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prejudices of a perfect heaven to color his judgment. Consequently, when he discovered that the spots did not return to the same position and order, he was quick to announce that he could “free the Sun from the injury of spots.” His premise of celestial immutability constrained his imagination and prevented him from considering the possibility that the spots could change, even beyond recognition. He therefore concluded that the spots had to be stars orbiting the Sun. Galileo’s course of attack on the question of the distance of the spots from the Sun’s surface was entirely different. He identified three observations that needed an explanation: First, the spots appeared to be thinner when they were near the edge of the solar disk than when they were near the disk’s center. Second, the separations between the spots appeared to increase as the spots approached the center of the disk. Finally, the spots appeared to travel faster near the center than close to the edge. Galileo was able to show with a single geometrical construction that the hypothesis—that the spots were contiguous to the surface of the Sun and were carried around by it—was consistent with all the observational facts. His detailed explanation was based on the visual phenomenon of foreshortening on a sphere—the fact that shapes appear thinner and closer together near the edge (figure 19 demonstrates the effect for circles drawn on a spherical surface).

The importance of Galileo’s demonstration for the foundations of the scientific process was tremendous. He showed that observational data become meaningful descriptions of reality only when embedded in an appropriate mathematical theory. The same observations could lead to ambiguous interpretations unless understood in a broader theoretical context.

Figure 19

Galileo never gave up an opportunity for a good fight. His most articulate exposition of his thoughts on the nature of mathematics and of its role in science appears in another polemic publication—The Assayer. This brilliant, masterfully written treatise became so popular that Pope Urban VIII had pages from it read to him during his meals. Oddly enough, Galileo’s central thesis in The Assayer was patently wrong. He tried to argue that comets were really phenomena caused by some quirks of optical refraction on this side of the Moon.

The entire story of The Assayer sounds a bit as if it were taken from the libretto of an Italian opera. In the fall of 1618, three comets appeared in succession. The third one, in particular, remained visible for almost three months. In 1619, Horatio Grassi, a mathematician from the Jesuit Collegio Romano, anonymously published a pamphlet about his observations of these comets. Following in the footsteps of the great Danish astronomer Tycho Brahe, Grassi concluded that the comets were somewhere between the Moon and the Sun. The pamphlet might have gone unnoticed, but Galileo decided to respond, having been told that some Jesuits took Grassi’s publication as a blow to Copernicanism. His reply was in the form of lectures (largely written by Galileo himself) that were delivered by Galileo’s disciple Mario Guiducci. In the published version of these lectures, Discourse on the Comets, Galileo directly attacked Grassi and Tycho Brahe. This time it was Grassi’s turn to take offense. Under the pseudonym of Lothario Sarsi, and posing as one of his own students, Grassi published an acrimonious reply, criticizing Galileo in no uncertain terms (the response was entitled The Astronomical and Philosophical Balance, on which the opinions of Galileo Galilei regarding Comets are weighed, as well as those presented in the Florentine Academy by Mario Guiduccio). In defense of his application of Tycho’s methods for determining distances, Grassi (speaking as if he were his student) argued:

Let it be granted that my master followed Tycho. Is this such a crime? Whom instead should he follow? Ptolemy [the Alexandrian originator of the heliocentric system]? Whose followers’ throats are threatened by the out-thrust sword of Mars now made closer. Copernicus?