisors may also have played a role. As to the place value system, the permanent importance of which has been compared to that of the alphabet (both inventions replaced a complex symbolism by a method easily understood by a large number of people), its history is still considerably obscure. The next group of cuneiform texts dates back to the first Babylonian Dynasty, when King Hammurabi reigned in Babylon (1950 B.C.) and a Semitic population had subdued the original Sumerians. In these texts we find arithmetic evolved into a well established algebra. Although the Egyptians of this period were only able to solve simple linear equations, the Babylonians of Hammurabi’s days were in full possession of the technique of handling quadratic equations. They solved linear and quadratic equations in two variables, and even problems involving cubic and biquadratic equations.

The strong arithmetical-algebraic character of the Babylonian mathematics is also apparent from its geometry. The texts show that the Babylonian geometry of the Semitic period was in possession of formulas for the areas of simple rectilinear figures and for the volumes of simple solids, although the volume of a truncated pyramid had not yet been found. The so-called theorem of Pythagoras was known, not only for special cases, but in full generality. The main characteristic of this geometry was, however, its algebraic character. This is equally true of all later texts, especially those dating back to the third period, that of New Babylonian, Persian, and Seleucid eras (from 600 B.C. – A.D. 300).

6.1.3 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.

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