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."