The Greeks divided the field of mathematics into arithmetic (the study of “multitude,” or discrete quantity) and geometry (that of “magnitude,” or continuous quantity) and considered both to have originated in practical activities. Proclus, in his Commentary on Euclid, observes that geometry, literally, “measurement of land,” first arose in surveying practices among the ancient Egyptians, for the flooding of the Nile compelled them each year to redefine the boundaries of properties. Similarly, arithmetic started with the commerce and trade of Phoenician merchants. Although Proclus wrote quite late in the ancient period (in the 5th century AD), his account drew upon views proposed much earlier, by Herodotus (mid-5th century BC), for example, and by Eudemus, a disciple of Aristotle (late 4th century BC).

However plausible, this view is difficult to check, for there is only meagre evidence of practical mathematics from the early Greek period (roughly, the 8th through the 4th centuries BC). Inscriptions on stone, for example, reveal use of a numeral system the same in principle as the familiar Roman numerals. Herodotus seems to have known of the abacus as an aid for computation by both Greeks and Egyptians, and about a dozen stone specimens of Greek abaci survive from the 5th and 4th centuries BC. In the surveying of new cities in the Greek colonies of the 6th and 5th centuries, there was regular use of a standard length of 70 plethra (one plethron equals 100 feet) as the diagonal of a square of side 50 plethra; in fact, the actual diagonal of the square is 50√2 plethra, so this was equivalent to using ^{7}/_{5} (or 1.4) as an estimate for √2, which is now known to equal 1.414 . . . . In the 6th century BC the engineer Eupalinus of Megara directed an aqueduct through a mountain on the island of Samos, and historians still debate how he did it. In a further indication of the practical aspects of early Greek mathematics, Plato describes in his Laws how the Egyptians drilled their children in practical problems in arithmetic and geometry; he clearly considered this a model for the Greeks to imitate.

Such hints about the nature of early Greek practical mathematics are confirmed in later sources, for example, in the arithmetic problems in papyrus texts from Ptolemaic Egypt (from the 3rd century BC onward) and the geometric manuals by Hero of Alexandria (1st century AD). In its basic manner this Greek tradition was much like the earlier traditions in Egypt and Mesopotamia. Indeed, it is likely that the Greeks borrowed from such older sources to some extent.

What was distinctive of the Greeks' contribution to mathematics—and what in effect made them the creators of “mathematics,” as the term is usually understood—was its development as a theoretical discipline. This means two things: mathematical statements are general, and they are confirmed by proof. For example, the Mesopotamians had procedures for finding whole numbers a, b, and c for which a^{2} + b^{2} = c^{2} (e.g., 3, 4, 5; 5, 12, 13; or 119, 120, 169). From the Greeks came a proof of a general rule for finding all such sets of numbers (now called Pythagorean triples): if one takes any whole numbers p and q, both being even or both odd, then a = (p^{2} - q^{2})/2, b = pq, and c = (p^{2} + q^{2})/2. As Euclid proves in Book X of the Elements, numbers of this form satisfy the relation for Pythagorean triples. Further, the Mesopotamians appear to have understood that sets of such numbers a, b, and c form the sides of right triangles, but the Greeks proved this result (Euclid, in fact, proves it twice, in Elements, Book I, proposition 47, and in a more general form in Elements, Book VI, proposition 31), and these proofs occur in the context of a systematic presentation of the properties of plane geometric figures.

The Elements, composed by Euclid of Alexandria, around 300 BC, was the pivotal contribution to theoretical geometry, but the transition from practical to theoretical mathematics had occurred much earlier, sometime in the 5th century BC. Initiated by men like Pythagoras of Samos (late 6th century) and Hippocrates of Chios (late 5th century), the theoretical form of geometry was advanced by others, most prominently the Pythagorean Archytas of Tarentum, Theaetetus of Athens, and Eudoxus of Cnidus (4th century). Because the actual writings of these men do not survive, knowledge about their work depends on remarks made by later writers. While even this limited evidence reveals how heavily Euclid depended on them, it does not set out clearly the motives behind their studies.

I. Read the text and define the key sentence in each paragraph.

II. Analyse the structure of paragraphs. How are they built?

III. Answer the questions:

1. What did the Greeks divide mathematics into?

2. Where have they originated in?

3. How did geometry arise?

4. What did arithmetics start with?

5. Why is this view difficult to check?

6. What was distinctive of the Greeks' contribution to mathematics?

7. What does this mean?

8. What work is the main contribution to theoretical geometry?

9. Who was theoretical geometry initiated by?

10. Was it advanced by others?

11. Did Euclid depend on them?

12. Does the evidence set out the motives behind their studies?

III. Single out the key information from the passage and present its summary.

Part 2

It is thus a matter of debate how and why this theoretical transition took place. A frequently cited factor is the discovery of the irrational. The early Pythagoreans held that “all things are number.” This might be taken to mean that any geometric measure can be associated with some number (that is, some whole number or fraction; in modern terminology, rational number), for in Greek usage the term for number, arithmos, refers exclusively to whole numbers or, in some contexts, to ordinary fractions. This assumption is common enough in practice, as when the length of a given line is said to be so many feet plus a fractional part. However, it breaks down for the lines that form the side and diagonal of the square. (For example, if it is supposed that the ratio between the side and diagonal may be expressed as the ratio of two whole numbers, it can be shown that both of these numbers must be even. This is impossible, since every fraction may be expressed as a ratio of two whole numbers having no common factors.) Geometrically, this means that there is no length that could serve as a unit of measure of both the side and diagonal; that is, the side and diagonal cannot each equal the same length multiplied by (different) whole numbers. Accordingly, the Greeks called such pairs of lengths “incommensurable.” (In modern terminology, unlike that of the Greeks, the term “number” is applied to such quantities as √2, but they are called irrational.)

This result was already well known at the time of Plato and may well have been discovered within the school of Pythagoras in the 5th century BC, as some late authorities like Pappus of Alexandria (4th century AD) maintain. In any case, by 400 BC it was known that lines corresponding to √3, √5, and other square roots are incommensurable with a fixed unit length. The more general result, the geometric equivalent of the theorem that √p is irrational whenever p is not a rational square number, is associated with Plato's friend Theaetetus. Both Theaetetus and Eudoxus contributed to the further study of irrationals, and their followers collected the results into a substantial theory, as represented by the 115 propositions of Book X of the Elements.

The discovery of irrationals must have affected the very nature of early mathematical research, for it made clear that arithmetic was insufficient for the purposes of geometry, despite the assumptions made in practical work. Further, once such seemingly obvious assumptions as the commensurability of all lines turned out to be, in fact, false, then in principle all mathematical assumptions were rendered suspect. At the least, it became necessary to justify carefully all claims made about mathematics. Even more basically, it became necessary to establish what a reasoning has to be like to qualify as a proof. Apparently, Hippocrates of Chios, in the 5th century BC, and others soon after him had already begun the work of organizing geometric results into a systematic form in textbooks called “elements” (meaning “fundamental results” of geometry). These were to serve as sources for Euclid in his comprehensive textbook a century later.

The early mathematicians were not an isolated group but part of a larger, intensely competitive intellectual environment of pre-Socratic thinkers in Ionia and Italy, as well as Sophists at Athens. By insisting that only permanent things could have real existence, the philosopher Parmenides (5th century BC) called into question the most basic claims about knowledge itself. In contrast, Heracleitus (c. 500 BC) maintained that all permanence is an illusion, for the things that are perceived arise through a subtle balance of opposing tensions. What is meant by “knowledge” and “proof” thus came into debate.

Mathematical issues were often drawn into these debates. For some, like the Pythagoreans (and, later, Plato), the certainty of mathematics was held as a model for reasoning in other areas, like politics and ethics. But for others, mathematics seemed prone to contradiction. Zeno of Elea (5th century BC) posed paradoxes about quantity and motion. In one such paradox, it is assumed that a line can be bisected again and again without limit; if the division ultimately results in a set of points of zero length, then even infinitely many of them sum up only to zero, but, if it results in tiny line segments, then their sum will be infinite. In effect, the length of the given line must be both zero and infinite. In the 5th century BC a solution of such paradoxes was attempted by Democritus and the “atomists,” philosophers who held that all material bodies are ultimately made up of invisibly small “atoms” (the Greek word atomon means “indivisible”). But in geometry such a view came into conflict with the existence of incommensurable lines, since the atoms would become the measuring units of all lines, even incommensurable ones. Protagoras and Democritus puzzled over whether the tangent to a circle meets it at a point or a line. The Sophists Antiphon and Bryson (both 5th century BC) considered how to compare the circle to polygons inscribed in it.

The pre-Socratics thus revealed difficulties in specific assumptions about the infinitely many and the infinitely small, about the relation of geometry to physical reality, as well as in more general conceptions like “existence” and “proof.” Philosophical questions such as these need not have affected the technical researches of mathematicians, but they did make them aware of difficulties that could bear on fundamental matters and so made them the more cautious in defining their subject matter.

Any such review of the possible effects of factors such as these is purely conjectural, since the sources are fragmentary and never make explicit how the mathematicians responded to the issues that were raised. But it is the particular concern over fundamental assumptions and proofs that distinguishes Greek mathematics from the earlier traditions. Plausible factors behind this concern can be identified in the special circumstances of the early Greek tradition—its technical discoveries and its cultural environment—even if it is not possible to describe in detail how these changes took place.

I. Read the passage . How will you divide it into parts? How many parts will you divide it into?

Define the main idea of each part.

II.Tasks to Part 1

1. Analyse all the uses of the pronoun "it" . What does it stand for in each case? How do we translate it into Russian?

2. What is the meaning of "that" and "these " in the sentences? Study the context and translate them into Russian.

1. In modern terminology, unlike that of the Greeks, the term “number” is applied to such quantities as √2, but they are called irrational.

2. These were to serve as sources for Euclid in his comprehensive textbook a century later.

3. Analyse the use of modal verbs in the sentences. Translate them into Russian.

1. This might be taken to mean that any geometric measure can be associated with some number…

2. … it is supposed that the ratio between the side and diagonal may be expressed as the ratio of two whole numbers, it can be shown that both of these numbers must be even.

3. …there is no length that could serve as a unit of measure of both the side and diagonal; that is, the side and diagonal cannot each equal the same length multiplied by (different) whole numbers.

4. This result …may well have been discovered within the school of Pythagoras in the 5th century BC…

5. The discovery of irrationals must have affected the very nature of early mathematical research…

6. … it became necessary to establish what a reasoning has to be like to qualify as a proof.

7. These were to serve as sources for Euclid in his comprehensive textbook a century later.

4. How are the words "since" and "for"used in the sentences? Translate them into Russian

1… for in Greek usage the term for number, arithmos, refers exclusively to whole numbers…

2. This is impossible, since every fraction may be expressed…

3… for it made clear that arithmetic was insufficient for the purposes of geometry…

5. Analyse the sentence:

This assumption is common enough in practice, as when the length of a given line is said to be so many feet plus a fractional part.

What syntactical construction is used in the subordinate clause? How will you translate it into Russian? What is the meaning of "as" in it?

6. What is the meaning of "the irrational" in the sentence "A frequently cited factor is the discovery of the irrational."? Is it the same as in the phrase "… contributed to the further study of irrationals… "?

7. Is the word 'factor' used in the same meaning in the sentences:

1. A frequently cited factor is the discovery of the irrational

2…since every fraction may be expressed as a ratio of two whole numbers having no common factors.

8. The word "incommensurable" means "íåñîèçìåðèìûé". How will you translate the word "commensurability " later in the passage? Explain why? Present other words of the same word family.

9. Suggest the Russian counterparts for the following words and expressions:

geometric measure , whole number, rational number, an ordinary fraction, a fractional part, a side and diagonal of a square, a ratio of two whole numbers, a common factor, a unit of measure, multiplied by, a square root, a unit length, a rational square number, a substantial theory, mathematical research, even and odd numbers

10. Work in pairs on the consecutive translation of the part.

III.Analyse the rest of the passage in terms of morphological, syntactic and lexical difficulties and present its translation at sight.

IV. What conclusions are drawn about the development of mathematics in Greece?