Mathematics from Diophantus to Leonardo of Pisa - 2

With the Greeks geometry was regarded with the utmost respect, and consequently none were held in greater honour than mathematicians, but we Romans have delimited the size of this art to the practical purposes of measuring and calculating.

Towards the end of the Roman Empire, a few elementary mathematical works were produced in Rome, and some of these survived to be used extensively in medieval Western Europe. The arithmetic and geometry, written perhaps by the Roman aristocrat Boethius, who was executed in 524, are the most important examples in this respect, but they are considered to be poor synopses of much deeper ancient work. The Roman translation of the Elements, attributed to Boethius, contains all the definitions of the first three books, the postulates, and most of the axioms. The enunciations, with diagrams but no proofs, are given of most of the propositions of the first, second and fourth books. Nobody in Western Europe continued the Greek tradition of abstract mathematics. The Greeks themselves, both in their homeland and in their colonies, such as Alexandria or Asia Minor, had

ceased to invent new ideas in mathematics by the 5th century. Some Greeks, including Proclus, wrote commentaries or histories of their predecessors, whose chance survival has enabled us to learn the content of much ancient Greek work and a little about its original discoverers

In the early 7th century, around 620-630, the creation of Islam and the subsequent expansion of Arab conquerors into much of North Africa and Asia led to the demise of the greater part of the Byzantine Empire, which was the inheritor of the Greek tradition, and of the remains of the Western Roman Empire. The Arabs came into contact with Greek learning, especially in Alexandria. Initially, they were without intellectual interest in this unfamiliar knowledge, rather as the Romans had felt with regard to their Greek predecessors, but by 750, the Arabs began to absorb the Greek legacy of mathematics and astronomy. Indeed, in the early years of Islam, secular knowledge was not in conflict with religious knowledge, but was seen as a way towards the latter (this was not the case during the early centuries of Christianity in the West). Learning was encouraged in the Islamic world, and contributions were made to practical applications, as well as theory. A centre of learning, known as the House of Wisdom, was established in Baghdad in the early 9th century, under the support of a series of enlightened caliphs. Here, translations into Arabic were made of Greek classics, such as Euclid’s Elements. Without these translations, many of the most famous and original of the Greek treatises might have been lost during periods of political confusion, barbarian invasion and intellectual sterility in Europe.

One of the scholars working in Baghdad at this time was Mohammed ibn-Musa al-Khawarizmi. Details of his life are few, but he is thought to have died around 850. He has achieved considerable fame in the history of mathematics, as a number of basic ideas have entered common usage thanks to his exposition of them. Al-Khawarizmi wrote two books on arithmetic and algebra, the first of which has come down to us only in a Latin translation De numero indorum (On the Hindu art of reckoning). In this work, he gave an explanation of the Hindu or Indian system of numerals. We presume that al-Khawarizmi became acquainted with the system by reading Arabic translations of the work of the Indian mathematician Brahmagupta (who flourished c 625). Due to confusion over sources, later readers came to believe that the Hindu numeral system had been devised by the Arabs, although there is no question of its Hindu origin. Nowadays, we refer to it as the Hindu-Arabic system. The new notation was associated with the name of al-Khawarizmi and became known as algorismi. Eventually, it was called algorism or

algorithm. We do not have exact details of what al-Khawarizmi wrote about the numerals. The later Latin versions of his work state that Al-Khawarizmi introduced nine characters to designate the first nine numbers, and a circle to designate zero. He demonstrated how to write any number using these characters, via the place-value system. Our word zero probably derives from the Arabic sifr. A medieval translation of sifr later led to the word cipher, which is still used. The significance of the zero symbol was considerable, since its presence made precise earlier notations where the meaning of the number intended by symbols was not always clear.

Arab historians have credited al-Khawarizmi as the first Islamic author to solve problems by al-jabr, which has given rise to our word algebra, and al-muqabala. These words describe techniques that we regularly use in algebra. The literal meaning of al-jabr is restoration, and it is understood to refer to the adding of equal terms to both sides of an equation to eliminate negative terms. It can also refer to the multiplication of both sides of an equation by the same number to eliminate fractions. Both these procedures are still vital today. Al-muqabala means comparing and it is understood to refer to the reduction of positive terms by subtracting equal amounts from both sides of an equation. The Arabic phrase al-jabr wal-muqabala means performing algebraic operations, or more generally, the science of algebra.

Al-Khawarizmi’s algebra is much less advanced than that of the Greeks or Hindus. It covers only simple linear and quadratic equations. It is also purely rhetorical in style, unlike Diophantus, and it is presumed that he learnt much from Hindu and possibly Mesopotamian sources. The title of the work in English is ‘The Compendious Book of Calculation by al-jabr and al-muqabala’. To give the flavour of the rhetorical style, we will describe one of his problems.

One square and ten roots of the same amount to thirty nine dirhems; that is to say, what must be the square which, when increased by ten of its own roots, amounts to thirty nine.

Here is his solution.

You halve the number of roots, which in the present instance equals five. This you multiply by itself; the product is twenty five. Add this to thirty nine; the sum is sixty four. Now take the root of this, which is eight, and subtract from it half the

number of roots, which is five. The remainder is three. This is the root of the square you sought; the square is nine.

The problem to be solved is, in our modern algebraic notation:

x² + 10x = 39.

The procedure to be used is completing the square. As the equation begins with the term x², take half the coefficient of x, which is 5, and square it to get 25. Add this to the right hand side to get 64. Then we have

x² + 10x + 25 = (x + 5)² = 64.

Taking square roots, we get x + 5 = 8, and we subtract 5 from 8 to obtain x = 8 - 5 = 3. The negative solution is not considered, as negative numbers were not admitted as genuine numbers.

Al-Khawarizmi justifies his solution by a geometric construction using squares and rectangles. A similar, easier solution was already given by Euclid. In the second chapter of his algebra book, al-Khawarizmi explains rules for computing areas and volumes. Of particular interest are various rules for calculating the circumference of a circle. Let d be the diameter of a circle and p its circumference. Then three formulae are given for p in terms of d:

Now the rules really represent approximations to π, and none is exactly correct. The first rule amounts to the approximation, due to Archimedes. The second rule approx-

imates π by an approximation known to the Hindu mathematician Brahmagupta.

The third is a very accurate approximation π = 3.1416, already known in China in the third century.

In his book on algebra, al-Khawarizmi uses the word shay’, meaning thing or something, for the unknown quantity which is to be found. This was translated literally into Latin as cossa, meaning thing, and as a consequence, algebra became known in some countries as coss, signifying the method of finding the unknown quantity or thing. For

instance, in England, algebra was known initially as the Cossike Art and in Italy as Regola de la cosa (cosa is the Italian for thing). Al-Khawarizmi’s achievements in mathematics are not great compared with his Greek and Indian predecessors, but their timeliness made them highly influential and his writings stand as the starting point for much further development of algebra in Islamic countries. Translations of his work into Latin served as a basis for many popular medieval treatises on arithmetic.

The Arabs had performed a vital service in preserving much of the legacy of Greek mathematical culture, and by collecting and explaining many of the mathematical discoveries made in Asia. While it cannot be said that their own achievements in creating original mathematics were great, Western Europe owes them a great debt, as it was not until the 12th century that Western scholars became acquainted with the many classical Greek texts, through translations from Arabic sources. No significant contributions to mathematical theory had been made in Western Europe until the end of the 12th century, when things began to change. We may trace this turn in the pattern of events to the growth of trade between Italy and elsewhere. Banking assumed a key role in trade, as the bartering of goods was gradually replaced by monetary transactions. Merchants travelled further, and became acquainted with such Arabic scientific achievements as the Hindu-Arabic numeral system, as well as with their medicines and machines.

Merchants had to familiarize themselves with the elementary arithmetic of commerce. Traditionally, they employed the abacus, a board with counters or pebbles, for calculation, and the results they obtained were written into account books using Roman numerals. The Roman numerals were inefficient, although they remained popular well into the 14th century. They were however slowly replaced by Hindu-Arabic numerals, which proved to be much easier to manipulate. It is difficult to discover details of how the Hindu system of numerals evolved but the use of nine symbols and a dot to designate zero seems to have been established in parts of India at the beginning of the 7th century CE. The system had spread to China and the Middle East by the 8th century. The Arabs brought the system to Spain (much of it under the control of the Arabs) by the 10th century, and the numerals are found in a Spanish manuscript of 976. Their progress into the rest of Europe was delayed, as, for example, they are not recorded in France until 1275. Hindu-Arabic numerals were employed by Italian merchants from the 14th century, and they may be seen in existing Florentine account books of the early 15th century, in conjunction with Roman numerals. It was not until the late 15th century that Roman numerals ceased to

be used in these account books.

The towns of Florence, Genoa, Milan, Pisa and Venice were all centres of Italian trade, from which merchants ventured forth into Africa and the Near East in the 12th and 13th centuries. The beginnings of modern trade and capitalism can be traced to these times and activities. As we mentioned, traders working abroad became exposed to different cultural and scientific influences, and they brought news of their discoveries to their home cities. One especially noteworthy example of this cultural enrichment occurs in the work of Leonardo of Pisa, known as Fibonacci.

Leonardo was a member of the Bonacci family, and he styled himself filio Bonacci, which became Fibonacci. He travelled to Algeria with his father, who was in charge of a Pisan trading company there. He subsequently visited Egypt, Syria and Constantinople. During these business trips, he learned the method of calculating with Hindu-Arabic numerals. On his return to Pisa, in about 1200, Leonardo spent the next 25 years composing works that would explain the use of the Hindu-Arabic numerals in commercial activity, and also introduce what was known of algebra and geometry in the East. His work had thus a two-fold purpose: instruction in the practicalities of arithmetic for everyday life, and also to serve as an introduction to the theory and aesthetics of more advanced mathematics. The following five of his works have come down to us:

• Liber abbaci (1202)

• Practica geometriae (1220)

• Flos (1225)

• Letter to the philosopher Theodorus

• Liber quadratorum (1225)

Of these, the Liber abbaci was edited and printed in modern form by Baldassare Boncom-pagni in Rome between 1857 and 1862. Much of Leonardo’s work had been forgotten prior to this rediscovery. The name Liber abbaci refers to the art of calculation. (Note that in Italy, the calculating masters were called maestri d’abbaco.) The first seven chapters of the Liber abbaci introduce the new Hindu numerals. Following Arabic usage, the units are on the right, the fraction on the left of whole numbers. Examples teach the basic

operations of arithmetic. It takes a modern reader considerable effort to recognize what is signified in Leonardo’s notation. For example,

which we would certainly not imagine on a first look.

The next few chapters provide problems appropriate to merchants, for example, on currency conversion. There follow problems and puzzles leading to linear equations– sometimes determinate, sometimes indeterminate. Here is an example.

A man buys 30 birds: partridges, doves and sparrows. A partridge costs three silver coins, a dove two and a sparrow 1/2. He pays with 30 silver coins. How many birds of each type did he buy?

This is a problem of linear equations in three unknowns. Suppose that he bought x partridges, y doves and z sparrows. Then we have

x+y +z = 30

We are looking for solutions in integers for these two linear equations. Although not explicitly stated, we need to assume that x > 0, y > 0, z > 0. We in fact have a Diophantine system of linear equations. Multiplying the second equation by 2 we get

x +y +z = 30

6x + 4y + z = 60

and subtracting, 5x + 3y = 30. As 3 divides 30 and 3y, it divides 5x, and since 3 and 5 are relatively prime, 3 divides x. Similarly, 5 divides y. As we assume that x > 0, y > 0, the only possible solution is x = 3, y = 5 and then z = 22. If we allowed solutions with various unknowns equal to 0, there are two other possibilities. Problems of this type were popular for many centuries, and indeed had appeared in Asian sources centuries before Leonardo wrote his work.

Leonardo also gives a method for approximating square roots, namely,

approximately, provided that r is small compared with a. This follows since

and r²/4a² will be quite small if r/2a is small. For example, if we wanted to approximate , we could take a = 3, r = 1 and get 3.16 (compared with a more accurate value of 3.14). This approximation is a special case of Newton’s method, although much more elementary.

Leonardo of Pisa is best known today for what we call the Fibonacci sequence. In modern terminology, the sequence arises from the following problem posed in the Liber abbaci.

Suppose that at the beginning of the year, a pair of rabbits is left on an island. These rabbits take two months to mature and after two months they produce another pair. Thereafter, they continually produce another pair each succeeding month. Each newborn pair takes two months to mature and after two months they also produce another pair each succeeding month. Assuming that no deaths occur, find the number of pairs of rabbits on the island after n weeks.

Let u_n denote the number of pairs after n months. Starting at n = 0, we have u₀ = 1. Similarly, u₁ = 1, but u₂ = 2, as the initial pair have produced a new pair. Then u₃ = 3, but u₄ = 5, as both the original pair and the new pair have produced a new pair each. It is not too difficult to see that u_n+2 satifies the equation

un+2 = un+1 +un,

which we call a recurrence relation. From this, we can see that the next number in the sequence is found by adding its two immediate predecessors, and the Fibonacci sequence is

1, 1, 2, 3, 5, 8, 13, 21,...

While Leonardo did not investigate this sequence in generality using abstract mathematics, it may be worthwhile to detour briefly and describe some of its background, as it is quite significant. It has been studied in enormous detail and numerous research papers have been written about the Fibonacci and related sequences. One very special feature is the following. If we look at the ratio of a term in the sequence to its immediate predecessor, we obtain the sequence of fractions

and in decimal form this is

1, 2, 1.5, 1.66, 1.6, 1.62, 1.615, . . .

It is not hard to see that the ratios actually converge to a number equal to 1.618 .... We can be more precise about this number. It is actually equal to

We can see why this number occurs. Let’s assume that. Then we also will have

. Taking the recurrence relation in the form u_n+1 = u_n + u_n-1, we obtain on division by u_n that

Taking limits as n → ∞,

which implies that

Multiplying by r, we obtain the quadratic

r² - r - 1 = 0,

whose solution is. As the limit r is clearly bigger than 1, we must take the

solution, as explained. An explicit formula for the terms of the Fibonacci

sequence can be given in terms of the roots of the quadratic above. Indeed, a standard method for solving such recurrence relations shows that

This formula is difficult to use directly and it is often possible to use other techniques to prove facts about the Fibonacci sequence.

The roots of the quadratic x² - x - 1 = 0 had already occurred in a celebrated problem of ancient Greek geometry. Suppose that we have a line segment AB. A point C on this line segment, chosen so that AC is greater than CB, is said to divide the segment in extreme and mean ratio if

Thus the ratio of the whole segment to the larger segment equals the ratio of the larger segment to the smaller. Let |AB| = a and |AC| = b. Then |CB| = a - b and we obtain

Hence a² - ab = b², and setting r = a/b, we get r² - r - 1 = 0, with the value for r as before. We choose C so that r|AC| = |AB|, so that the length of AC is 1/r times the total length. Now 1/r equals r - 1 by the properties of the quadratic, so |AC| is about 0.618 times the total length. The Greeks said that the line segment had been divided according to the golden section. This golden section, equal to 0.618 approximately, or

precisely, was especially appealing to the Greeks. It occurs in another form in designing rectangles of pleasing proportions. Consider a rectangle with sides of length a and b, where a > b. The Greeks held that such a rectangle was especially elegant if the ratio of the sum of the two lengths to the larger length equalled the ratio of the larger to the smaller length, in other words, if

We can again solve for a/b to obtain the numbers occurring in the golden section. It is claimed that the Greeks built temples to reflect the golden section, for observers, on being asked to choose a rectangular design which they liked best, often decided upon that which diplayed the golden section. The ratio is also seen in designs in other civilizations, and occurs in a variety of mathematical contexts. In 1509, the Italian mathematician Fra Luca Pacioli published a book entitled De divina proportione, which is devoted to the study of the golden section and the five regular solids. The book is illustrated by noteworthy illustrations, perhaps produced by Leonardo da Vinci. The golden section was also employed in the work of such Renaissance artists as Piero della Francesca.

It might be added that the Greeks discovered a way of dividing a line segment in extreme and mean ratio by use of a ruler and compass. It is related to their construction of a regular pentagon, in which any two diagonals cut each other at points dividing the diagonal in extreme and mean ratio. It is unlikely that Leonardo was aware of any connection between his sequence and this problem of classical geometry.

In the Liber quadratorum, Leonardo recounts his solution to a problem proposed to him by John of Palermo, when he visited the court of the Emperor Frederick II. The question was: find a rational number x such that x² + 5 and x² - 5 are both squares.

Leonardo’s solution is to take

which leads to

This type of problem was not new, as it has the flavour of Diophantus, and Arabian mathematicians had already considered similar problems. Note that the solution given by Leonardo is certainly not unique.

We can sum up Leonardo’s work by noting that he learned much from his contact with the Arabs and Byzantines, and he transmitted this knowledge to his fellow countrymen. His examples owe much to Diophantus, and to Euclid, as well as to such Islamic mathematicians as al-Khawarizmi. We even find traces in the Liber abbaci of old Egyptian problems, retaining their original numbers. His works were copied for several centuries, and he was frequently quoted by later mathematicians such as Pacioli and Cardano. Writing as late as 1500, indeed, Pacioli acknowledged that he depended almost entirely on Leonardo’s work. Leonardo’s exposition of the methods of computation had an immediate influence, and he is considered to be the teacher of the masters of calculation in Italy (maestri d’abbaco). The more theoretical parts of his algebra and number theory were mainly ignored until a greater interest in mathematics for its own sake developed a few centuries later. Leonardo was formerly credited with making many personal discoveries, but this is now discountenanced. Nonetheless, his importance is great in view of the depth of ignorance of mathematics prevalent in Western Europe at the time he wrote. Leonardo is considered to be the first great mathematician of Christian Western Europe.