Mathematics from Diophantus to Leonardo of Pisa - 2
Greek mathematics lacked the notational devices that enable us to think quickly and easily on problems that we conceptualize through the use of algebraic symbols. We may solve practical problems through algebra, using letters to denote unknown quantities, and the same method enables us to investigate abstract problems. We will discuss the development of improved notation later in the lectures. We do not associate the Greeks with the study of algebra. This is partly because their notation was not sufficiently flexible to allow them to make progress by symbolic manipulation, but also because geometry was the dominant theme in their work. There is an exception to the generalization that the Greeks were primarily geometers, provided by Diophantus of Alexandria, whose work we will now discuss. The golden age of Greek geometry may be held to have occurred between the sixth and third centuries BC. There then occurred a decline, which was arrested by a flurry of activity around 250-350 AD by Greek mathematicians working at Alexandria.
It was in Alexandria that Diophantus produced his Arithmetica, a work comprising thirteen books (or chapters), of which six have survived. It should be noted that Alexandria had a famous mathematical library, which was destroyed after the capture of the city in 641 by the Arabs. Copies of parts of the original work, written out at later times by scribes, survived into the age of printing, when eventually scholars were able to give a modern version of the work, in both Latin and Greek. In this way, almost by chance, the work of Diophantus was able to influence the thought of such modern mathematicians as Pierre de Fermat. He wrote the problem which has become known as Fermat’s Last Theorem into his copy of a Latin translation of Arithmetica.
We do not know exactly when Diophantus lived, but it is surmised that he was active around 250 AD. As we have mentioned, the subject matter of the Arithmetica is not the more usual geometry, but is rather algebra, relating to the study of numbers. Diophantus was interested in the exact solution of equations, both determinate and indeterminate, not simply in approximate solutions. The solutions often had to be given in terms of integers. This has led to the notion of a Diophantine equation, which is a subject of great interest today. For example, Fermat’s Last Theorem concerns a Diophantine equation: given an integer n ≥ 3, then there do not exist non-zero integers x, y and z that satisfy
xn +yn = zn. Although Fermat thought that he could prove this, and wrote down his claim in his copy
of Arithmetica, it was not until 1994 that Andrew Wiles actually gave a complete proof. Thus, although Diophantus did not himself pose the problem known as Fermat’s Last Theorem, his ideas have certainly influenced the progress of mathematics and provided stern challenges for the abilities of individual mathematicians.
The Arithmetica is a collection of about 150 problems. It is not a systematic attempt to teach algebra. A modern version of the text, by Thomas Heath, is available in English. Anyone who tries to study the original manuscript versions would have great difficulty, as the symbolism of algebra as we know it now was not employed. Thus we would need to be trained to decipher the true meaning of the text. Historians of mathematics have identified three stages in the development of algebra:
• a rhetorical or early stage, in which everything is written out in words. This survived in part into the 16th century.
• a syncopated stage, in which some abbreviations appear.
• a symbolic or final stage.
Diophantus is held to have moved to the second stage, away from the first stage customary at the time he lived. His Arithmetica is the beginning of literal algebra, that is, algebra where letters are used to denote quantities. In particular he introduced a symbolic notation for his unknown, in the form of an s. The unknown was called arithmos, meaning number in Greek. He also introduced symbolic notation for the first six positive and negative powers of the unknown, and for subtraction and equality.
A representative example of the collection is: find two integers x and y whose sum is 20 and whose squares sum to 208. We might solve this by setting x+y = 20, x2 +y2 = 208 and substituting for y in terms of x to obtain
2x2 - 40x + 192 = 0,
which is a standard quadratic. Diophantus proceeds slight differently by letting the two numbers be 10 - x and 10 + x, whose sum is certainly 20. Then we get
(10 - x)2 + (10 + x)2 = 208,
which leads to 2x2 = 8, and hence x = 2. The numbers are 8 and 12. This solution is neater, as the quadratic is much easier to solve.
The problem above is called determinate, as it leads to a definite solution, or to a small number of solutions. A problem of an indeterminate type, with an infinity of solutions, that was proposed by Diophantus is the following. Find two numbers such that when either is added to the square of the other, a perfect square results. This is more typical of what is now considered to be a Diophantine problem. If the numbers are x and y, we want
x+y2 = a2
x2 +y = b2 for suitable numbers a and b. It is required that x, y, a and b should all be rational numbers, or preferably integers, if possible. There are infinitely many rational solutions but it is not clear if there are any integer solutions.
Diophantus adopts the following specific approach to the problem, rather than attempting a general analysis. Take the second number y to be 2x + 1. Then, as
x2 + 2x + 1 = (x + 1)2,
the second equation is automatically satisfied. To satisfy the first equation, we need
x + (2x + 1)2 = a2.
Diophantus then chooses a to equal 2x - 2. This then means that the x2 terms in the equation disappear and we get
x+4x2 +4x+1=4x2 -8x+4 and hence.
This gives a rational solution to the problem. There are no solutions to the equation x + (2x + 1)2 = a2 if we want both x and a to be integers greater than 0.
Problem 24 of Book IV of Arithmetica is particularly prophetic, although it is the only example of this kind in the entire work. The problem is: to divide a given number into two numbers such that the their product is a cube minus its side.
If we let a denote the given number, we seek numbers x and y so that
y(a - y) = x3 - x.
Diophantus gave a solution for a = 6 by substituting x = ky - 1, where k is to be chosen. The equation becomes
y(6 - y) = (ky - 1)3 - (ky - 1) = (ky - 1)(k2y2 - 2ky).
Assuming that y is different from 0, we can divide by y to obtain
6 - y = k(ky - 1)(ky - 2).
Diophantus then chose k to equal 3, with a view to simplify the quadratic equation in y that results. We obtain
6 - y = 3(3y - 1)(3y - 2) = 27y2 - 27y + 6.
From a modern point of view, in the equation y(6 - y) = x3 - x, replace y by z + 3 and x by -w. Then we obtain
z2 = w3 -w +9
which is a classic form of indeterminate equation known as an elliptic curve. The Dio-phantine problem is to find integral or rational solutions for z and w. A vast amount of theory was developed about such elliptic curves, and Wiles used key parts of this theory in his solution of the Fermat problem.
Writing in his famous book Disquisitiones mathematicae (1801), the German mathematician Gauss said of Diophantus:
The celebrated work of Diophantus, dedicated to the problem of indeterminateness, contains many results which excite a more than ordinary regard for the ingenuity and proficiency of the author, because of their difficulty and the subtle devices he uses, especially if we consider the few tools that he had at hand for his work. However, these problems demand a certain dexterity and skillful handling rather than profound principles and, because the questions are too specialized and rarely lead to more general conclusions, Diophantus’s book seems to fit into that epoch in the history of Mathematics when scientists were more concerned with creating a characteristic art and a formal algebraic structure than with attempts to enrich Higher Arithmetic with new discoveries. The really profound discoveries are due to more recent authors like those men of immortal glory P. de Fermat, L. Euler, L. Lagrange, A. M. Legendre (and a few others).
A classical example of an indeterminate Diophantine equation is: find positive integers a, b and c that satisfy the equation
a2 +b2 = c2.
As we mentioned in Chapter 1, we refer to solutions of this equation as Pythagorean triples (a, b, c). We can investigate this Diophantine equation as follows. Rearranging, we obtain
a2 = c2 - b2 = (c + b)(c - b).
We can look for a solution so that
Then we obtain
Now assuming that a is an integer, in order that c and b are also integers, we need a to be odd, say a = m, where m is odd, and then we have the solution attributed to Pythagoras, as described in Chapter 1:
where m is odd. Alternatively, assuming that a is even, say a = 2n, then we have
4n2 = (c + b)(c - b).
This time we try a solution c - b = 2, c + b = 2n2. This leads to a solution
a=2n, b=n2 -1, c=n2 + 1.
The commentator Proclus attributes this method of generating Pythagorean triples to Plato. Here the smallest side has even length, and the other sides have odd length, whereas in the solution due to Pythagoras, the smallest side has odd length. It should be understood that these two methods of generating Pythagorean triples are just special cases of a complete solution of the problem, as follows.
Suppose that a, b and c are relatively prime– which is a reasonable assumption, since we can remove any common factors and descend to this case. Suppose that c = 2z is even. Then c2 = 4z2 is divisible by 4. By assumption, a and b cannot both be even–otherwise 2 is a common factor of a, b and c. Now if say a is even and b odd (or vice versa), it is clear that a2 + b2 is odd, and so cannot equal c2 which is even by assumption. Thus a and b are both odd, and so we can write
a = 2x+1, b = 2y+1,
where x and y are integers. Then multiplying out
a2 + b2 = 4x2 + 4x + 1 + 4y2 + 4y + 1 = 4z2.
This leads to
2 = 4(z2 - x2 - y2 - x - y),
which is a contradiction, since the right hand side is four times an integer and so cannot equal 2. We deduce that c cannot be even, and hence is odd. Therefore, by the previous argument a and b cannot both be even or both odd, since otherwise a2 + b2 is even and so cannot equal c2, which is odd. Thus exactly one of a and b is odd, and we choose notation so that a is odd and b even.
We now have
a2 = (c + b)(c - b).
Next, we claim that c - b and c + b are relatively prime. For let d be their gcd. Then certainly d divides 2c and 2b. As c is odd and b even, c - b and c + b are both odd. Hence d is odd, and it follows that d is a common divisor of b and c. On the other hand, as (c + b)(c - b) = a2, we deduce that d2 is a divisor of a2. Elementary prime factorization shows that d must be a divisor of a. Thus d divides all of a, b and c and is hence 1, since a, b and c are assumed to be relatively prime. Thus c - b and c + b are relatively prime, as claimed. Now, again appealing to the theory of prime factorization, if a product of two relatively prime integers is a square of an integer, each factor is a square. So we can write c - b = r2, c + b = s2, a2 = r2s2, where r and s are odd integers. Hence a = rs, and. This completely describes the solution. It is extraordinary that no such
solution can be found for any higher order Diophantine equation of the form
an +bn = cn, as the solution of Fermat’s problem shows.
The initial influence of Diophantus was probably not great. He was working during the late classical period, when the last great discoveries of Greek mathematics were made, and there was little tradition of work in algebra. Thus, he is unlikely to have had many followers who could build on his foundation of algebra. Eventually, Arab mathematicians continued the themes of his work in the 10th century. Problems of the type considered in the Arithmetica later appeared in Leonardo of Pisa’a Liber abbaci of 1202, a key work in medieval mathematics. It is assumed that Leonardo derived his information from Arab
sources. A manuscript of the Greek text of Arithmetica was brought to Italy from Constantinople in the 15th century, just before the city was captured by the Turks. The manuscript was deposited in the Vatican Library, where it could be studied by Renaissance scholars. For example, Bombelli’s Algebra of 1572, a work of great importance as we shall see, contains 147 problems taken from Diophantus, many using the same numerical values. The celebrated French algebraist Fran¸cois Vi`ete also used 34 problems from Diophantus in his Zetetica of 1593. A translation of the Arithmetica into Latin was made by the German mathematician Xylander in 1575. In 1621 a Greek version of the text was prepared by Bachet de M´eziriac, incorporating Xylander’s Latin translation. The foundation of modern number theory dates from the appearance of this text, and mathematicians such as Fermat, Euler and Gauss drew inspiration from it.
The work of Diophantus appeared at a time when political and military power in much of Europe and the Near East resided in the Roman Empire, although that power was already waning. The Romans had taken over Greece and its colonies several centuries before, and were exposed to the beauties of Greek mathematics and astronomy, but they made few original contributions to mathematical thought and theory. They were essentially a practical people, interested in building, architecture and engineering. They required mathematics for measurement, but remained largely indifferent to purely speculative or theoretical mathematics. As the Roman orator Cicero wrote in his Tusculan Disputations: