Archimedes' law, "Eureka!", Archimedes screw, "Give me a fulcrum and I will turn the Earth!" And finally: "Don't touch my blueprints!" These words and expressions exhaust almost everything that we learn about the famous Syracuse in high school.
We know that Archimedes is a great mechanic of antiquity and a hero of resistance to the Romans. But this legendary man, above all, was one of the greatest Greco-Roman mathematicians.
Ancient mathematician
Far from being self-taught, he received an excellent education in Alexandria, the main scientific center of the time. Archimedes spent all his life in correspondence with scientists from there. And in the legendary Alexandria of the 3rd century BC, the achievements were collected not only of the peoples of the Mediterranean basin, but, thanks to the campaigns of Alexander the Great, also of many mysterious civilizations of Mesopotamia, Persia and even the Indus valley.
However, even before the Renaissance, when interest in serious mathematics arose for the first time in many hundreds of years, very few original works by Archimedes survived. Not ancient Greek manuscripts, but at least copies, translations, or just quotes. Not to mention the detailed proofs of formulas and theorems. For a long time, Archimedes the mathematician was known to scientists no more than Einstein was to a schoolboy: he was very smart, did a lot of something very important - and that was all.
Scant information has been preserved that in the treatise "The Method of Mechanical Theorems" Archimedes explained in detail his most amazing mathematical discoveries. Only now this treatise for about a thousand years has been listed among those forever lost to mankind.
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The first glimmer of hope
One of the famous 19th century Bible scholars, Konstantin von Tischendorff, worked in the libraries of Constantinople in the 1840s. From there, he brought home a page of a manuscript that interested him, on which he found some half-erased complex mathematical calculations in Greek, similar to the work of Archimedes.
Alas, the scientist simply tore a page out of the book when the librarian was looking the other way. This act of vandalism was in vain - neither Tischendorf nor anyone else attached any particular importance to the text.
Finally found
The real merit of the opening of the book, noticed by Tischendorf and later famous as Archimedes' Palimpsest, belongs to an obscure Turkish librarian. He cited an excerpt from strange mathematical calculations in a catalog sent around the world, which fell into the hands of the Danish historian and philologist Johan Ludwig Heiberg. He was so intrigued that he immediately left and got acquainted with the book in person in 1906. What he saw shocked him to the core.
At first glance, a rather ordinary liturgical book of the 13th century from the monastery of Mar Saba in the Jerusalem desert. But if you look closely, across the liturgical text were barely noticeable lines in earlier Greek, replete with scientific and philosophical terms.
The term "palimpsest" means "newly scraped". Due to the value of parchment in the Middle Ages, unwanted books were often split into separate sheets, cleaned of ink, then stitched together and a new text was written. In Archimedes' Palimpsest, each of the sheets was still folded in half to form a smaller book.
Therefore, the new text was written across the old. As a writing material, an unknown monk used Byzantine collections of scientific works from about 950. But the cleanup was not very thorough and the original text was visible.
Heiberg's joy knew no bounds when he realized that a greater number of the original texts are copies of the works of Archimedes and that among them the longed-for "Method …" is present almost in full.
The library forbade taking the manuscript out of its premises (who can blame them after Tischendorf's visit?), So the scientist hired a photographer to reshoot the entire book for him.
Then, armed with nothing but a magnifying glass, Heiberg set about deciphering the photocopy in detail. The final result, and then the English translation, was published in 1910-1915. The discovery caused a lot of noise and even got on the front page of the New York Times.
Continuation of the adventure
But then the First World War began, at the end of which the Ottoman Empire ceased to exist. Amid the devastation in Constantinople, which soon became Istanbul, there was absolutely no time for ancient manuscripts. In the 1920s, a huge amount of Turkish values moved to Europe. Only much later was it possible to establish that a certain Frenchman was able to acquire and take out Palimpsest to Paris, where the book for a long time became just a collectible wonder.
Interest in the works of Archimedes was revived only in 1971. Nigel Wilson, a specialist in ancient Greek culture from Oxford, drew attention to some words in a document from the Cambridge Library (the same page of Tischendorf), which, in his opinion, were used only by Archimedes.
Wilson received permission to study the document more thoroughly and not only confirmed that the page belongs to Palimpsest, but also proved that with the help of previously unavailable technologies (such as ultraviolet lighting) the text can be completely restored. The only thing left to do was to find the code that had sunk into oblivion. The academic world began intensive searches, but they did not lead to anything.
Final return
In 1991, an employee of the Christie's auction house received a letter from a French family wishing to put up for auction the alleged Palimpsest. The news was received with a fair amount of skepticism, but the subsequent examination gave an unexpectedly positive verdict. As a result of a sensational auction, the document was sold to an anonymous billionaire for $ 2 million.
All the scientists of the world held their breath - after all, at the will of the new owner, the book could simply be closed in the safe forever. Fortunately, the fears were in vain. When Dr. Will Noel, manuscript curator at the Walters Museum of Art in Baltimore, USA, asked the owner's agent for the code to be reviewed, his initiative was enthusiastic. The billionaire made his fortune on high technologies, and therefore he himself was not so far from science and its interests.
From 1999 to 2008, a whole group of specialists worked with Archimedes' Palimpsest. The document, which by that time turned out to be in a monstrously poor condition, was carefully restored. When the codex was embroidered into separate sheets, it was discovered that many lines of Archimedes' text were hidden inside the binding and therefore previously inaccessible. Among them were key points in theorem proving. And the latest scanning methods (ranging from infrared to X-ray) and computer processing have helped to restore everything that is possible, even invisible letters.
But why is it so important ?! It was known a long time ago that Archimedes often combined large numbers and very small quantities. For example, to calculate the length of a circle, he inscribed it in a polygon with a large number, but a small length of sides. This brings us closer to the important in mathematics infinitely large and small quantities. But was Archimedes able to operate with true mathematical infinity?
Infinity seems to be just an abstraction. But it underlies mathematical analysis, which is fundamental to virtually any modern engineering, physical, and even economic calculations. Without it, it is impossible to build a skyscraper, design an airplane, or calculate a satellite's entry into orbit. Modern mathematical analysis was pioneered by Newton and Leibniz at the end of the 17th century, and almost immediately the world began to change.
It was working with infinity that gave our civilization its technological power. Thanks to the discovery and restoration of Palimpsest, today we know for sure that for Archimedes, infinity was a verified working tool. His calculations are flawless, and the proofs withstand the rigorous testing of modern mathematicians. It's funny, but Archimedes quite often uses what in modern mathematics is called Riemann sums, in honor of the famous mathematician … of the XIX century.
True, some of his methods clearly came "from another world", for the modern scientist they are alien and unnatural. They are no worse and no better than the current ones, they are just different. This is higher mathematics, "genetically" in no way connected with modern mathematics.
What have we lost?
It is a pity, but the discovery of the forgotten Archimedes manuscript was too late. In the 20th century it became a sensation, but only in the history of science. What would have happened if this manuscript had fallen into the hands of scientists hundreds of years earlier? If Newton had read it while still at school? Or Copernicus? Or Leonardo da Vinci?
Even for mathematicians of the 19th century, this work would be of more than academic interest. For scientists of the XVII-XVIII centuries, its significance would be enormous. And in the Renaissance, having fallen into the right hands, he would simply have produced the effect of an exploding bomb, completely redrawing the future development of mathematics and engineering.
What have we lost, having lost access to just one ancient book for centuries? Cities on Mars, interstellar spaceships, environmentally friendly thermonuclear reactors? One can only guess about this.
Georgy KHALETSKY