Figure 1: Polygonal arrays. Encyclopædia Britannica, Inc.

Although Euclid handed down a precedent for number theory in Books VII–IX of the Elements, later writers made no further effort to extend the field of theoretical arithmetic in his demonstrative manner. Beginning with Nicomachus of Gerasa (fl. c. AD 100), several writers produced collections expounding a much simpler form of number theory. A favourite result is the representation of arithmetic progressions in the form of “polygonal numbers.” For instance, if the numbers 1, 2, 3, 4, . . . are added successively, the “triangular” numbers 1, 3, 6, 10, . . . are obtained; similarly, the odd numbers 1, 3, 5, 7, . . . sum to the “square” numbers 1, 4, 9, 16, . . . , while the sequence 1, 4, 7, 10, . . . , with a constant difference of 3, sums to the “pentagonal” numbers 1, 5, 12, 22, . . . . In general, these results can be expressed in the form of geometric shapes formed by lining up dots in the appropriate two-dimensional configurations. In the ancient arithmetics, such results are invariably presented as particular cases, without any general notational method or general proof. The writers in this tradition are called neo-Pythagoreans, since they viewed themselves as continuing the Pythagorean school of the 5th century BC, and in the spirit of ancient Pythagoreanism they tied their numerical interests to a philosophical theory that was an amalgam of Platonic metaphysical and theological doctrines. With its exponent Iamblichus of Chalcis (4th century AD), neo-Pythagoreans became a prominent part of the revival of pagan religion in opposition to Christianity in late antiquity.

An interesting concept of this school of thought, which Iamblichus attributes to Pythagoras himself, is that of “amicable numbers”; two numbers are amicable if each is equal to the sum of the proper divisors of the other (for example, 220 and 284). Attributing virtues such as friendship and justice to numbers was characteristic of the Pythagoreans at all times.

Of much greater mathematical significance is the arithmetic work of Diophantus of Alexandria (active at an unknown time between the 2nd century BC and the 3rd century AD). His writing, the Arithmetica, originally in 13 books (six survive in Greek, another four in the medieval Arabic translation), sets out hundreds of arithmetic problems with their solutions. For example, Book II, problem 8, seeks to express a given square number as the sum of two square numbers (here and throughout, the “numbers” are rational). Like those of the neo-Pythagoreans, his treatments are always of particular cases rather than general solutions; thus, in this problem the given number is taken to be 16 and the solutions worked out are ^{256}/_{25} and ^{144}/_{25}. In this example, as is often the case, the solutions are not unique; indeed, in the very next problem Diophantus shows how a number given as the sum of two squares (e.g., 13 = 4 + 9) can be expressed differently as the sum of two other squares (for example, 13 = ^{324}/_{25} + ^{1}/_{25}).

To find his solutions, Diophantus adopted an arithmetic form of the method of analysis. He first reformulated the problem in terms of one of the unknowns, and he then manipulated it as if it were known until an explicit value for the unknown emerged. He even adopted an abbreviated notational scheme to facilitate such operations, where, for example, the unknown is symbolized by a figure somewhat resembling the Roman letter S. (This is a standard abbreviation for the word number in ancient Greek manuscripts.) Thus, in the first problem discussed above, if S is one of the unknown solutions, then 16 - S^{2} is a square, supposing the other unknown to be 2S - 4 (where the 2 is arbitrary but the 4 chosen because it is the square root of the given number 16), Diophantus found from summing the two unknowns ([2S - 4]^{2} and S^{2}) that 4S^{2} - 16S + 16 + S^{2} = 16, or 5S^{2} = 16S, that is, S = ^{16}/_{5}. So one solution is S^{2} = ^{256}/_{25}, while the other solution is 16 - S^{2}, or ^{144}/_{25.}

I. Read the passage and answer the questions:

1.Did writers after Euclid try to follow his manner?

2. How did they explain the number theory?

3. How were the results of arithmetic progressions represented in ancient arithmetics?

4. Was there any general notational method or general proof?

5. What are the writers in this tradition called? Why?

6. What is an interesting concept of this school of thought?

7. How are amicable numbers defined?

8.What was characteristic of the Pythagoreans?

9. Why is the arithmetic work of Diophantus of Alexandria significant?

10. Are his treatments particular cases or general solutions?

11. What did he adopt to find his solutions?

II. What do the words "writers" and " writing " mean in the passage? Suggest the way of translating them into Russian.

III. Some words and word combinations are difficult to translate. Translate the sentences paying special attention to italicized words.

1. Beginning with Nicomachus of Gerasa (fl. c. AD 100), several writers produced collections expounding a much simpler form of number theory

2.…later writers made no further effort to extend the field of theoretical arithmetic in his demonstrative manner.

3. For instance, if the numbers 1, 2, 3, 4, . . . are added successively, the “triangular” numbers 1, 3, 6, 10, . . . are obtained; similarly, the odd numbers 1, 3, 5, 7, . . . sum to the “square” numbers 1, 4, 9, 16, . . . , while the sequence 1, 4, 7, 10, . . . , with a constant difference of 3, sums to the “pentagonal” numbers 1, 5, 12, 22, . . . .

4.…in the spirit of ancient Pythagoreanism they tied their numerical interests to a philosophical theory

5. With its exponent Iamblichus of Chalcis…

6. Like those of the neo-Pythagoreans, his treatments are always of particular cases rather than general solutions.

7. … he then manipulated it as if it were known until an explicit value for the unknown emerged.

8. He even adopted an abbreviated notational scheme to facilitate such operations, where, for example, the unknown is symbolized by a figure somewhat resembling the Roman letter S.

9. A favourite result is the representation of arithmetic progressions in the form of “polygonal numbers.”

IV. What do the pronouns "that" and "those" stand for in the sentences? Translate them into Russian.

1. An interesting concept of this school of thought, which Iamblichus attributes to Pythagoras himself, is that of “amicable numbers”

2. Like those of the neo-Pythagoreans, his treatments are always of particular cases rather than general solutions

V. Can you explain the use of the definite article with numbers in the sentence? Translate it into Russian.

…(where the 2 is arbitrary but the 4 chosen because it is the square root of the given number 16)…

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

number theory, an arithmetic progression, to make no effort, in the form of,a square number, a geometric shape, a school of thought, to attribute to, a proper divisor, to be characteristic of, to be of great significance, to set out a problem, to work out a solution, to find a solution, a method of analysis, in terms of, an explicit value, a square root.

VII. Suggest the vebs corresponding to the following nouns: