**By Joseph W. Dauben**

A replica of Galileo's workroom, as recreated at the Deutsches Museum in Munich, Germany. |

Galileo's early work seems to have concentrated upon arguments against Aristotle, involving a sustained pattern of observation and demonstration requiring little in the way of mathematics, concentrating instead on physical experience. His most revolutionary observational discoveries came with the telescope, and these provided, for the first time, about 1610, a number of good physical arguments in favor of the Copernican theory. Later, Galileo's impressive discovery of the parabolic nature of projectile motion-elaborated fully in 1638-seemed to display the essentially mathematical character of physical phenomena. Galileo believed that nature was inherently mathematical, that mathematics was the language of nature, and that mathematics was the key to understanding the reality behind the appearance of natural phenomena-for example, accelerated and parabolic motions.

What Galileo achieved in revolutionizing physics was to show how observation, careful measurement, and attention to the structure of a given event all led to an appreciation of hidden causes that ultimately expressed the pervasive mathematical unity of all nature. Yet he was not the first to have done this, although in terms of astronomy and physics he was clearly a pioneer. Renaissance artists-painters, sculptors and architects-had been observing nature with a special interest in depicting it realistically from the early 15th century on. In fact, by turning to the problem of art and science in the Renaissance, it is possible to find what I believe are important roots for Galileo's own peculiarly realistic-and idealistic-approach to nature. For the values and attitudes Galileo held were ones he shared with Italian humanists, including philosophers, artisans, even musicians.

The first steps towards a new sense of artistic reality were taken early in the 15th century. One of the earliest masters of the human form was Masaccio. In his "Expulsion from Paradise," painted about 1427 in Florence, one can easily see the results of careful observation of human anatomy. Masaccio's Adam reflects the underlying structure of skeletal frame and superimposed muscle, as does his fresco of "Peter Baptizing the Neophytes." A century later, art and anatomy combined dramatically in the studies of Andreas Vesalius, the Flemish physician whose major work, On the Structure of the Human Body, was illustrated by one of Titian's students. The illustrations provide a graphic, detailed record of the musculature and skeletal framework of the human body, and literally seem to pare away layer upon layer of muscle to reveal the hidden structure underneath.

The first artist to perform actual dissections to improve his anatomical knowledge may well have been Antonio Pollaiuolo. His painting of the "Martyrdom of St. Sebastian," his last work completed in 1475, is a tribute to his virtuosity. In much the same way, Luca Signorelli undertook to complete a series of frescoes in Orvieto between 1499 and 1504. In both "The Damned Consigned to Hell" and in the "Resurrection of the Dead," Signorelli seems to have adopted unnatural and contrived positions for many of his figures, again to show his skill in representing the human body.

The ultimate achievement of this sort, however, was brought to perfection by Michelangelo, whose mighty Adam in the Sistine Chapel seems a direct evolution, artistically, from the Signorelli in Orvieto. In fact, we know that artists of the late 15th century like Michelangelo and Leonardo actively pursued anatomical dissections to perfect their understanding of the human form.

In the same year that Galileo's Dialogue Concerning the Two Chief World Systems was published in 1632, Rembrandt painted his famous "Anatomical Lecture", graphically representing what physicians had learned from Renaissance artists: that nature was accurately representable only by virtue of careful observation, through anatomical dissection revealing the hidden structure underlying the human form. What Renaissance artists had clearly achieved through careful observation of nature, including studies of anatomical dissections, was a means to recreate the 3-dimensional physical reality of the human form on 2-dimensional surfaces. In part, the key to this achievement lay in understanding the underlying, hidden structure of the human body which then enabled the artist to produce realistic representations of what he saw on the flat surface of a wall in the case of frescoes, or on a wooden panel or paper in the case of drawings.

If artists in the 15th century had learned to portray with faithful accuracy the human body through careful observation and anatomical dissection-a similar inspiration occurred to those seeking a corresponding dramatic reality in the representation of physical space. A means was devised early in the 15th century for translating the reality of 3-dimensional natural phenomena onto 2-dimensional surfaces, producing virtually realistic copies. A correspondence was thus made possible, through mathematics, between the representational reality of the artist and the physical reality of nature.

The first to carry out a series of optical experiments that led to a mathematical theory of perspective was the Florentine architect and engineer Filippo Brunelleschi. His most stunning accomplishment, in fact, is the stupendous dome which crowns the cathedral in Florence, a work which occupied him intermittently from 1417 to 1434. The technical difficulties involved in erecting the new dome underscore an important aspect of his talents: he was a daring innovator, with a solid knowledge of mathematics and mechanics.

Mathematics was equally important to Renaissance artists in determining the correct proportions for the figures they drew. Leonardo da Vinci followed such principles explicitly, measuring not only the proper proportions of the human head, but the dimensions of the various parts of the anatomy of the horse as well. This was all embodied literally in his most complete visual statement of the harmony between mathematics and nature, his famous drawing of the human figure, proportioned in keeping with the architectural perfection of the square and circle.

It is no coincidence, I think, that the artistic Renaissance and the scientific Renaissance should have both developed at first largely in Italy. Scientists like Galileo were doing exactly what Renaissance artists had been doing all along, with growing skill and increasingly sophisticated techniques, in their depictions of nature in realistic terms since the late 15th century. The veracity of their mathematical vision was well-established by the time Galileo began thinking about the mathematics-the geometry-of space nearly a century later. What Renaissance artists had discovered was that in addition to careful observation and attention to underlying physical structure-often this meant anatomical structure-mathematics was an especially useful tool for translating the physical reality of 3-dimensional objects in 3-dimensional space into realistic illusions of that same reality on only 2- dimensional, flat surfaces.

Galileo, like Renaissance artists of the 15th century, was interested in form, in the underlying reality of the natural world. He, too, was interested in the sort of physical reality that he felt his mathematics and the telescope were making clear for the first time. Light, optics, mathematics-all were as important keys for Galileo as they had been for Brunelleschi, Alberti and Piero della Francesca. From Plato Galileo took his faith in the ultimate rationality of nature, and the fact that the key to understanding nature was to be found in the ideal, perfect world of mathematics; but from Aristotle Galileo also understood that to understand nature, one must also be a systematic observer, and that it is only through experience and careful study of nature that the hidden secrets-the mathematical structures underlying the appearance of physical events and phenomena-can be discovered.

Galileo achieved a synthesis of observation and theory in a way that was strikingly modern and yet was also a product of the centuries of Italian humanism and the tremendous burst of energy we associate with the artistic Renaissance. New discoveries advanced the arts as well as the sciences, and many of these were due to new instruments and methods, especially ones related to mathematics.

Renaissance artists had contributed greatly to man's knowledge by the time Galileo was doing his first work at Pisa. The humanist artists of the Italian Renaissance had performed their own dissections to promote the study of anatomy, they had invented mathematical perspective to make possible the accurate, realistic portrayal of physical space. The literary humanists had managed to revive all sorts of classics, in particular the works of Plato. Christopher Columbus had directly challenged the limits to the finite European world of Ptolemy's geography. The bounds of human knowledge were expanding at a rapid rate. Renaissance artists were seeking a new world, thanks in part to mathematics and the new perspective, literally, that mathematics provided.

Renaissance artists and architects had already succeeded in translating physical space into the mathematical terms of proportion and perspective to produce works that tricked the eye and rivaled nature. Galileo used mathematics with equal skill to reveal the underlying structure of physical space and motion to show that these, too, could be reduced to mathematical analysis. In connecting physical space and real motion-which could be observed experimentally-with the ideal and uniform change of his neo-platonic, mathematical world, Galileo also served to bridge the early stages of the scientific revolution in Europe-featuring figures like Copernicus and Kepler-with the later unifying achievements of Descartes, Newton and Leibniz.

*Joseph W. Dauben is a professor of history and the history of science at Lehman College of the City University of New York, and is a member of the PhD program in history at the Graduate Center, CUNY. This article is adapted from his online exhibit at http://www.pd.astro.it/ars/arshtml/arstitle.html.*

*Copies of "The Art of Renaissance Science," a videotaped presentation of this material produced by Science Television, are available from the American Mathematical Society and through Amazon.com.*

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