Johannes Kepler was the first to apply the new mathematics to divine the laws of celestial motion. Kepler adopted a Copernican, heliocentric view of the universe from his earliest days. He focused on the number, size, and relation of the planets, seeking some grand design. After years of calculations and attempts to fill the gaps in his theories, he finally created a model of the universe that satisfied him. He noted that there were six known planets, thus five intervals between them, and noted that there were five possible regular solid figures (that is, figures with equal sides and angles)--cubes, tetrahedrons, dodecahedrons, icosahedrons, and octahedrons. By layering these solid figures, one inscribed within the next, in this order from largest to smallest, he believed he could map the orbits of the planets. This scheme was soon discredited, but Kepler continued to search for some divine plan, explicable by simple mathematics, to explain the structure of the universe.
In 1609, Kepler published New Astronomy with Commentaries on the Motions of Mars. The work clearly sets forth two of the tenets of modern astronomy: 1) the planets move around the sun not in circles, but ellipses; 2) planets do not move uniformly, but in such a manner that a line drawn from a planet to the sun sweeps out an equal area of the ellipse of its orbit in equal time, even if the ellipse is not perfectly centered on the sun. In 1618, Kepler presented the third of his laws of planetary motion, stating that the squares of the periods of the planets' orbits are proportional to the cubes of their distances from the sun. These observations were all at least somewhat accurate, and led to the final discarding of Aristotelian cosmology by the academic world.
Galileo Galilei was the most well known and successful scientist of the Scientific Revolution, save Isaac Newton. In 1604, by observing the appearance of a new luminous body in the remote region of space for which no motion of the stars could be detected, he demonstrated that the remote and, according to Aristotelian cosmology, static region of space was not actually static. In 1609, Galileo introduced both the telescope and the microscope. His first observations with the telescope were published in 1610, in a 24-page booklet entitled Messenger of the Heavens. The first half of the booklet described Galileo's observation of the surface of the moon, which he proved was rough rather than smooth. He professed the existence of up to ten times as many distant, seemingly fixed stars than were currently known. The second half of the book is largely devoted to the moons of Jupiter.
In 1612, Galileo announced that through the observation of dark spots on the sun, he had concluded that the sun itself was revolving. This announcement spawned one of his first conflicts with the Church, which considered these findings contrary to Church doctrine. In 1616, the Inquisition warned Galileo to "abandon these opinions." A few days later, the works of Copernicus were "suspended till corrected."
By 1630 Galileo had completed his magnum opus, Dialogue on the Two Chief Systems of the World, comparing the Ptolemaic, or geocentric and the Copernican, or heliocentric systems, and finding the heliocentric model far superior. In the work Galileo discussed at length the doctrine of uniformity, proposing the view that corresponding causes produce corresponding affects throughout the universe, thus leading to the recognition that terrestrial physics may be used to explain the motion of heavenly bodies. This philosophy was in direct opposition to the Church-sponsored Aristotelian system, which aligned itself with a geocentric view of the universe and differentiated between terrestrial and celestial physics.
The Dialogue brought matters to a head for Galileo. In August 1632 the sale of the book was prohibited, and its contents examined by a special commission. Galileo was found guilty of heresy and forced to sign a recantation of his theories, after which he was sentenced to house arrest for the remainder of his life. Galileo signed the recantation to save his own life, but legend has it that as he signed the prepared document, under his breath he muttered "the Earth does move, however."
Kepler's idea of the universe was based on his study of the ancient theorists, which, combined with years of indoctrination, had instilled in him a sense of mysticism. He was convinced that the arrangement of the universe must correspond with some concept of geometric harmony and beauty. Thus he sought a simple mathematical solution to the problem of mapping the universe, trusting that the laws of nature would provide one. This was a regression in the attitudes of the seventeenth century astronomers, most of who were prepared to accept an irregularly shaped universe, or at least one that did not fit any simple geometric scheme. However, Kepler persisted in his efforts to discover a universe that worked under a unified grand scheme. Though he attempted to map the solar system in this method, he repeatedly failed due to his convictions. Kepler expressed his astronomical goal as to replace arbitrary hypotheses with mathematical explanation. He was hindered in this task by his belief in mysticism and astrology. Astrology was widely practiced and widely believed to be accurate by most Europeans in the seventeenth century, and Kepler was no exception. He attempted, through his studies of the solar system, to confirm with reason the astrological influence of heavenly bodies, something neither he nor anyone has ever been able to do conclusively.
Galileo, devoid of any adherence to mysticism, posed a sharp contrast to Kepler. Kepler was a German Protestant, a mystic and a dreamer, while Galileo was an Italian Catholic who sought evidence, explicable facts, and was unrivaled in his time in experimental acumen. However, in their genius, the two astronomers were peers, and between the two of them, created the modern view of a mathematical universe. Galileo, especially, contributed to this progress, confirming with observation the fact that the ancients had not known enough to formulate theories on the universe, and combining his in depth knowledge of mathematics and physics with that observation to begin to formulate a new view of the universe.
The discovery of Jupiter's moons was significant because it provided, in effect, a small model of the solar system. Galileo used the model of Jupiter and its moons to explore the manner in which the planets might orbit the sun. The implication of his first observations was to call the traditional Aristotelian system, advocated by the Church, into question. Thus a number of forces joined together to oppose Galileo. Academic Aristotelians had long despised him. Added to these were Jesuits, who were actively engaged in teaching the beliefs of the church. Many pious Europeans joined their ranks, and a mass of common citizens unwilling to entertain novel ideas united with this oppositional group as well. Thus Galileo was advised to be cautious if he valued his life and his work, and he resigned himself to remain somewhat reticent until he had gathered enough evidence and built up strong enough theories to fully unleash his view of the universe upon civilization. This moment came in 1630, and upon the release of his theories he was soundly punished by his opponents.
Galileo, more than any other scientist of the era, introduced the change in thought that broke with the ancients and led to modern science. The Dialogue presents a character convinced of the Aristotelian system, who is made to look hopelessly stupid, in a symbol of Galileo's conviction that the Aristotelian system, based on arbitrary and mystical hypotheses, was outdated in the world of exact science. Perhaps his greatest theoretical contribution was the argument that the laws of physics operated equally everywhere, a conclusion that vastly expanded the possibility for better understanding of astronomy through terrestrial experimentation. Further, Galileo developed tools, both mathematical and physical, to explore the universe on all of its levels. The telescope allowed for magnification and better resolution of objects at a great distance, and the microscope allowed scientists to observe the complexity of nature on a smaller scale than ever before. However, there was still much more work to be done in the exploration of the heavens. Galileo had not fully integrated his physics with Kepler's theories on the motions of celestial bodies. That was left to Isaac Newton.