From "silicon valley" of antiquity to the inventor of analytical geometry
Are we being asked to turn the equation ax+b=0 to solve, we hardly think about it: immediately we solve for x, although a or. B could have been the unknown we were looking for. Certainly: it has become ingrained in mathematical problems, the letter x as the coarse to be determined. But why actually? Where does this convention come from that we automatically pass on from generation to generation in school?? To understand this, we have to take a trip into the history of mathematics and to a legendary city.
Our present mathematics draws from many sources: from the astronomical observations of the babylonians, from the geometrical knowledge of the egyptians, but above all from the investigations of the greeks, who found their way back to europe via the arab world. The hellenistic world left behind not only the philosophical treatises of a plato or an aristotle, but also the mathematical results of pythagoras or eratosthenes.
While athens had the greatest cultural influence on the european continent in antiquity, another mediterranean city played an equally important role in the development of mathematics and astronomy. It was alexandria, located on the nile delta and founded by alexander the rough himself around the year 331 b.U.Z. Founded. The macedonian had conquered egypt at that time and expelled the persian rulers. He brought to power the ptolemaic dynasty, which was.H. Greek kings who henceforth ruled over the pharaonic empire until the death of cleopatra. It was also in alexandria, perhaps less than a hundred years after the founding of the city, that euclid wrote the 13 books of the "elements" the first work in which the axiomatic method was used in a fundamental way.
Alexandria was a kind of "silicon valley" of antiquity. The city could boast one of the seven wonders of the world: the pharos of alexandria towered about 150 meters into the air. For centuries, only the pyramids of giza stood even higher. In the famous library of alexandria were kept all the important books (actually manuscript scrolls) of antiquity and, when a ship docked at the port, all the manuscripts brought with it were immediately copied in the library. Next door, scholars conducted their research in the so-called museion, which was the "first university in the world" has been called.
The library was used for.B. The old testament was translated into greek and the first philological studies were undertaken. Alexandria was the largest and most dynamic city of the ancient world at that time, until rome overtook it. It was in this context that mathematics and natural sciences flourished. Alexandria was also the stage of world history: the dramatic triangle between cleopatra, julius casar and marcus antonius began and ended here, before egypt and the entire middle east became roman provinces. Such a crude drama later shakespeare did not leave untreated.
Geometrization of mathematics
Euclid’s "elements" are so important for the history of mathematics, because they show a clear systematic way for the solution of several numerical problems, namely the axiomatic geometrization. Instead of solving numerical problems by equations, one can set an equivalent geometrical task. The unknown coarseness can be equated with the length of a distance. Your square corresponds to the flatness of a square with that distance as edge. The volume of a cube with the same edge length is equal to the third power of the unknowns, etc.
If one reads the "elements", especially in modern editions, which show everything in color "modernity" of the test methods.1 one can only marvel at the fact that, although more than 22 centuries separate us from euclid, we solve many geometric problems today in exactly the same way as we did in the "elements" shown.
However, the geometrization of mathematics, despite all its successes, has led to impasses. Some mathematicians e.G.B. Did not allow to add the square of a number with the number, because you can not mix a long with a flat "should" (if one thinks of mab units). Especially the negative numbers caused headaches. Although they can appear as a solution of equations, z.B. In x+4=2, sometimes the imagination was not sufficient to handle such numbers with segments or elements. To identify mathematical objects. By the way, one should not believe that negative numbers were difficult to handle only in ancient times. Until after the renaissance, working primers were sold especially for working with negative numbers.