Paper: Strange Turns in Renaissance Mathematics


World History 219, Section 802
Dr. Monica E. Podbielski

As far as I can tell, mathematicians are an odd lot. I’ve known a few in my varied undergraduate majors of physics, engineering, and computer science. There is always a surprise to be found in the minds of these people, whether it be the amount of uncontainable excitement that dizzying feats of logic can induce in one, or the belief of another that mathematics is now obsolete because computers provide all the mathematical tools people need. I think that a statement like that, even if it turns out to be totally wrong, reflects something about our time.

In one sense mathematicians have a grand consensus in which they share a worldwide language and philosophy. A lot of the history of math has to do with how this consensus was arrived at, and studying this would be one way to investigate mathematics in history. On a different level mathematics has always drawn to it creative and unusual people. Surely many wonderful ideas have not made it into our current views. Any strange turns taken by the mathematicians of history that aren’t part of our textbooks may have been equal or better illustrations of historical forces at work in the Renaissance than those that stuck around.

In Renaissance times the driving force behind mathematics was shifting from mercantile arithmetic to a disciplined pursuit of Euclid, Pythagoras, Apollonius, and Archimedes. (Struik 109) The printing press began to make these works more accessible in Western Europe after 1447. (Boyer 297) For a century European mathematicians struggled with the formidable task of playing catch-up with the Greeks and Arabs. Then, in the mid-16th century, significant advances, especially in algebra, began to appear in print. (Swetz 378)

Some of these advances were made in Germany by Pastor Michael Stifel. His well known achievements in mathematics include the discovery of logarithms and the use of modern notation for positive, negative, and square root. (O’Connor 1) There is plenty more to learn from him, though, by taking a look at his lifelong convictions that were not destined to be a part of modern science.

After earning an M.A. at the University of Wittenberg Stifel became a monk, and was ordained a Catholic priest in 1511. It was costly in those times for Catholics to ensure their place in Heaven through the Church, and when Stifel began to grant absolution without the customary charge, he became unpopular with his brothers. His admiration of Luther did not go unnoticed either, and in 1522 he felt the need to depart for the Taunus mountains. (Westfall 1)

As a Lutheran pastor Stifel’s religious beliefs did not wane as he moved or fled from parish to parish, nor were they independent from his mathematical endeavors. He tied the two together in the 1520’s in what he called wortrechnung, or word calculus. (Katz 324) This involved assigning a series of numbers called triangular numbers to the letters of the alphabet, and performing calculations with them. (Otermat 1)

To Stifel, wortrechnung was just as valid as any of his other mathematical work. He applied it to passages of the Bible and studied the results. By January of 1533 he was utterly convinced that he had calculated the date of the End of the World: October 18 of that very same year. (Katz 324)

Stifel shared his newfound knowledge with his congregation, and was soon forbidden to preach. In his respectful silence, he began to give away all his books. It was too much for people to ignore. The residence of Lochau became a hive that buzzed with those who came to prepare for The End. No one worked the fields, merchants gave away their goods, and the town historian crept away with the treasury. (Otermat 1) Surely mathematics has never again created the likes of this little town in 1533.

On what may have been the worst day of his life, Michael Stifel called his congregation together to usher in the Apocalypse. As the day wore on, he must have sensed that at least he would meet his end, but even this possibility vanished when the authorities from Wittenberg took him into custody and out of danger. (Katz 324)

To his credit, Stifel kept his promise to never prophesy again. He was not, however, cured of his interest in wortrechnung. Before he died he authored two books about it. Nor did he refrain from applying it to religious subject matter. In one example he performed operations on the words “Leo Decimus” that equated this pope with the number 666, showing him to be the Antichrist. (O’Connor 1)

These examples reveal a lot about the kind of beliefs that must have been prevalent in 16th century Europe. It’s clear that Stifel’s thought was analytical and quantitative, undeniably a part of the early Scientific Revolution. So much faith was growing in logic and reason that people understandably believed it was capable of anything. If math could predict an eclipse, why not the Apocalypse? If it could prove that parallel lines never meet, why not the meanings of the Bible? It’s also clear that some parts of the Scientific Revolution had not yet arrived. Wide practice of empirical methods, for example, would have to wait for Brahe, Kepler, and others. (Eves 194)

Stifel’s strange turns also clearly demonstrate the spread of the Reformation. Not just because Martin Luther bailed him out a few times, but also because he convinced others of things like the identification of the pope with the Antichrist using mathematics. (Westfall 1) His choice of subject matter goes on to suggest a clear Protestant interest in the cryptic Book of Revelation.

A year after Stifel published his most famous work, Arithmetica Integra, a budding English mathematician named John Dee graduated from Cambridge University. (Boyer 309) Dee would spend much of his life entranced by the possibilities of mathematics, much like Stifel. And like Stifel some of his contributions would become a permanent part of mathematics, and some would be remembered for other reasons.

John Dee first gained notoriety in Paris where he boldly gave some of the first lectures heard there on Euclid’s Elements. At 23 years old, most of his enthralled audience were scholars older and more experienced than himself. (Fell-Smith 19) Regardless of his age the lectures were a huge success. He went on to write his Mathematical Preface, an introduction to the English translation of the Elements. This work made the difficult material more widely accessible, and its popularity increased proportionally. (Katz 354)

In addition to his publications, evidence of Dee’s mathematical abilities can be inferred from the testimony of the large group of scientists, experimenters, and other mathematics practitioners that praised his advice. (Calder II.2) But Dee’s interests were far from limited to mathematics. To catalog them would be similar to the task of cataloging his 4,000-volume library. (Fell-Smith 22) He investigated all aspects of religion, the occult, astrology, and alchemy, which helped lead to the popular opinion that he was a sorcerer and magician. At various times in his life he was the victim of angry mobs, prison sentences, and witch trials due to these opinions. (Fell-Smith 19) His curiosity never rested though, nor did his faith in mathematics. The combination of the two led him to some strange turns as he tried to apply mathematical principles to vast and broad subjects.

One of the great sciences awakening in Dee’s time was astronomy. Copernicus had proposed a heliocentric universe in an attempt to explain astronomical observations, and mathematics was used to explain and predict phenomena. (Katz 369) But astronomy did not interest the general public sufficiently to provide adequate funding for its study. Astronomers were therefore also astrologers that provided the popular service of revealing the influence of the heavens on the lives of people on Earth. Dee was a careful astrologer and very precise in his calculations. It was more than just a way to pay the bills for him, he saw in it a universal physical law. If physical law governs the behavior of bodies both on Earth and in the heavens, he reasoned, surely there is some kind of verifiable harmony between the two. (Calder III.11)

Another of Dee’s interests, the cabalistic teachings that spread with the popularity of Hebrew studies in general, provided yet another possibility of harmony between various systems. The doctrines of cabalah often viewed a symbol as a higher reality than the object it represents. (Calder III.24) Manipulating symbols, which comprises much of mathematics, can in this context be interpreted as a method of education superior to mere observation of the physical world.

Cabalistic methods gave Dee the opportunity to apply his mathematical abilities directly to other systems of symbols. All the symbols of the Hebrew alphabet were interpreted as derivations of a single letter “yod”. Several interpretive techniques existed for text written in this alphabet. They involved substituting words for numerically equivalent ones and other numerical manipulations of text, similar in concept to Stifel’s wortrechnung. Dee took the ideas and attempted all sorts of mathematical operations on texts in various languages. (Calder III.26)

All these various influences seem to have culminated in 1564 when, after years of preparation, Dee wrote his Monas Hieroglyphica in twelve days. Although he claims in the title page that this work is entirely compatible with his previous works, it seems to represent a reversal of some of his previous tendencies. (Calder VI.7)

Dee had previously been a proponent of experiment and observation, sometimes making detailed astronomical observations daily or even hourly. But in Monas he no longer praises experimental method, discarding it as unnecessary. (Calder VI.7)

The most apparent example of a shift of purpose in Monas is the style in which it’s written. In contrast to the Preface which was written in the vernacular so as to be comprehensible to the layman, Monas is written in bewildering puzzles and codes. Both the Greek and Jewish traditions that Dee was familiar with were oral traditions. If esoteric knowledge was written, it was written to be understood only by the initiated. It might even contain false figures and statements to throw unwanted readers off the track. (Calder VI.7) Perhaps Dee still felt that mathematics should be accessible to everyone, but respected the secret aspects of his other areas of study that employed math. In any case it seems possible that the fine points of Monas may never be understood, and we are fortunate that Dee provided an overview of its purpose in a letter to the emperor Maximillian. (Calder VI.8)

The letter reveals Monas Hieroglyphica as a very ambitious work. It is centered around the development and understanding of the monad – a hieroglyph that is to the planetary symbols as the letter “yod” is to the Hebrew alphabet. That is to say that all the planetary symbols can be generated from the monad. This process is presented in mathematical fashion in twenty-four theorems. According to the letter, understanding the theorems is the key to understanding all the secrets of nature. The monad is the symbol for God’s act of creation. (Calder VI.8)

This statement shows the incredible force that Dee perceived in mathematics as a way of using and learning from symbols. The power of it rests in the cabalistic idea that symbols are not just arbitrary pictures assigned to designate something, but geometric representations of the essence of the thing. Mathematics is a method of manipulating the symbols that preserves their meaning. A real understanding of a symbol and the method used to arrive at it is therefore an understanding of the essence, or true nature, of that which the symbol designates.

It seems that Dee believed that the monad and the mathematics used to generate the planetary symbols from it constitute a truer understanding of Creation than observation could ever yield. The following passage from Monas seems to link this view with his loss of interest in the experimental method.

“Will not the astronomer regret his experimentation, his endurance of cold and night, when by means of this procedure within closed doors, at every, or any moment he can observe the exact motions of the heavenly bodies without the aid of machines or instruments?” (Calder VI.9)

The strange turns that Dee took provide their own historical information about 16th century Europe. The Scientific Revolution was beginning to value empiricism and experimentation more and more, as Dee did early on. Great confidence was also put in mathematics, and eventually Dee decided that math was powerful enough to completely replace experimentation. This decision was influenced heavily by his cabalistic studies, which are themselves evidence of a growing Jewish presence in Europe combined with an increased availability of the printed materials of various cultures and their translations. Dee’s huge library is in itself a tribute to the massive effects of the printing press a century after its introduction. Economically we see that Dee depended somewhat on the popularity of astrology for his living, and that funds were generally not available for research into pure mathematics and astronomy. His ultimate motive for these studies was religious – he wanted to use them to understand God’s act of creation.

More insights come from a look at the public opinion of Dee as a sorcerer or magician. We might still agree to these labels for some of his practices today, but we also must admit that Dee was a sound mathematician and scientist as well. His critics in his day made no such distinction. People perceived all of Dee’s researches as magical because they did not understand them. Their opinion had enough clout to see Dee put on trial for witchcraft, sent to prison, and his house burned to the ground.

It may be possible that if Dee had not taken such pains to encrypt his theories in Monas that great observers of his time like Tycho Brahe would today be remembered as fools or not at all. As it happened, if anyone was ever successful in implementing Dee’s procedure for divining the true nature of the universe, they didn’t let on. So it was left to mathematicians like Johann Kepler to use the detailed observations Brahe provided to assemble a publicly available mathematical model of the solar system.

Kepler was fueled throughout his life by essentially the same motivation as John Dee. He wanted to find the mathematics that God used to create the world and planets. (Katz 373) Unlike Dee, some of the answers he found are still in astronomy textbooks today.

There is no need to question the validity of Kepler’s place in history as a mathematician and scientist. Of all the various models of the solar system since Ptolemy’s in the 2nd century, the one he presented in 1609 in The New Astronomy was by far the closest to that we now use. (Swetz 403) His works contained elements that anticipated the calculus, and are generally appreciated for the beauty of their applications both practical and fanciful. (Struik 129)

For an advocate of a heliocentric solar system Kepler was quite fortunate when it came to avoiding the wrath of the Church. He was never imprisoned or forced to recant like Galileo, or burned at the stake like Bruno. When all Protestants were exiled from his school in Graz, Austria, a special exception was made for him. (Katz 374) His life was not trouble-free, though. He did have to flee the church on occasion, and he came to his mother’s defense when she was accused of witchcraft. But he never betrayed his Protestant roots or his belief in a heliocentric solar system.

The need to make a living made Kepler a practicing astrologer as well, but he was never fascinated by it the way Dee was. He labeled astrology “the foolish daughter of the respectable, reasonable mother of astronomy.” (Swetz 400)

It was in Kepler’s first book, Mysterium Cosmographicum, that we see some ideas that are usually disregarded as mystical today. He works with Copernicus’ model of the solar system in this book, which places the sun in the center of the circular orbits of the six planets that were known then. The model was geometrically simple and beautiful, but Copernicus had given no geometrical explanation for the various distances between the planets. In an attempt to improve the model, Kepler added the five regular convex polyhedra.

A regular polyhedron is a solid three-dimensional shape made out of regular polygons. Regular polygons are two-dimensional shapes whose sides are all the same length. A square is a regular polygon, and the regular polyhedron made out of squares is a cube. The wonderful thing about regular polyhedra is that in a universe of Euclidean geometry, it is only possible to construct five. They look like five multi-sided dice with four, six, eight, ten, and twenty sides respectively. (Kepler 1011)

Kepler figured that if each planet’s orbit is imagined as a great circle on a sphere, it would make six spheres around the sun. The five regular polyhedra would then fit right in between the six spheres, filling the space between the planets with their geometrical perfection. He concluded that it must have been so by design, and did not live to see the discovery of Uranus, the seventh planet. (Swetz 401)

In addition to filling the space in the solar system with the pleasing shapes of geometry, Kepler filled it with the sounds of music in his later book, Harmonies of the World. As a student of music Kepler knew the simple mathematical ratios that are the basis of an interval between two notes. Comparing the daily movement of Saturn along it’s orbit at the point closest to the sun with the daily movement at the point farthest from the sun he found the ratio 4:5, a major third in music. Doing the same with the other planets he found other intervals. (Kepler 1031) Then he compared the different planets with each other and found still more intervals. (Kepler 1032) Taking a subset of all these intervals and transposing them into one key, he discovered a major scale. Another ordering revealed a minor scale. (Kepler 1037) Finally, using the individual notes corresponding to each planet he worked out some multipart harmonies. And his model of the solar system bloomed into a dance of heavenly geometries and melodies. (Kepler 1040)

The world that witnessed Kepler’s strange turns was irrevocably immersed in the Scientific Revolution. The fruit that Kepler reaped from the unequaled observations of Tycho Brahe showed how profitable carefully recorded experiments could be. A stronger Reformation movement gave Kepler enough shelter to avoid being tortured or killed by the hostile Catholic Church for his ideas. Astrology continued to put pennies in the pockets of astronomers, and a mathematician with some clout could defend his mother at a witch trial.

Above all else was an unwavering faith in the power of mathematics that Kepler shared with Michael Stifel and John Dee. Each of these Renaissance mathematicians was so deeply awed by the amazing ability of mathematics to describe, explain, and predict, that they placed no limits on the possibilities it enabled. This is evidenced in part by the contributions each made to modern mathematics. But we are so used to our own ideas of math, its limits, and its uses, that it is easy for us to overlook the deep significance and potential mathematics held for these men, and the forces that motivated them to make their discoveries.

By contrast it is better to look at the strange turns the Renaissance mathematicians took with their endeavors, the attempts at theories of such grand scale that our current ideas are challenged and knocked out of context. This adjusts our perspective and hopefully creates a richer picture of them and the historical world they lived in.


Works Cited

Boyer, Carl B. A History of Mathematics. Princeton: Princeton University Press, 1985.

Calder. “John Dee Thesis.” John Dee Society. Http://www.johndee.org/ calder/html/TOC.html (3 March 1998).

Eves, Howard. Great Movements in Matematics Before 1650. United States: Mathematical Association of America, 1983.

Fell-Smith, Charlotte. John Dee. London: Constable & Company, 1909.

Katz, Victor J. A History of Mathematics. New York: HarperCollins College Publishers, 1993.

Kepler, Johannes. “Harmonies of the World.” Ptolemy: The Almagest; Nicolaus Copernicus: On the revolutions of the heavenly spheres; Johannes Kepler: Epitome of Copernican astronomy: IV-V (and) The Harmonies of the World: V. Ed. Maynard Hutchins. Chicago: Encyclopaedia Britannica, 1990.

Otermat, Megan. “Lutherstadt Wittenberg, Pastor Michael Stifel.” KDG Wittenberg. http://www.wittenberg.de/seiten/personen/stifel.html (25 February 1998).

O’Connor, John J., and Edmund F. Robertson. “Michael Stifel.” Rice University. http://www-groups.cs.st-and.ac.uk/~history/ Mathematicians/Stifel.html (25 February 1998).

Struik, Dirk J. A Concise History of Mathematics. New York: Dover, 1948.

Swetz, Frank J. ed. From Five Fingers to Infinity. Chicago and La Salle: Open Court, 1994.

Westfall, Richard S. “Stifel [Styfel], Michael.” Galileo Project. http://es.rice.edu/ES/humsoc/Galileo/Catalog/Files/stifel.html (25 February 1998).

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