Leonardo Pisano Fibonacci

The "Middle Ages" in our consciousness associate with the concept of inquisition orgy, campfires, on which witches and heretics are incinerated, and crusades for "the body of God". The science in those times obviously was not "in a center of society attention". In these conditions appearance of the mathematical book "Liber abaci" ("the book about an abacus"), written in 1202 by the Italian mathematician Leonardo Pisano (by the nickname of Fibonacci) was the relevant event in the "scientific life of society".

Who was Fibonacci? And why his mathematical works are so important for the West-European mathematics? To answer these questions it is necessary to reproduce the historical epoch, in which Fibonacci lived and worked.

It is necessary to note that the period since the 11th until the 12th centuries was the epoch of brilliant flowering of the Arabian culture but at the same time and the beginning of its downfall. To the end of the 11th century that is to the beginning of the Crusades the Arabs were, doubtlessly, the most educated people in the Glob surpassing in this respect of their Christian enemies. Even before the Crusades the Arabian influence penetrated to the West. However the greatest infiltration of the Arabian culture to the West began after the Crusades, which weakened the Arabian world, but on the other hand, boosted the Arabian influence on the Christian West. Not only the Palestinian cotton and sugar, pepper and black wood of Egypt, self-color rocks and specifies of India are searched and appreciated by the Christian West in the Arabian world. He starts to assess properly the cultural heritage "of the great antique East", the saver of which became the Arabian culture. The world opened by the West researchers could not blind them by the paints and scientific achievements and in the West world the demands for Arabian geographical maps, tutorials on algebra and astronomy, Arabian architecture rise rapidly.

One of the most interesting persons of the Crusades epoch, a harbinger of the Renaissance epoch, was the emperor Fridrich Gogenstaufen, an apprentice of the Sicilian Arabs and an admirer of the Arabian culture. At his palace in Pisa the greatest European mathematician of the Middle Ages Leonardo Pisano (by the nickname of Fibonacci that means the son of Bonacci) lived and worked.

Leonardo Pisano Fibonacci (1170-1228)
Leonardo Pisano Fibonacci (1170-1228)

About Fibonacci life it is known a little. Even the exact date of his birth is obscure. It is supposed, that Fibonacci was born in the eighth decade of the 12th century (presumptively in 1170). His father was a merchant and a government official, the representative of the new class of the businessmen generated by the "Commercial Revolution". In that time the city of Pisa was one of the largest commercial Italian centers actively cooperating with the Islam East, and Fibonacci's father traded in one of the trading posts, founded by Italians on the northern coast of Africa. Due to this circumstance he can give his son, the future mathematician Fibonacci, good mathematical education in one of the Arabian educational institutions.

One of the known historians of mathematics Moris Cantor called Fibonacci "as the brilliant meteor flashed past on the dark background of the West-European Middle Ages". He supposed that, probably, Fibonacci perished during one of the Crusades (presumptively in 1228), accompanying the emperor Fridrich Gogenstaufen.

Fibonacci wrote several mathematical works: "Liber abaci", "Liber quadratorum", "Practica geometriae". The book "Liber abaci" is the most known of them. This book was published at the life of Fibonacci in two issuings in 1202 and 1228. The book consists of 15 sections, which sequentially treat: about the new Hindu ciphers and how with their help to represent numbers (section 1); about multiplication, addition, subtraction and division of numbers (the sections 2-5); about multiplication, addition, subtraction and division of fractional numbers (the sections 6-7); about finding of the prices of the goods and about their exchange, about the rule of comradeship and about the rule of the "double false situation" (the sections 8-13); about finding of square and cube roots (the section 14); and, at last, about the rules relating to geometry and algebra (the section 15). Note that Fibonacci conceived the book as the manual for traders, however by its significance the book came out far beyond the trade practice and, in essence, presented by itself the peculiar mathematical encyclopedia of the Middle Ages epoch. Since this point of view the 12th section is of an especial interest. In this section Fibonacci formulated and solved a number of mathematical problems interested since point of view of general perspectives of mathematics development. This section takes an almost third part of the book and, apparently, Fibonacci gave to it the greatest significance and in this section Fibonacci shown the greatest originality.

The problem of "rabbits reproduction" is the most known among the problems formulated by Fibonacci. This one resulted in discovery of the numerical sequence 1, 1, 2, 3, 5, 8, 13, ..., called Fibonacci numbers.

The second problem is interesting in historical sense and is called the "problem about seven old women". The old women go to Rome, each woman has 7 mules, each mule bears 7 bags, in each bag 7 breads lie, for each bread 7 knives lie, each knife cut 7 chunks of bread. How much a total number of all listed is equal? In historical relation this problem is interesting by the fact that it is similar to the problem, which met in Rind papyrus (Egypt).

The next problem considered by Fibonacci is called the "problem about choice of the best system of standard weights for weighing on the beam balance" or simply the "weighing problem". In the Russian historical-mathematical literature the "weighing problem" is known under the title of "Bashet-Mendeleev's problem", called so in honor of the French 17th century mathematician Bachet de Mesiriaque, who placed this problem in the "Collection of pleasant and entertaining problems" (1612), and of the outstanding Russian chemist Dmitry Mendeleyev, who interested in this problem when he worked as a director of the Main Standard and Weight Bureau of Russia.

The essence of Bashet-Mendeleev's problem consists of the following: for what system of standard weights, taking them by one, it is possible to weigh every possible freight Q from 0 up to maximum freight Qmax that the value of maximum freight Qmax, would be greatest among all possible variants? Two variants of this problem are known: (1) when we can lay the standard weights on the free cup of the balance; (2) when we can lay the standard weights on the both cups of the balance.

For the former case the "optimal system of standard weights" is reduced to the binary system of weights: 1, 2, 4, 8, 16, ... and the "optimal" measurement algorithm, arising in this case, "generates" the classic binary number system underlying modern computers.

For the latter case the "optimal system of standard weights" is reduced to the "ternary" system of weights: 1, 3, 9, 27, 81, ... and this case "generates" so-called ternary symmetrical number system used in the "ternary" computer "SETUN" designed in the 50th of the 20th century at the Moscow university.

The methodological significance of the "weighing problem" consists first of all of the fact that it is one of the first "optimization" problems in the history of mathematics. Secondly, it concerns to the "problem of measurement ", that is, to one of the fundamental mathematical problems. In third, it is connected to the problem of number systems, one of the fundamental problems of modern computer science. Just the development of this problem from this points of view resulted in development of so-called "algorithmic measurement theory"; we will tell later about this theory.

But let's return again to Fibonacci and his mathematical works. Though Fibonacci was one of the brightest mathematical minds in the history of the West-European mathematics, however his contribution in mathematics is belittled undeservedly. The significance of Fibonacci's mathematical creativity for mathematics is assessed properly by the Russian mathematician Prof. Vasiljev in his book "Integer Number" (1919):

"The works of the learned Pisa's merchant were so above the level of mathematical knowledge even of the scientists of that time, that their influence on the mathematical literature becomes noticeable only in two centuries after his death at the end of the 15th century, when many of his theorems and problems are entered by the Leonardo da Vince friend, professor of many Italian universities Luca Pachioli in his works and in the beginning of the 16th century, when the group of the talented Italian mathematicians: Ferro, Cardano, Tartalia, Ferrari by the solution of the cubical and biquadrate equations gave the beginning of the higher algebra".

It follows from this statement that Fibonacci almost for two centuries anticipated of the West-European mathematicians of his time. Like to Pythagor who got his "scientific education" in the Egyptian and Babylonian science and then promoted to transfer of the obtained knowledge to the Greek science, Fibonacci got the mathematical education in the Arabian educational institutions and many from the obtained there knowledge, in particular, the Arabian-Hindu decimal notation he attempted to introduce to the West-European mathematics. And like to Pythagor the historical role of Fibonacci for the West science consists of the fact that he by his mathematical books promoted to transfer of the Arabian mathematical knowledge to the West-European science and by that he created fundamentals for further development of the West-European mathematics.