Archimedes ‘Law, Eureka !, Archimedes’ screw, “Give me a point of support, and I will turn the Earth!” And, finally: “Do not touch my drawings!” These words and expressions exhaust almost everything that we learn about the famous Syracuse in the middle school.

We know that Archimedes is a great mechanic of antiquity and a hero of resistance to the Romans. But this legendary man, first of 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 his entire life in correspondence with scientists from there. And in the legendary Alexandria of the 3rd century BC, the achievements of not only the peoples of the Mediterranean basin were collected, but, thanks to the campaigns of Alexander the Great, as well as the many mysterious civilizations of Mesopotamia, Persia and even the Indus Valley.

However, even before the Renaissance, when for the first time in many hundreds of years there was an interest in serious mathematics, very few original works of Archimedes reached. Not Greek manuscripts, but at least copies, translations or just quotations. Not to mention detailed proofs of formulas and theorems. Archimede-mathematician was known to scientists for a long time no more than Einstein to a schoolboy: very intelligent, he did a lot of something very important – and that’s it.

The scanty information about the fact that in the treatise “Method of mechanical theorems” Archimedes explained in detail his most amazing mathematical discoveries. Only here this tract for about a thousand years was listed among the forever lost to mankind.

First glimmer of hope

One of the famous researchers of the Bible of the XIX century, Constantine von Tischendorf, in the 1840s he worked in the libraries of Constantinople. From there he brought home the page of the manuscript that interested him, on which were found some half-complicated mathematical calculations in Greek, similar to the work of Archimedes.

Alas, the scientist just tore the page from the book when the librarian looked the other way. This act of vandalism was vain – neither Tischendorf nor anyone else gave the text any special significance.

Finally found!

The real merit of the opening of the book, seen by Tischendorf and later known as the Palimpsest of Archimedes, belongs to the unknown Turkish librarian. He quoted excerpts from strange mathematical calculations in a catalog distributed throughout the world, which fell into the hands of the Danish historian and philologist Johan Ludwig Heyberg. He was intrigued enough that immediately left and got acquainted with the book personally in 1906. What he saw shook him to the core.

At first glance, a fairly mediocre liturgical book of the 13th century from the Map Saba monastery in the Jerusalem desert. But if we look closely, across the liturgical text there were barely noticeable lines on the earlier Greek, abounding in scientific and philosophical terms.

The term “palimpsest” means “newly scraped”. Because of the value of parchment in the Middle Ages, unnecessary books were often divided into separate sheets, cleaned of ink, then stitched and written a new text. In the Archimedes Palimpsest, each of the sheets was still folded in half to get a smaller book.

Therefore, the new text was written across the old one. As a writing material, an unknown monk used Byzantine collections of scientific works around 950. But the cleaning was not very thorough, and the original text was noticeable.

There was no limit to the joy of Heyberg when he realized that more original texts were a copy of Archimedes’ works and that there was almost a coveted “Method …” among them.

The library banned the manuscript from its premises (who can blame them after Tischendorf’s visit?), So the scientist hired a photographer who photographed the whole book for him.

Then, armed only with a magnifying glass, Heiberg took up the detailed decoding of the photocopy. The final result, and then the English translation, were published in 1910-1915. The opening caused a lot of noise and even got on the editorial “The New York Times”.

Continuation of adventure

But then the First World War began, at the end of which the Ottoman Empire ceased to exist. Among the devastation in Constantinople, which soon became Istanbul, it was completely down to the ancient manuscripts. In the 1920s, a huge number of Turkish values moved to Europe. Only much later it was possible to establish that a certain Frenchman was able to acquire and take out Palimpsest to Paris, where the book for a long time had become simply a collector’s curiosity.

Interest in the works of Archimedes revived only in 1971. An expert on ancient Greek culture from Oxford, Nigel Wilson drew attention to some words in a document from the Cambridge Library (that 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 inaccessible technologies (such as ultraviolet illumination), the text can be completely restored. It remained a small matter 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 auction house “Christie” received a letter from a French family who wished to auction the alleged Palimpsest. The news was received with a fair amount of skepticism, but the subsequent examination made an unexpectedly positive verdict. As a result of sensational bidding, the document was sold to an anonymous billionaire for $ 2 million.

All scientists of the world held their breath – after all, at the behest of the new owner, the book could be simply closed in a safe forever. Fortunately, fears were in vain. When Dr. Will Noel, curator of the manuscripts of the Walters Art Museum of Art in Baltimore (USA), asked the owner’s agent to provide a code for study, his initiative was received with enthusiasm. The billionaire earned his fortune on high technologies, and therefore he was not so far from science and her interests.

From 1999 to 2008, a whole group of specialists worked with Archimedes’ Palimpsest. The document, which by that time was in a monstrously bad condition, was carefully restored. When the code was expanded into separate sheets, it was found that many lines of Archimedes’ text were hidden inside the binding and therefore previously inaccessible. Among them were the key points in the proof of theorems. And the newest methods of scanning (ranging from infrared to X-ray) and computer processing helped restore all that is possible, even invisible letters.

But why is it so important ?! Once upon a time it was known that Archimedes often combined large numbers and very small values. For example, to calculate the length of a circle, he inscribed it into a polygon with a large number, but a short side length. This brings us to infinitesimal and small values important in mathematics. But was Archimedes able to operate with true mathematical infinity?

It seems that infinity is just an abstraction. But it lies at the basis of mathematical analysis, fundamental for virtually any modern engineering, physical and even economic calculations. Without it, you can not build a skyscraper, design an aircraft, or calculate the satellite’s exit into orbit. Modern mathematical analysis was laid down by Newton and Leibniz in the late 17th century, and almost immediately the world began to change.

It was work 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 impeccable, and the evidence can be checked by modern mathematicians. Funny, but Archimedes quite often applies what in modern mathematics is called Riemann sums, in honor of the famous mathematician … XIX century.

True, some of his methods have clearly come “from another world”, for a modern scientist they are alien and unnatural. They are no worse and no better than the present, they are just different. This is the higher mathematics, “genetically” in no way connected with the modern.

What have we lost?

It’s a pity, but the discovery of the forgotten Archimedes manuscript is too late. In the XX century it became a sensation, but only in the history of science. And what would happen if this manuscript fell into the hands of scientists hundreds of years earlier? If it were still on the school bench read Newton? Or Copernicus? Or Leonardo da Vinci?

Even for mathematicians of the XIX century, this work would be more than academic interest. For scholars of the 17th and 18th centuries, its significance would be enormous. And in the Renaissance, having got into the right hands, he would simply have produced the effect of a bomb exploding, completely redesigning the future development of mathematics and engineering thought.

What have we lost, losing for ages access to just one antique book? Cities on Mars, interstellar spaceships, environmentally friendly thermonuclear reactors? This can only be guessed.