## Table of contents:

- The wisdom of the ancients
- Archimedes' law
- Knowledge of the Ecumene
- Lost evidence
- Archimedes method
- Biblical scholar
- Amazing document
- Codex from oblivion
- Real nightmare
- Grains of sand in the universe
- Integrals of Archimedes
- Exhaustion method
- A bitter lesson

It is useful to look at this history from the point of view of the New Chronology, which is used by the entire scientific world without exception. Yes, this is not a misprint, the modern official history is the result of the New Chronology of Scaliger and Petavius, who worked on compiling the historical annals of the planet in the 16-17 centuries.

## The wisdom of the ancients

One has only to look at the portraits or busts of respectable pundits, which often illustrate the relevant paragraphs: high foreheads, wrinkled faces, serious eyes, solid disheveled beards - and then compare them with what is presented in the same paragraphs as the highest achievement of these scholars to chuckle with a mixture of arrogance and contempt.

Ha! They pondered and worked all their lives, read countless works of other thinkers, argued with their own kind in order to create some kind of Thales' theorem or Pascal's law, which now any child of not the highest grades learns in a few lessons. Isn't this clear evidence of progress?

No, no, such a disdainful attitude is never presented explicitly, on the contrary, in words our books extol the wisdom of the ancients in every possible way. However, it is worth adding two and two, and even the most lagging schoolchild will realize: if this is wisdom, then what was stupidity in those days ?! How primitive our ancestors were!

It is in this light that the notions that a few thousand years ago around the world rode savages in loincloths with crudely carved stone axes, for whom even a bow and arrow seemed the pinnacle of technological genius, seem very plausible. And even earlier? Forget it! Monkeys, just monkeys. Some contradictions with this picture of the development of civilization - for example, the "dark ages" of medieval Western Europe or the amazing "seven wonders of the world" seem to be nothing more than exceptions that prove the rule.

## Archimedes' law

But how justified is such an exaltation over the geniuses of the past centuries? Is it really that if one of them got in somehow in our day, then any high school student would easily compare with him in terms of mental development? And he could have hit him on the spot with some kind of logarithm or integral?

Let us turn to one of the most familiar thinkers of the ancient world. Archimedes. Everyone knows his story, right? He is featured in countless books and popular science films, even in several children's cartoons. A funny old man who ran around the city naked screaming "Eureka!"

With the help of this principle, later called "Archimedes' law," he learned to measure the volume of bodies of arbitrarily complex shapes. And along the way, he helped the tyrant Syracuse bring to the surface a deceiving jeweler who made a custom-made crown not of pure gold, but of an alloy of gold and silver. He was also a famous mechanic, the author of "Archimedes' screw" and numerous military machines and mechanisms that terrified the ancient Roman invaders. Those, however, despite all the cunning combat devices still somehow took Syracuse, and poor Archimedes died at the hands of an ignorant Roman soldier for demanding "not to touch his blueprints."

And, here, he also said: "Give me a fulcrum, and I will turn the Earth!" - which, despite its impressive sound, was nothing more than an illustration of the simplest mechanical principle of the lever. Well, that's probably all, right?

## Knowledge of the Ecumene

Alas, and not nearly so. Any more or less serious biography will tell us that Archimedes was not only an outstanding philosopher, naturalist and inventor, but, above all, one of the greatest mathematicians of the Greco-Roman era. He was far from self-taught, but received an excellent education in Alexandria of Egypt, the main scientific center of that time, and all his life was in correspondence with scientists from there.

The amount of knowledge available in Alexandria of the 3rd century BC exceeds any imagination, since there were collected not only the achievements of all the peoples of the Mediterranean basin, but, thanks to the campaigns of Alexander the Great, also many mysterious civilizations of Mesopotamia, Persia and even the Indus valley. So, through Archimedes, we can hope to at least slightly touch the knowledge of almost the entire "Oycumene".

Moreover, historians of science reasonably believe that we know much more about Archimedes than about any other ancient mathematician. True, they immediately add that we know practically nothing at all about the others. So we know very little about Archimedes either. Of course, the excellent mathematical reputation of Archimedes did not raise doubts among anyone for millennia, but the further the more questions arose as to what results and, most importantly, HOW they were achieved.

## Lost evidence

The fact is that very few of the original works of Archimedes have survived not only to our days, but even to the Renaissance, when, for the first time in many hundreds of years, interest in serious mathematics arose. This, of course, is not about manuscripts written by his own hand, but at least about reliable copies of copies or full-fledged translations into other languages.

Unfortunately, a huge part of the heritage of antiquity was preserved only in quotations cited by other, sometimes much later authors, and this applies not only to Archimedes, but also to absolutely all other remarkable ancient scientists and philosophers. What we think we know about them is only a very small part of what they actually achieved. In addition, this small part contains a myriad of accidental and deliberate distortions of many scribes, translators and commentators, not all of whom were equally honest and conscientious.

Moreover, like many mathematicians of the early eras, Archimedes did not always provide detailed proofs of his formulas and theorems in his works. This was due both to the fact that no proof is required for practical application, and to the fact that there has always been a circle of envious people who want to appropriate a significant result for themselves. Keeping the method of proof in secret made it possible to confirm the authorship or to deny the authorship of the impostor, if the need arose. Sometimes, to further confuse the situation, false evidence was released with deliberately introduced inaccuracies and errors.

Of course, when the result received general acceptance, the correct evidence was still published, but, for obvious reason, the number of manuscripts that recorded them was much less than the number of those where only the final decision was given. It was complicated by the fact that in ancient Greek mathematics, drawings not only illustrated the text of the proof, but were themselves an essential part of it - and not every scribe was skillful enough in copying complex geometric shapes. Because of this, much of the evidence was lost forever.

## Archimedes method

For about a thousand years, among such works forever lost to mankind, there was also the treatise of Archimedes "The Method of Theorems of Mechanics", often known simply as "Method". It was in it that Archimedes explained in detail how he achieved some of his most surprising results.

Its significance for understanding the legacy of this ancient Greek thinker is so great that historians of science sometimes call this treatise "a cast of the brain of Archimedes." Without access to at least excerpts from this text, it was considered almost impossible to determine the true level of Archimedes' mathematical knowledge and skills.

The first glimmer of hope that this work may have survived, appeared by the middle of the 19th century. The capture of Egypt by the Napoleonic army and the export from there to Europe of a huge amount of cultural values awakened among enlightened people an interest in the study of the Ancient East. At that time, the Bible was considered the quintessence of all ancient history, but its authority was to some extent undermined by criticism of thinkers of the Enlightenment.

Direct study of the monuments of bygone civilizations opened up the opportunity to confirm the biblical text with facts, and many Europeans and Americans enthusiastically took up this matter. Someone traveled to the Middle Eastern countries in search of lost works of art, someone at their own expense unearthed the ruins of dead cities, and someone looked for long-forgotten manuscripts in the libraries of the Middle Eastern countries.

## Biblical scholar

Alas, although many of these "biblical scholars" of the 19th century achieved amazing results, for the most part they were very far from professionalism. Which is perfectly illustrated by the following episode. The well-known German "biblical scholar" Konstantin von Tischendorf 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 noticed some half-erased complex mathematical calculations in Greek.

Sadly to admit it, he apparently just ripped it out of the book when the librarian was looking the other way. This page is now kept in the Cambridge University Library, at the same time as evidence of an amazing accidental discovery and the barbaric attitude of some Western "scientists" to the heritage of antiquity.

Although a little later this page played a role in the acquisition of the legacy of Archimedes, the real merit of the discovery of the book, which later became known as Archimedes' Palimpsest, does not belong to Tischendorf, but to an obscure Turkish librarian. While compiling the catalog, he also drew attention to the lines of mathematical calculations and gave an excerpt from them in the library catalog, which was published and sent around the world.

## Amazing document

At the beginning of the 20th century, this catalog fell into the hands of the Danish historian and philologist Johann Ludwig Heiberg, who was so intrigued that he was not too lazy to get to Constantinople, and got acquainted with the book personally in 1906. What he saw shook him to the core.

It turns out that an amazing document fell into his hands. At first glance, this is a rather ordinary liturgical book from the deserted monastery of Mar Saba, near Jerusalem, copied in the 13th century. But if you look closely, across the liturgical text were barely noticeable lines in earlier Greek, replete with scientific and philosophical terms. Any specialist familiar with the culture of the Middle Ages, it was immediately clear what this meant.

Alas, the parchment on which medieval books were written was made of calfskin and was an expensive item. Therefore, the lack of this material was often resolved in a rather straightforward way: less needed books were divided into separate sheets, ink was peeled off these sheets, then they were stitched again and a new text was written on them. The term "palimpsest" just denotes a manuscript over a cleaned-up text.

In the case of Archimedes' Palimpsest, each of the original sheets was also folded in half to create a smaller book. Therefore, it turned out that the new text was written across the old.As a writing material, an unknown scribal monk used collections of scientific and political works compiled in the Byzantine Empire around the 950s. Fortunately, the cleanup was not very thorough, which revealed the original code.

A preliminary examination by Khyberg showed that the authorship of a large number of 10th century texts belongs to none other than Archimedes, and, most importantly, the longed-for "Method" is present almost in full among them! Unfortunately, the library forbade taking the manuscript out of its premises (after meeting characters like Tischendorf, who can blame them?), So the scientist hired a photographer to reshoot the entire codex for him. Then, armed with nothing more than a magnifying glass, Khyberg set about painstakingly deciphering the photocopy. He managed to make out a lot, and the final result was published in 1910-15, and the English translation was published quite quickly. The discovery of Archimedes' lost labor caused quite a stir and even made it to the front page of the New York Times.

But the difficult fate of Palimpsest Archimedes did not end there. During the First World War (as a result of which the Ottoman Empire ceased to exist) and during the devastation immediately after it, there was absolutely no time for ancient manuscripts in Constantinople. As in the days of Napoleon from Egypt, in the 1920s a huge stream of Turkish values flowed into Europe. It was only much later that it was established that a certain private collector was able to acquire and export the Palimpsest to Paris. Where he for a long time became just a curiosity, revolving in a world very far from knowledge.

## Codex from oblivion

Interest in the book was revived only in 1971, and again thanks to the library catalog. Nigel Wilson, a specialist in ancient Greek culture from Oxford, drew attention to an interesting document from the Cambridge Library, a page already familiar to us, roughly torn out by Tischendorf.

The fact is that a search in ancient Greek dictionaries indicated that some of the terms used on the page were characteristic precisely of the works of 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 of the 10th century 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. Finally, in 1991, an employee of one of the leading auction houses in the world, Christie's, received a letter from a certain French family stating that they wanted to put up the Palimpsest for auction. 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 locked in the safe forever.

## Real nightmare

Fortunately, the fears were in vain. When Will Noel, curator of manuscripts at the Walters Museum of Art in Baltimore, USA, approached the owner's agent for permission to restore and study Palimpsest, his initiative was received with enthusiasm. They say that the billionaire made his fortune on high technology and therefore he himself was not so far from science and its interests.

1999 to 2008 a whole group of specialists from various fields, from philology and art history to spectroscopy and computer data analysis, were engaged in the restoration and scanning of Archimedes' Palimpsest. It was not an easy job.

Noel himself describes his first impression of the manuscript as follows: “I was horrified, disgusted, this is an absolutely disgusting document, it looks very, very, very ugly, completely unlike a great artifact.Just a nightmare, a real nightmare! Burnt, with an abundance of PVA glue along the end, under the drips of this glue, much of the text of Archimedes, which we were going to restore, is hidden. Stationery putty everywhere, pages pasted over with paper strips. There are simply no words to describe the poor condition of Archimedes' Palimpsest."

In the monastery, the book was actively used in divine services, so in many places it is smeared with candle wax. In the mysterious period 1920-1990. someone has falsified colorful "Old Byzantine" miniatures on some pages in an attempt to raise the cost of the manuscript. But the main trouble was that the entire codex was seriously damaged by mold, in some parts of the page that had gone through.

## Grains of sand in the universe

But there were also joys. When the codex was embroidered into separate sheets, it was discovered that many lines of Archimedes' text were hidden inside the binding and therefore inaccessible to Khyberg - sometimes these were key points in theorem proving.

Shooting in different ranges of the electromagnetic spectrum, from infrared to X-ray, with subsequent computer processing of images, made it possible to reconstruct the letters of the 10th century text even where they were hidden or completely invisible to the naked eye.

But why all this painstaking work? Why long-term searches? What in the text of the works of Archimedes, and, in particular, the "Method" hidden from us for a millennium, can one find such that would justify the enthusiasm of scientists in relation to Archimedes' Palimpsest?

It was known a long time ago that Archimedes was interested in very large numbers and very small quantities, and connecting one with the other. For example, to calculate the length of a circle, he inscribed it in a polygon with a large number of but small sides. Or he was interested in the number of the smallest grains of sand in the Universe, which was represented as a huge number. This is an approximation to what today is called infinitely large and infinitely small quantities. But was Archimedes able to operate with mathematical infinity in the true, modern sense of the word?

## Integrals of Archimedes

At first glance, infinity is nothing more than an abstract mathematical abstraction. But only after mathematicians learned to operate with this category, the so-called "mathematical analysis" appeared, a mathematical approach to describing any changes and, in particular, movement. This approach underlies almost any modern engineering, physical and even economic calculations; without it, it is impossible to build a skyscraper, design an airplane, or calculate the launch of a satellite into orbit.

The foundation of our modern mathematical analysis, differential and integral calculus, was created by Newton and Leibniz at the end of the 17th century, and almost immediately the world began to change. Thus, it is the work with infinity that distinguishes the civilization of horse-drawn and windmills not only from the civilization of computers and spaceships, but even from the civilization of steam engines and railways.

So the question of infinity has enormous, one might even say "civilizationally determining" significance. And after the works of Khyberg at the beginning of the 20th century and, in particular, after the work of Noel's team a few years ago, which put many dots on the "i", the answer to this question is very unambiguous and emphatic: yes, Archimedes knew the concept of infinity very well, and not only theoretically operated on it, but also practically applied it in calculations! His calculations are flawless, his proofs stand up to rigorous testing by modern mathematicians. It's funny, he quite often uses what in modern mathematics is called "Riemann sums", in honor of the famous mathematician … XIX century.

When calculating volumes, Archimedes uses a technique that cannot but be called integral calculus.True, if you read his calculations in detail, you get the feeling that this is an integral calculus "from another world." While much overlaps with what is familiar to us today, some approaches seem completely alien and unnatural. They are neither worse nor better, they are just different. And from this a frost creeps through the skin: this is the highest mathematics, genetically in no way connected with modern! Millennia after Archimedes, scientists of modern times invented all this from scratch, anew, with the same content, but in a slightly different form.

## Exhaustion method

Unfortunately, Archimedes' Palimpsest does not and cannot provide an answer to another intriguing question: to what extent were such methods of calculation unique to Archimedes and reflecting his own genius, and to what extent were they typical of Greco-Roman mathematicians and engineers in general? At least one method of calculation, such as mathematical analysis, which Archimedes is fluent in, can be traced back to about the 5th century BC. e. This is the "method of exhaustion", the development of which in ancient Greece is usually associated with the name of Eudoxus of Cnidus, although there is evidence that he was known earlier.

Of course, later this method was also either reinvented or reconstructed in the 17th century. The experience of mathematics in recent centuries tells us that scientists who are fluent in applied mathematics are very rarely responsible for theoretical breakthroughs. Archimedes is, first of all, an applied scientist, he is interested in problems of calculating specific lengths, areas, volumes.

So, it may well be that his technique for working with infinite quantities was not so much developed as modified or revised by him. But if the scientists of the Alexandria or some other scientific school of the ancient world were fluent in mathematical analysis, the key to modern technologies, what else could they know and be able to do? It captures the spirit from the horizons that such an assumption opens.

## A bitter lesson

Now, knowing the history of Archimedes' Palimpsest, you can step back and think. Yes, to our deep regret, its opening was late. In the 20th century, it became a sensation, but a sensation only among specialists in the history of science. But what would have happened if its history had been different? If this manuscript had fallen into the hands of scientists 100, 300, 500 years earlier? What if Newton had been reading this book while still at school? Or Copernicus? Or Leonardo da Vinci?

Modern researchers confidently argue that even for mathematicians of the 19th century, this work would be of more than academic interest. For mathematicians of the 17th-18th 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? We will never know …

But this bitter lesson should not be wasted. How many equal and possibly more valuable books and documents are still hidden from us? Is it on dusty shelves in archives and libraries, tucked away in museum storerooms, locked in collectors' fireproof cabinets? How many secrets are kept in undeciphered cuneiform tablets and inscriptions on the walls of ancient structures?

If a text written in the 200s BC, no less than two thousand years later, could still be considered revolutionary, are there not ancient works that can give a significant impetus to science and technology today? We risk and will never know if we do not get rid of the arrogant and ignorant idea of the "primitiveness" of our ancestors.