^{th}February 1891. We give a translation by Stefanie Eminger.

Ladies and gentlemen!

For those who pursue the development of mathematics and its related disciplines from a cultural-historical point of view, the age of the Renaissance will always be of very particular interest: it is the age in which the consolidating process, from which our modern mathematical sciences emerged as an international cultural factor, took place.

The Renaissance! The rebirth of the arts and the sciences! This word conjures up such a wealth of images in our soul! Before our inner eye the grand masters of Italian art rise up; we imagine seeing the works of Leonardo da Vinci, of Raphael, of Michelangelo, which have been constituting a common and inexhaustible source of the most pure and noble pleasure for civilized people across the globe for centuries. We feel transported to the illustrious courts of the art-loving Italian princes, in particular the Medici, and partake in the literary efforts that tie in with the Italian national literature established by Dante, Petrarch, and Boccaccio.

And then again we remember that at the same time the age of the Renaissance is the age where sciences, in particular classical studies, flourished again; that it coincides with the enlightening and liberating activities of Humanism and the Reformation. We let our gaze wander northwards, away from Italy, and meet, apart from the stalwart figures Zwingli and Luther, the great humanist Melanchthon, Erasmus of Rotterdam, and Reuchlin. In particular, we fondly remember the appearance of the unresting fighter^{1} whose glorious and prolific life ended on the close-by Ufenau, and to whom we have taken such a liking due to David Friedrich Strauss and not least due to the heartfelt poetry of Conrad Ferdinand Meyer. And by surrendering to these imaginations, we suddenly find ourselves in the middle of the struggle that liberated the people from the spells of the Mediaeval Ages, right in the centre of the great, magnificent humanist movement. Everywhere the spirits stir, everywhere fresh scientific activities blossom -- it seems as if a warm spring breeze is wafting across the country, as if we can hear the shout of joy with which Ulrich von Hutten concluded his ever memorable letter to Willibald Pirckheimer: "Oh century! Oh sciences! It is a joy to live!"

And, I hear you ask, does mathematics have a share in all this glory? Mathematics: this science, which is always regarded with respect, out of consideration for the largely impressive advantages that it warrants modern technology alone, but not always with sympathy; this science, which is so often characterized, and so unjustly so, as a dry, prosaic one, which addresses the intellect, and analysis, but not the mind, nor imagination; this science, which one, therefore, so likes to put in a certain conflict with creativity and art, or indeed with the most ideal pursuit of all^{2!
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Hewho would go through the trouble of dispelling such prejudices would only have to point out Plato, one of the most qualified -- and at the same time most visionary -- representatives of Antiquity, who began his lectures in the Academy with the sentence: "No-one ignorant of geometry may enter my house!" Plato loved and appreciated mathematics so much not because of any material advantages, but precisely because it is particularly well suited to removing one's mind from the world of sensations and render it amenable for philosophy, due to its inherent power of abstraction. And for those who do not wish to let Antiquity convince them that mathematics has a part in the grand, ideal challenges of humanity that is not to be underestimated, that conviction will suggest itself during thorough and comprehensive study of the cultural activities of the Renaissance. Today I will briefly sketch out some of these activities for you.

Ladies and gentlemen, it hardly seems possible to appreciate the contribution that a science as abstract as mathematics has made to the culture of any era in any other way than by putting it in context with the overall historical development of this discipline. Mathematics, like in fact any pure scientific research, sees its purpose not so much in the applicability of its results, but much more in satisfying very particular philosophical questions posed by the intellect. Incidentally, with regard to results, it is very hard to predict when, in what form and to what extent they will influence the civilization process at some future point. Hence, you will expect me to present the cultural factors of the Renaissance that we are interested in specifically not in an isolated manner, but as the products of a historical development. And if I go back a little further than would seem necessary at first glance, then please do not assume that the reason for this is the difficulty of speaking about mathematics without being able to assume any special knowledge. Rather, although the questions I will consider first are very distant to the Renaissance in time, they are linked directly to it in contents and are therefore vital in order to understand these factors.

Mathematics had been flourishing in Italy already once before. This was at the time of the Greek colonies in Lower Italy and in Sicily. And the last Greek mathematician on Italian soil, Archimedes of Syracuse, was also the most brilliant of mathematicians in Antiquity. He received his deathblow, deeply immersed in his studies and not worrying about anything other than "do not disturb my circles" in the face of an assailing warrior when the Romans conquered his hometown in 212 BC. This put the final nail in the coffin for the science that he represented so brilliantly; along with him, mathematical research in Italy was wiped out for more than a millennium! For over the centuries, the great Roman nation did not produce a single noteworthy mathematician, at least by Greek standards. Indeed, we can hardly regard it as a coincidence that this nation, which showed little originality and was wholly dependent on the Greeks with respect to arts and literature, also showed such an incredibly poor predisposition for any mathematical speculation.

In the last centuries BC and the first centuries AD, it was almost exclusively the scholars of the Academy in Alexandria who cultivated mathematics. Already under the first Ptolemy, at around 300 BC, Euclid had written his famous "Elements", which you all know, here. Since the Renaissance they have been forming the basis of the teaching of geometry, even today. Here lived Eratosthenes, the geographer and chronologer, who mathematicians know due to the so-called "sieve of Eratosthenes"; and Apollonius, the great geometer who conducted in-depth studies of the theory of conic sections and introduced the terms ellipse, hyperbola, and parabola. Here Hipparchus, the actual originator of scientific astronomy to whom we owe the introduction of longitude and latitude in order to determine the position of a point on a sphere, conducted his famous observations of the moon in about the year 150 BC. And it was here, in Alexandria, where Claudius Ptolemy wrote his immortal work * μεγάλη Σνταξις*, the

*Great Treatise*, about 300 years later.

The name of Ptolemy, the founder of the world-view named after him, is connected with the culture of an era spanning almost one and a half millennia, and particularly with the culture of the Renaissance so profoundly that I cannot but dwell on this magnificent figure for a while.

You all know that the Ptolemaic system, which was regarded and adored like a Gospel until the mid-16^{th} century, is based on the geocentric principle, i.e. on the fundamental view that the Earth hovers motionless in the centre of the universe and that the celestial bodies, including the Sun, rotate around his centre. But this view alone, which, after all, eventually suggests itself to any naïve observer, did not suffice to establish the everlasting fame of Ptolemy. In fact, he deserves credit for describing the movements of the celestial bodies as exhaustively as possible, using trigonometric tools that he largely crafted himself and whose perfection remained unsurpassed for centuries.

Hipparchus had already observed that the seasons are of different durations 300 years before Ptolemy. The only explanation for this phenomenon that he could find was that although the Sun rotated uniformly around the Earth in a circle, the Earth was not in the exact centre of this circle. Now, Hipparchus could give a sufficient explanation for the apparent motion of the Sun based on this so-called eccentric circle, but the much more involved motions of the Moon and the planets presented insurmountable difficulties. The apparent, i.e. as seen from Earth, annual motion of the planets is characterised by a loop: although the planets move in one particular main direction, i.e. from West to East on the whole, they occasionally slow down, stop, move backwards for a short period of time, stop again, and continue to move in the original direction. Now, how should one go about explaining this complicated motion without infringing the fundamental view of Antiquity that only allowed the use of uniform motion on a circle in order to explain celestial motions, and to which people adhered partly due to metaphysical reasons? Ptolemy solved this problem by assuming that every planetary motion was composed of two circular motions. According to Ptolemy, every planet initially moves uniformly along a circle, the so-called epicycle, whose centre, in turn, moves along a second circle, the so-called deferent, which is circumscribed eccentrically around the Earth. By describing the planetary orbits as so-called epicycloids, he could explain the planetary motion, complicated greatly by antecedence in particular, without violating the fundamental principle mentioned above.

This is a brief sketch of the basis of Ptolemy's famous theory of epicycloids, which testifies to commendable ingenuity. But now I have also touched upon the mathematical content of the world-view, which was the predominant one for most of the Renaissance and which started to give way to a more sophisticated understanding of the worldview only towards the end of the era. I will explain later how this significant change happened.

There is a widespread belief that after the Arabs conquered Alexandria in 641 Greek culture found a new home in Constantinople, where many Alexandrian scholars had moved, and that here, in the capital of the Eastern Roman emperors and protected by them, Greek science was maintained and developed throughout the Mediaeval Ages. When scholarly Greek refugees brought classic manuscripts to safety to Italy, due to the Turks' advance and the conquest of Constantinople in 1453, the Occident became acquainted with the precious treasure of Greek culture, which initiated the renaissance of the sciences.

As far as mathematics is concerned, this belief is inaccurate. I do not want to disavow the ardour and the strong impulse caused by those invaluable manuscripts. A tale according to which the well-known Italian humanist Pico della Miranbola paid for a single Livius with an entire estate illustrates the enthusiasm with which those manuscripts were received, and how badly the intellectuals wanted to possess old Greek and also Roman manuscripts! But knowledge of Greek mathematics had already been brought to the civilized people of Europe via a different route. In addition, and just to mention it straight away: Greek mathematics only formed one arm of the gigantic stream of mathematical thoughts that flooded the Occident towards the end of the Mediaeval Ages.

The Arabs adopted the legacy of Antiquity initially. As is well known, they created an empire that stretched towards the Indus in the East and the Ebro in the West within an incredibly short amount of time. However, what we marvel at the most is the flexible minds that this people must have possessed, so as to enable them to develop a primitive nomadic lifestyle into a civilization such as under the splendid governance of Harun al-Rashid^{3} in just under 150 years. It was under this ruler, with whom we have been so familiar since our childhood through the tales of "Arabian Nights", but even more so under the Caliph Al-Ma'mun that a fruitful period of translating began, due to which we know many a Greek writing that might have been lost otherwise. The first Greek manuscripts that were translated to Arabic were Ptolemy's *Σύ νταξις*, Euclid's

*Elements*, Apollonius's conic sections and Archimedes's treatises on measuring the circle and on the sphere and cylinder. The name "Almagest", which we use for Ptolemy's work even today, stems from this period. It derives from the Arabic article

*al*and the Greek superlative

*, "greatest", which*

*μ**ε**γ*ι*σ**η*

*Μ*("great")

*ε**γ*ά*λ**η**Σύ*had gradually turned into.

*ν**τ**α**ξ*ις
Ladies and gentlemen, human history is full of peculiar discrepancies! Christendom trembled when the Arabs triumphantly conquered all of Spain at the beginning of the 8^{th} century and even advanced into the Frankish Empire across the Pyrenees; they hailed Karl Martell after the victorious battle between Tours and Poitiers for having heroically liberated them from the Barbarians. And under the same Barbarians, particularly under the Umayyad dynasty, sciences and arts gradually reached their heyday, their golden age, in Spain. Moreover, Christendom received the most precious contribution of its own intellectuality from the very same people whom they considered had obliterated all civilization!

In the 12^{th} and 13^{th} centuries scholars from all over Europe flocked to the academies in Toledo, Seville, Cordoba and Granada to study the Greek classics and, most importantly, to translate them from Arabic to Latin. That way, and in particular due to the work of diligent translators such as Gerhard of Cremona, Adelard of Bath and others, the Christian Occident gradually gained an insight into the sophisticated mathematics of Antiquity.

But I have already mentioned that Greek mathematics only formed *one* of the sources that were to amalgamate and develop into our modern mathematics. Indeed, the significance of the Arabs for civilization, who primarily communicated the ideas of different peoples, was of a much more universal nature than was assumed previously. For, apart from knowledge of Antiquity, we also owe to them a first insight into the intellectual life of a people whose approach to mathematics was completely different to that of the Greeks, but not less sophisticated, and which complemented it very well: I am talking about the Indians.

Due to their highly developed sense of aesthetics, the Greek almost exclusively investigated mathematical problems that could easily be visualized, i.e. problems in geometry. In contrast, the Indians' exceptionally accomplished sense of numbers and an unparalleled love of calculation, spread across all social classes from ancient times, led them to dealing with problems in arithmetic and algebra for the most part.

After all, India is the home of chess, the arithmetic game par excellence; poetic musing and dreaming was connected with mathematical speculation so intimately that poets were fond of indulging in charming intellectual games, and, conversely, mathematicians liked to verse their treatise. Arithmetic riddles and competitions formed part of the Indians' social amusements; in fact, it is even reported that Buddha, when courting a girl, had to sit an arithmetic exam in order to beat his rival in love!

We owe an invention to our predescessors at the banks of the Ganges, whose sophisticated mathematics was brought to us by the Arabs at the same time as that of the Greeks. This invention has been in the possession of European scholars since the beginning of the 13^{th} century, but its results started to benefit the whole human race only very gradually and only since the intellectual upsurge that took place towards the end of the 15^{th} century. Despite the classic simplicity of this invention -- or perhaps precisely because of it -- we may add it to the collection of influential cultural factors that shaped the Renaissance: I am referring to the invention of Indian numerals and the Indian place-value system.

Contrary to popular belief, the nature of the Indian numeral system does not rely on the fact that the number 10 forms the basis of the system. Both the Greeks and the Romans, in fact, all Indo-Germanic peoples, used the decimal system. In addition, according to studies by Alexander von Humboldt and other scholars we may argue that all the peoples of the world, except for a few isolated cases, use 5, 10 or 20 as a base for counting, with respect to the arrangement of the human body. Rather, the Indian numeral system is characterized by the fact that one can express any arbitrarily large number by using no more than ten symbols and positioning them next to one another. One of these symbols, zero, denotes absence, whereas the other symbols denote the numbers from 1 to 9. In order to express any arbitrary number it suffices to simply state that in addition to a numerical value, each of these symbols also has a certain place-value depending on its position. This is done in such a way that, reading from right to left, the first digit always represents the ones, the second digit represents the tens, the third one represents the hundreds, etc.

Ladies and gentlemen, you may think that it can hardly be of any importance to speak about such elementary things that each and every one of us knows from school. I do not even wish to remind you that this simple assumption, which forms the basis of our calculation today, has escaped a mathematically gifted people such as the Greeks. We have to travel back only a few centuries to find completely unfamiliar conditions, which we would barely be able to cope with. Indeed, it is only with history's help that we can conceive the major benefit of this Indian invention today, which has such a profound impact on all aspects of scientific and everyday life.

For instance, it was impossible for the Romans, using their cumbersome numerals that you are all familiar with, to do arithmetic in a similar manner to us today, using our digits. You can easily see this for yourselves by adding or even multiplying two arbitrary large numbers written in Roman numerals. Admittedly it would prove difficult to invent a more uncritical and more awkward notation for numbers than the Roman one. The Greek notation on the other hand is in a different league entirely. They had a specific symbol for each one, each ten and each hundred: the letters of their alphabet (using a few letters of an outdated alphabet). In arranging these symbols they already tried out the place-value numeral system. But even so, doing arithmetic by our standards was not really possible with these numerals either, especially when dealing with large numbers, possibly except for exceptionally gifted intellectuals like Archimedes and Apollonius. Thus, the people of Antiquity had to rely on using their fingers to do calculations; incidentally, particularly when doing complicated computations this requires a certain subtlety and dexterity, which the Italians display in the Morra game nowadays. Alternatively, they had to resort to computing on a so-called abacus, a tool that is basically identical to the calculating frame that you are all familiar with from our elementary schools. Apart from a few modifications, which I will not mention here, these are essentially the computation methods that were used throughout the Mediaeval Ages.

In light of these primitive tools the question suggests itself: Would our whole modern culture, all of our scientific and social life, and our polymorphic infrastructure even be conceivable without this inconspicuous invention that is the Indian numerals?

These remarks will give you a rough idea of how important calculating with digits is in the historical development of civilization. Now please allow me to make a few comments referring to the origins of our numerals. Nowadays there are no more doubts that they originated in India -- including the *zero*, without which a place-value numeral system would not be possible. They definitely existed in the second century CE, and in all likelihood they emanated from the first letters of the Sanskrit names for the respective nine numerals. The invention of zero is likely to be more recent, but its first appearance can be traced back to 400 AD after all. The Arabs learned this new computation method from the Indians; specifically, after Muhammad ibn Musa al-Khwarizmi, a mathematician who lived at the beginning of the ninth century under the Caliph Al-Ma'mun, had made Indian arithmetic accessible to his fellow countrymen in several treatises. As has been shown by Reinaud and Boncompagni, the sobriquet al-Khwarizmi, meaning "he who comes from Khwarizmi" (today's Khiva) has been preserved in the words "algorism" and "algorithm", which is the name for all regular calculation procedures in mathematics.

We already know to some extent how the Indian numerals eventually reached the Occident in the 12^{th} century. In particular, I have to highlight the outstanding contribution of Leonardo Pisano, called Fibonacci, undoubtedly the most important mathematician of the entire Christian Mediaeval Ages. He deserves credit for introducing the Indian digits in his seminal work *Liber abaci*, written in 1202. The fact that we know of the Indian numerals due to the work of the Arabs later often led to the erroneous term "Arabic numerals". However, both the Arabs themselves and also the Italians in the Renaissance were well aware of the Indian origins of their arithmetic.

Thus, at the beginning of the 13^{th} century we see the Christian Occident being in possession of the mathematics of two highly gifted people; one of which, the Greeks, represents the mathematics of Antiquity, and the other one, the Indians, represent the mathematics of the Mediaeval Ages. I am happy to add that the contribution of the Indians to mathematics is by no means limited to elementary arithmetic. In fact, their most outstanding achievements can be found in algebra and higher number theory (especially relating to indefinite equations of first and second degree). Mathematicians like Aryabhata, Brahmagupta and Bhaskara are rightly placed among the most distinguished number theorists of all times.

One should be right in thinking that such great insights, particularly the magnificent example of Fibonacci, would have led to a tremendous intellectual upsurge, an epoch of greatest mathematical productivity. However, this was hardly the case until the middle of the 15^{th} century. One gets the impression that the nations were almost overwhelmed by the scores of new intellectual material opened up to them, and that they could process it only little by little.

But the situation radically changed in the middle of the 15^{th} century: the invention of the printing press, which caused unequalled social and intellectual upheavals, joined the momentous stimulation originating in Constantinople, which we have already met. It was only now that mathematics had a more profound impact on Western culture, and that we can talk about a revival of the Greek and Indian spirit. The wisdom of the Brahmins now begins to bear fruit. The Indian place-value system becomes widely accepted -- I will explain how -- and henceforth forms an integral part of the intellectual heritage of all civilized people. On the other hand we notice that Greek geometry, culminating in Ptolemy's theories, is booming, particularly in Germany and Italy; and we marvel at the global scientific process that comes to a close in the mid-16^{th} century, when the Ptolemaic world-view gives way to our modern one. As long as we look at the grand scheme of things only, we may regard these two developments -- the acceptance and permanent absorption of the Indian place-value system into our modern culture, and the fading of the Ptolemaic system and the foundation of a new world-view that emanated from it -- as the most important influences of mathematics on the culture of the Renaissance.

But let us ignore these two major areas of mathematical activity for a moment. There is a wealth of developments that testify to the extent of which the whole culture of the Renaissance was steeped in revived mathematical thinking. However, this can hardly come as a surprise given that we are talking about an age that was characterized by the notion of universality more than any other era afterwards. In fact, this does not only mean that people devotedly cultivated and advanced the individual sciences, and moreover, that the arts experienced a most glorious heyday, but also that this all-round, harmonic development of all human capacities recurred in certain *individuals*. These polymaths, perfect representatives of their era, illustrate the sprit of the Renaissance in a similar manner.

But it was not *solely* the pursuit of universal learning that made mathematics so compelling and attractive to the great masters of the era, like Brunelleschi, Leonardo da Vinci, Raphael, Michelangelo and, in particular, Albrecht Dürer. They were fully aware of the fact that notwithstanding the freedom of one's imagination, art was also subject to the law of necessity and conversely, that mathematics was also subject to the law of beauty, despite the rigour of logical reasoning. Thus they found mathematics to be related to their art and appreciated the benefits that it gained from their studying mathematics. The year 1420, justly given as the date for the renaissance of architecture, illustrates this well. The Florentine Cathedral had been completed, except for its dome. Architects from across the world gathered at a congress to study and solve the problem of constructing the dome. The most bizarre ideas were proposed; the suggestion to build the entire dome out of pumice, as this would reduce the strain, not even being the most foolish among them. Then Brunelleschi stepped forward, with the unheard-of proposal to close the enormous opening with a free cupola, without any sort of scaffolding. The suggestion was ridiculed, but Brunellesco did not budge until the project was assigned to him, and he executed it closely following the plan he had submitted. Thus he solved a problem that required an experienced architect and mathematician. The slender dome of Santa Maria del Fiore rises in a beautiful elliptical curve: an everlasting monument to its ingenious constructor.

As tempting as it may be for the mathematician to follow the footsteps of those great masters of the Renaissance, I do believe that, with respect to the topic I have chosen, the only way to do this age justice is by restricting myself to outlining the essence and the historic development of the great leading ideas, of which there was no shortage in this era. I will certainly not find it easy to pass over a man whose magical and powerful figure involuntarily reminds one of Goethe and whose mathematical ingenuity is comparable to that of Archimedes. But if I have to withstand appreciating the mathematician Leonardo da Vinci in more detail here, then I truly do not do this due to a lack of admiration for this all-embracing polymath, but because his grand scientific ideas, hidden away in handwritten notes that were not destined for publication, were virtually unknown to his contemporaries. Therefore, they unfortunately remained without any influence whatsoever on his era. Only our century had the privilege of discovering Leonardo to have been one of the most significant scholars, and certainly the most brilliant mechanic and physicist of his time. I will give you only a few random and incoherent examples: Leonardo knew the laws of free fall before Galilei did; he was the first to study, both in theory and in praxis, the influence of friction resistance on motion; he clearly stated that a *perpetuum mobile* and squaring the circle are impossible. He discovered capillary action; invented the *camera obscura*, and in doing so laid the foundations for an important area of optics; he invented an instrument to determine humidity; he identified scientific experiments as the true source of natural philosophy a whole century before Francis Bacon. Most remarkably perhaps, he developed a wave theory almost two centuries before Huygens, which he used to explain phenomena of sound and even of light! And all these scientific achievements are not just jotted down, but written out in full, in a clear, beautiful language and accompanied by excellent drawings. His notebooks are located in Milan and Paris. Furthermore, we learn about his work as an engineer, building fortifications and hydraulic structures; we learn that he built the great canal of Martesana, which connects Ticino with the river Adda and irrigates about 80,000 morgen of land^{4}; that he invented and constructed a variety of machines for various purposes, among them various aircrafts and even a steam-powered ship. And then we suddenly remember that this very same man created *The Last Supper* and was such a divine artist -- and we automatically remember the words of Jakob Burckhardt: "It will only ever be possible to catch a glimpse of the formidable outline of Leonardo's character!"

But there is *one* achievement, which resulted from the marriage of art and mathematics and can well and truly be called a child of the Renaissance, that I have to explain in a bit more detail: the foundation and the development of the theory of perspective. This is how one could go about understanding what it means to have an accurate perspective image of an arbitrary object: One puts a glass pane between the original object one wants to map and one's eye, which is situated at an arbitrary, but constant point in the space, the so-called point of view. The other eye is closed. If one assumes that the rays of light that travel from the points of the original in the direction of the eye, through the glass pane, leave a visible trace on the pane, then all of these traces will form an image, which is called the perspective view. The theory of perspective is simply the collection of rules according to which one can draw an accurate perspective image of a given object *without* using such a glass pane.

Many have debated whether or not the ancient Greek and Roman painters knew the art of perspective. Lessing has provided us with an in-depth study of this question in "Laocoon"^{5} and "Letters of Antiquarian Contents". His conclusion is to the disadvantage of the ancient painters, as "this part of art was completely unknown to the Ancients." His contemporary Johann Heinrich Lambert, the famous mathematician and architect at the court of Friedrich the Great, continued this historical critique in his treatise "Free Perspective", published here in Zurich. He claimed that one would have to consider Leonardo da Vinci as "the first to think of the true refinement of painting and of perspective." Admittedly, today we no longer associate only one single name with establishing perspective and consciously introducing it to painting and are happy to honour the great contributions of brothers Johann and Hubert van Eyck and, in particular, of the well-rounded Leon Battista Alberti in Italy, which were all made *before* Leonardo's time. However, everyone who knows a little bit about the development of painting will clearly recognize the milestone in the history of art, with regard to the use of linear perspective and particularly of the so-called aerial perspective, which is marked by the name of this great Florentine.

His great contemporary Albrecht Dürer, equipped with sound geometric knowledge, continued the work begun by Leonardo and his predecessors. Like Leonardo, Dürer has earned himself a place of honour in the history of mathematics. His fundamental work "Instructions for Measuring with Compass and Ruler", published in Nuremberg in 1525 and dedicated to his friend and benefactor Willibald Pirckheimer, was of the utmost importance for the development of art, and specifically for applied arts.

There is another reason why the year 1525 is of particular interest to us, which also leads me back to a topic I touched upon earlier. This year marks the beginning of mathematics teaching at German elementary schools. It is Luther to whom the Germans owe this important reform of elementary education. In the "Letter to the Councilmen of all German towns so that they may establish and maintain Christian schools", published at the end of the year 1524, he states: "I am speaking for myself: If I had any children and I could afford it, they would not only have to study languages and history, but also learn how to sing and make music and all of mathematics." And Luther did not talk in vain. In the 15^{th} century, the view that mathematics was not for public, but only for private education, still prevailed. Children in public schools were not even taught simple calculations with digits. But this radically changed when Luther appeared on the scene.

It fell to the re-organized primary schools of the 16^{th} century to solve a cultural problem: they were given the task of replacing the old methods of calculating and instead making the use of Indian digits such as we know it today an integral part of each and everybody's education. We gladly remember the men who successfully performed this significant task. Among them Adam Ries, whose name you are all familiar with and whose strenuous life came to an end in 1559, is entitled to be called a teacher of the German nation.

But radical reforms took place not only in the primary schools, but also in the secondary schools and universities during the first half of the 16^{th} century: at all such institutions, mathematics was adopted as a regular subject and independent professorships in mathematics were established. Those reforms happened not only in Germany, but also here in Switzerland, due to the tireless work of Zwingli. As early as in the 1520s, the humanist Myconius, a friend of Glarean, taught mathematics alongside classical languages at the school by the Fraumünster. Konrad Gessner, the future polymath and famous bibliographer, and Joachim Rheticus, future professor at Wittenberg and friend of Copernicus, were among his pupils. In Germany, the upswing of mathematics education in secondary schools and universities is closely linked to Melanchthon. An excellent mathematician himself, he deserves high credit for his efforts in spreading mathematics both orally and in writing. It was due to him, for example, that when the Gymnasium was founded in Nuremberg in 1525, a teaching post for mathematics was created straightaway as well; the first of its kind in Germany. It is easy to see why it was Nuremberg in particular that set an example when one realizes that this town was not only the centre of German art in the 15^{th} and 16^{th} centuries, but also led the way among German cities with respect to trade, industry, science, and literature. The convivial house of the one and only Willibald Pirckheimer, whose memory is honoured in the magnificent letter by Hutten, alone, was comparable to an academy of arts and sciences, which promoted extensive international scholarly intercourse and actively facilitated all of the ideal efforts of the time!

The lively interest in mathematics, which characterized not only Pirckheimer's circle of artists and scholars, but also the citizenry of Nuremberg, was passed on in this town by tradition. In order to get an impression of this, just check the comprehensive "Historische Nachricht von den Nürnbergischen Mathematicis und Künstlern, welche fast von dreyen *seculis* her durch ihre Schriften und Kunst-Bemühungen die Mathematic und mehreste Künste in Nürnberg vor andern trefflich befördert und sich um solche sehr wohl verdient gemacht"^{6}, compiled by Gabriel Doppelmayr in 1730. Already on the first page of this interesting compilation we stumble across the name of the man who laid the foundations to Nuremberg's mathematical fame and has justly been called the most active reformer of exact sciences in the 15^{th} century: Regiomontanus. In drawing your attention, ladies and gentlemen, to this distinguished figure, I have also arrived at the topic that will conclude my talk and which I have therefore looked at in some detail at the beginning (not at all coincidentally): astronomy.

Born in 1436 in the Franconian town Königsberg, Johannes Müller, known as Regiomontanus in the scientific world, went to Vienna as a fifteen-year old youth, in order to learn mathematics and astronomy from Peurbach. At the time, Peurbach was the most famous astronomer around and an exquisite humanist, who owed his excellent reputation partly due to an outstanding textbook on planetary theory, of course in the Ptolemaic style. By working together, the relationship between teacher and pupil soon became a very intimate one. This was exemplified during a stay in Vienna in 1460, when the scholarly Cardinal Bessarion, one of the Greek refugees who had come to Italy from Constantinople, invited Peurbach to continue his astronomic studies in Rome. Peurbach insisted that his young friend be allowed to accompany him, a condition to which Bessarion gladly agreed. Unfortunately, Peurbach, not yet 38 years old, died before they could set out on their journey. Even so, Bessarion held open his invitation, and when he returned to Rome in 1461 Regiomontanus was allowed to go with him. Regiomontanus spent seven busy years in Italy, keeping up an active scientific correspondence with the local scholars, in particular with the astronomer Bianchini who had been a teacher of Peurbach's. He plunged into the study of the Greek mathematicians with enthusiasm; as we know, these had only been known through translations from Arabic to Latin up until then, but now they were available to him in the original. When he left Italy in 1468, he was in possession of a whole collection of valuable Greek manuscripts; among them the manuscript of Ptolemy's συνταξιζ in particular, a gift from Bessarion. After sojourns in Vienna and Ofen, Regiomontanus chose to settle permanently in Nuremberg. There, Bernhard Walter, a man characterized not only by his wealth, but also by his great interest in science, set up an observatory, a mechanics workshop for producing scientific instruments, and even a private printing press for him. Here in Nuremberg Regiomontanus displayed positively astonishing scientific activities; marvellous plans occupied his mind. For example, he wanted to publish printed, accurate original editions of all the Greek manuscripts that he had brought with him from Italy. However, in 1475 he was honoured by an invitation from the Pope, who bestowed upon him the title of Bishop of Regensburg and appointed him to a post in Rome, where he was assigned the task of carrying out the absolutely necessary calendar reform. Regiomontanus only reluctantly complied with this request, which would prove to be so fatal for both him and science. Having barely arrived in Rome, he died of the plague, at the mere age of 40.

This is not the place to talk about Regiomontanus's scientific works or indeed demonstrate how he paved the way for Copernicus by means of improving trigonometric tools and the existing astronomical tables. I can only mention those achievements that played a direct part in the progress of civilization of the time. Among those, publishing the calendar and the so-called ephemerides rank among the most important ones, alongside the credit he deserves for spreading mathematics. Admittedly, hand-written catalogues where all the days of the year were separated by weeks and months and from which people could gather the dates of movable feasts or the beginning of an eclipse for example, existed before Regiomontanus's time. However, arranging the calendar in the perfected form that we know today and that we cannot do without anymore is due to Regiomontanus. He published the first printed German calendar in Nuremberg in 1474.

Publishing his ephemerides was even more significant. One can call them a kind of chronicle of the heavens, where Regiomontanus had listed consecutively all the phenomena that could be observed on the skies, i.e. the respective positions of the heavenly bodies, for the period from 1475 to 1506 -- done in advance and based on meticulous calculation. The ephemerides promptly caused a stir all over Europe, and became positively epoch-making in nautical sciences. Regiomontanus's ephemerides, together with the tools for astronomical observations that Regiomontanus had designed himself as well (the best of their kind before the invention of the telescope) now allowed sailors to determine the position of their ship out on the open sea according to the positions of the stars. The glory of Bartholomew Diaz, of Vasco da Gama, and of Christopher Columbus in particular, is not derogated by the fact that these audacious men had Regiomontanus's ephemerides on board their ships, which had been introduced to the Portuguese navy by the famous seafarer and cosmographer Martin Behaim from Nuremberg. However, by considering this fact, the great discoveries of the 15th century no longer appear as isolated events or as foolhardy endeavours, but as the results of continuous, purposeful efforts of the mind, which had their origins in the renaissance of the exact sciences.

Another work of Regiomontanus's, admittedly published long after his death, might be of interest to us: his paper on comets. Regiomontanus was the first to include comets in scientific studies. Unconcerned by the superstition that had been linked to the comets as dreaded celestial phenomena since time immemorial and inspired by the comet of 1472, Regiomontanus came up with the for his time completely novel idea of treating it like any other celestial body for once and carry out astronomical measurements. He was of the opinion that it was appropriate for those who wanted to talk about a comet to first know its position, its dimension, its distance from Earth, etc. Regiomontanus's views with regard to the physical composition of comets have also only been verified afterwards.

In order to get a complete and generally accurate picture of mathematics in the Renaissance, one cannot ignore another aspect, which is related to the superstitious beliefs about comets and which has often been referred to as an illness of those times. For the Renaissance was also a heyday of astrology; the alleged art of predicting events in the future based on the position of the stars. The belief in mysterious relationships between the stars and the destinies of human beings was such a widespread one that, up to the 17^{th} century, even the astronomers at universities could not elude its influence on their teaching. Very few were brave enough to campaign against astrology, but of course they were unsuccessful. However, we may count the frequently misjudged Swiss physician Theophrastus Paracelsus, unjustly nicknamed Bombastus, amongst them, alongside Italian scholars such as Paolo Toscanelli, Pico della Mirandola and others. A reason for this would be his beautiful sentence, which deserves to be preserved for future generations due to its classy simplicity: "A child needs neither stars nor planets, its mother is its planet and its star."

But we do not want to reproach this era because of astrology. In many cases astrology was just an expression of the boisterous quest for knowledge, of the immense, albeit not always properly channelled thirst for knowledge and education of the time, which is represented in the legend about Faust, for example. And having said that, astrology often also motivated and encouraged research in astronomy. Moreover, for as long as people considered the Earth to be the immovable centre of the universe, relating celestial phenomena to earthly events and asserting a causal connection between them seemed an obvious thing to do, as this was in accordance with the resulting significance of the Earth. Changing these assumptions was only made possible by a complete reform of the entire worldview.

You all know the name of the immortal man to whom we owe this truly grand achievement. However, I can barely hope that I will be able to do the merits of Nicolaus Copernicus justice in the short amount of time available.

Copernicus was born on 19 February 1473 in Thorn, Western Prussia. After having completed his school education he matriculated at the University of Cracow, where he studied humanities, mathematics and medicine, thus gaining a solid and eclectic education. In addition, he was well versed in arts, in drawing, painting and in music. At the age of 23 he went to Italy, in order to prepare for the post as Canon in Frauenburg, which his uncle, the future Bishop of Ermland intended for him, by studying theology and medicine in Bologna. Among the most famous teachers at the University of Bologna at the time was the astronomer Domenico Maria di Novara. Copernicus, whose favourite subject had always been astronomy, met this fine man, and soon a relationship akin to the one between Regiomontanus and Peurbach developed between them. Surely we can look for the origins of the bold ideas to which we owe our modern worldview in the stimulating scientific exchange that Copernicus experienced in Bologna and later on in Rome, Padua and Ferrara. However, we will attribute the unusual sense of aesthetics and beauty that Copernicus displayed in formulating his system to the influence of Italian art, which the young astronomer greatly appreciated.

Copernicus returned from Italy around the year 1505, filled with scientific and artistic impressions. Soon after he took on the post of Canon in Frauenburg, which he held until his death on 24 May 1543. His calm and serene life was filled only with the duties of his priesthood, the affiliated hospital for the poor, and his scientific studies, to which he devoted all of his leisure time. It was only late in his twilight years, and at the urging of his friends, that he decided to publish the ripe fruit of these studies: his great work "On the Revolutions", which he had kept back, to use his own words, not for *nine* years, but for *four times nine years*. When the first printed sheets arrived in Frauenburg, Copernicus was already in mortal agony. He is reputed to still have touched and looked at them before passing away.

Alexander von Humboldt says that: "The founder of our current world view was characterized by his courage and the confidence that he displayed to an almost higher degree than by his knowledge. He highly deserves the fine praise from Kepler, who called him the *man of free spirits*."

Indeed, imagine what a powerful scientific conviction was needed in order to confront the Ptolemaic tradition, held sacred for fourteen centuries! *The Earth rotates*! She is a planet like the others, like Mercury, Venus, Mars, Jupiter, and Saturn! *All* planets, and hence the Earth, too, rotate around a common fixed centre, the Sun! Moreover, in addition to the annual revolution around the Sun, the Earth also rotates around its axis daily! These are the phrases with which Copernicus caused a stir all over the world at the time, and which have been engrained in our brains so thoroughly nowadays, that we regard them as part of our intellectual identity. The Ptolemaic system was no doubt cleverly devised, as long as one would wish to base it on the geocentric principle, i.e. relating the movements of the celestial bodies to the Earth as a fixed centre. But it was precisely the assumption of this principle that had led to the greatest complications and insalubrities when trying to explain the motion of the planets. Order, simplicity and harmony appeared the moment that Copernicus replaced the geocentric principle with the heliocentric one. "I have not been able to find a more beautiful symmetry of the universe and a more harmonic link between the orbits by any other alignment, than by placing the Light of the World, the Sun, governing the whole family of rotating bodies, on a regal throne in the centre of the beautiful Temple of Nature!" he said enthusiastically.

Ladies and gentlemen! I will have to conclude my talk with the foundation of the Copernican system, which signified the beginning of a new, intellectually more liberal world in humanity's awareness of science. Of course, my talk can make no claims to be complete as the Renaissance was far too prosperous for that, also with regards to mathematics.

*Sammlung gemeinverständlicher wissenschaftlicher Vorträge*, edited by R Virchow and W Wattenbach; new series, volume

**6**, issue

**142**, Verlagsanstalt und Druckerei A.G., Hamburg, 1892.

^{1} Rudio refers to the German humanist and reformer Ulrich von Hutten, who died on Ufenau, an island in Lake Zurich, in 1523.

^{2} Given the context and the time of writing, Rudio probably means philosophy.

^{3} We have used the modern transcriptions of Arabic names here.

^{4} Morgen -- unit of measurement used in Germany and some other states, used until the 20^{th} century. The size of a morgen varied across the regions, from ½ acre to 2½ acres.

^{5} Published in 1766.

^{6} "Historic news about mathematicians and artists from Nuremberg, who have rendered outstanding services to mathematics and several arts in Nuremburg, splendidly promoting them through their writings and artistic efforts, for almost three generations."