Galileo's Muse

Galileo's Muse: Renaissance Mathematics and the Arts

Mark A. Peterson
Copyright Date: 2011
Published by: Harvard University Press
Pages: 352
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  • Book Info
    Galileo's Muse
    Book Description:

    Mark Peterson makes an extraordinary claim in this fascinating book focused around the life and thought of Galileo: it was the mathematics of Renaissance arts, not Renaissance sciences, that became modern science. Painters, poets, musicians, and architects brought about a scientific revolution that eluded the philosopher-scientists of the day.

    eISBN: 978-0-674-06297-9
    Subjects: History of Science & Technology, Mathematics, History

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-viii)
  3. Prologue
    (pp. 1-2)

    With something like these words, a mysterious author, to be encountered and perhaps unmasked in the last chapter, contemplates how foolish he will look when it turns out that his ambitions outstrip his abilities. He hesitates to attempt something that prudence warns him is risky, yet “though you drive it away with a pitchfork,” he says, quoting Horace, “it keeps coming back.”

    I know how he feels, introducing a book that might well be called overambitious. I can only say that it didn’t start out that way. The beginnings of this project were some observations about mathematics and the arts...

  4. 1 Galileo, Humanist
    (pp. 3-32)

    Following his condemnation by the Church in 1633, Galileo’s long and remarkable career should have been finished, but old as he was, nearly blind, under house arrest, and forbidden to publish, his last and ultimately most influential book suddenly appeared in print in 1638. My, my, what a surprise, he says, in effect, in the dedication.¹ What have we here? The book was Two New Sciences, a rambling dialogue full of highly original ideas and discoveries, including especially the famous parabola law: “It has been observed that missiles and projectiles describe a curved path of some sort; however no one...

  5. 2 The Classical Legacy
    (pp. 33-66)

    What did Galileo mean by praising Pythagoras “for his method of philosophizing”? At the very least Galileo must have had his own, private conception of Pythagoras, just as he had his own, private conceptions of Plato and Aristotle. Since Galileo’s “method of philosophizing” was his life’s work, Pythagoras is cited here in a way that demands attention.

    The Pythagoras of tradition founded a utopian community in Croton (in southern Italy), where he and his followers lived a life governed by philosophy.¹ The very words “philosophy” and “mathematician” are supposedly coinages of the Pythagoreans, the mathematikoi being the inner circle of...

  6. Poetry
    • [Introduction]
      (pp. 67-68)

      Dante’s description of being in love, in the New Life, is poetically experimental, clinically analytical, and emotionally extreme. Somehow it manages to be all these things at once as it tells the story of his love for Beatrice, whom he first saw at the age of nine and whom he loved from that moment on. For Dante, we realize, as we grapple with this strange book, poetry is not just a pretty arrangement of words on a page but rather an intellectual tool of vast capability, a method for the deepest problems, an instrument that he has begun to master....

    • 3 The Plan of Heaven
      (pp. 69-80)

      Dante’s Paradiso is built on the plan of Aristotle’s On the Heavens. That Christians might conceive of the world this way was established in the theology of Thomas Aquinas in Dante’s own lifetime. Aquinas doesn’t explicitly describe Aristotle’s universe, but he adopts its terms in speaking about the arrangement of things, which amounts to an implicit acceptance of it. He considers the question of the Earthly Paradise, for example, and whether it might be located as high up as just under the sphere of the Moon. He concludes that it could not be that high, because the element Fire rises...

    • 4 The Vision of God
      (pp. 81-94)

      Having described what was never described before, the structure of the Empyrean, the abode of God and the angels, Dante might have brought the Divine Comedy to a triumphant conclusion, but no. This was not enough. He continues upward into the angelic regions, apparently heading toward the point that is God. He is blinded by the light, only to find that his eyes are given new strength, so that even here, where he should be overwhelmed, he continues to see. He seems to know where he is going. The poem concludes with a vision of God that is completely surprising...

  7. Painting
    • [Introduction]
      (pp. 95-98)

      The artist Albrecht Dürer tells us how he first learned of the geometrical theory of painting, a problem that occupied him for the rest of his life. It happened in his own city of Nürnberg in 1500 in the company of an Italian visitor “named Jacopo, born in Venice, a good artist, who showed me a man and a woman that he had made in true proportion, so that I had rather have seen his method than a new kingdom, and if I knew it, I would have it printed to honor him, for the general benefit of all. I...

    • 5 The Power of the Lines
      (pp. 99-124)

      The few surviving frescoes of the ancient world do not tell a clear story about whether the Greek and Roman painters were trying to create realistic illusions or even whether they would have known how. Realism is just one possible aim for a painting. After a period of infatuation with geometrical methods in the Renaissance, European painters became tired of this idea. Islamic painters, who had access to all the same geometry as Europeans, never became interested in it at all.

      Still, it is common for paintings in many traditions and cultures to show more distant objects smaller, thus giving...

    • 6 The Skin of the Lion
      (pp. 125-148)

      In 1545 the artist Giorgio Vasari published his Lives of the Painters. It is a collection of biographical sketches of the great artists of the Renaissance by one of their own, together with critical and appreciative comments on their works. It was highly popular in its day, and has remained an invaluable and entertaining resource ever since, in spite of its frequent unreliability. Vasari seems to be writing largely from memory or from hearsay. If he had known how future generations would pore over his every word, he might have felt obliged to check his facts more carefully, but it...

  8. Music
    • [Introduction]
      (pp. 149-152)

      Galileo’s father Vincenzo Galilei knew very well the storied power of Greek music. He knew, for example, how Pythagoras had calmed “a Taoromenian lad who, after feasting by night, intended to burn the vestibule of the house of his mistress, on seeing her coming out of the house of his rival. The lad had been inflamed by a Phrygian song, but Pythagoras, happening to meet this Phrygian piper at an unseasonable time of night, persuaded him to change his Phrygian song for a spondaic one. Through this the fury of the lad was immediately soothed, and he returned home in...

    • 7 The Orphic Mystery
      (pp. 153-173)

      The music of the distant past might seem to be lost forever, but in the case of Greek music, the surviving evidence is tantalizingly abundant and informative. The intervals of Greek scales and modes are described mathematically in several sources, and there is a system for notating melodies that survives in tables. There are even a few surviving specimens of Greek music written in this notation. Vincenzo Galilei published four short ancient Greek hymns, ascribed to the second-century lyrist Mesomedes, in his Dialogue on Ancient and Modern Music from manuscript copies provided to him by Girolamo Mei.¹ Other fragments of...

    • 8 Kepler and the Music of the Spheres
      (pp. 174-196)

      Johannes Kepler was interested in music too, but for a very different reason. The music of the spheres, the music of the universe as a whole, was somehow the key to understanding the cosmos: Kepler was sure of that. And with his unparalleled ability to reduce large quantities of observational data to harmonious order, he strove to apply music theory to astronomy. His discovery that Ptolemy had done the same thing in the Harmonics fired his eagerness. In Kepler’s own voice from his 1619 Harmony of the World,

      My appetite was particularly intensified and my purpose stimulated by the reading...

  9. Architecture
    • [Introduction]
      (pp. 197-202)

      The dome of Florence’s cathedral Santa Maria del Fiore dominates the city today as it has done since it was completed in 1445. This architectural wonder not only eclipsed in size the Pantheon, which had been the largest dome in the world since Roman times, but it did so from a baseline elevation that was already 170 feet in the air. The man whose name has become almost synonymous with the great dome is Filippo Brunelleschi, the greatest architectural genius of the fifteenth century.

      Brunelleschi left no written document on his method for designing the dome. Not even drawings survive....

    • 9 Figure and Form
      (pp. 203-213)

      A proportion like 4 : 3 in the fifteenth century was not just a mathematical idea. It was musical (the perfect fourth), and it was geometrical. It was associated with the plan of the universe as a whole, with the music of the spheres, and now with the architecture of the ancients. Alberti follows Vitruvius in describing the dimensions of temples and their columns, but he supplements the Vitruvian account with his own observations and measurements of classical ruins. His detailed description of real things seldom follows a simple pattern, but when Alberti suggests his own favored proportions for ideal...

    • 10 The Dimensions of Hell
      (pp. 214-236)

      At the age of twelve Galileo became a novice of the order of Camaldolese monks at the monastery in Vallombrosa, where he had been sent to school. Ultimately he did not, of course, become a monk—his father put a stop to that. But something vaguely similar happened when he went to the university in Pisa to study medicine. In 1583, at the age of nineteen, under the influence of the geometer Ostilio Ricci, Galileo made what must have seemed to his father like another hopelessly otherworldly career choice: he decided to abandon medicine and devote himself to mathematics. In...

    • 11 Mathematics Old and New
      (pp. 237-254)

      A survey of Renaissance mathematics proper reveals a striking difference between its two subfields, the lively subject of algebra and the moribund subject of geometry. It is hard to avoid the conclusion that in the case of geometry, at least, philosophical preconceptions strongly limited what it could do. Algebra, on the other hand, is the success story of Renaissance mathematics. This branch of mathematics grew out of the arithmetic of the Arabs and Hindus, and it is not part of the classical legacy of Greece and Rome. It grew organically in a European context in response to real questions of...

    • 12 Transforming Mathematics
      (pp. 255-271)

      Galileo’s background in the humanities and the arts was formative for his approaches in mathematics and the emerging sciences. That is the idea I began with, and the idea I return to in this chapter. In particular I consider the importance of humanist history and literature for his understanding of the mathematical legacy, and the importance of standard artistic practice for what he was able to do in mathematical science. Neither history, literature, nor artistic practice were elements of a normal mathematical education, so any benefit he obtained from such a background was virtually unique to him. Conversely, a normal...

    • 13 The Oration
      (pp. 272-291)

      In August 1627, Galileo wrote his last letter to Kepler after a silence of seventeen years. The letter itself does not amount to much, just a recommendation for a young man named Johannes Stephan Boss, otherwise unknown, but in a postscript Galileo says, “I send, enclosed with this letter, an oration of Niccolò Aggiunti, an outstanding young man in serious and humane letters: I am sure that you will read him with great pleasure, and that you will find him wonderfully to your palate and taste.”¹ In view of the complicated relationship between Galileo and Kepler, it is natural to...

    • Epilogue
      (pp. 292-298)

      It is not easy to write about Galileo and yet to ignore the Copernican controversy, as I have done until now. In the end I feel a responsibility to record my view, namely that the importance of the Copernican controversy in Galileo’s biography is overstated. The archetypal dogmatic Copernican is an image that was constructed for Galileo by other people, it must be said, first by his persecutors, and later, when it appeared that he had been right after all, by those who regarded themselves as his successors. There is every reason to regard it with skepticism, and even to...

  10. Notes
    (pp. 299-320)
  11. Acknowledgments
    (pp. 321-322)
  12. Index
    (pp. 323-337)