Another exception was, in a certain measure, mathematics. Being the most abstract of the sciences, it quite early derived generally valid propositions. The multiplication table is just as true in the Andromeda Nebula and in the innumerable other parts of the universe as it is on Earth. Although mathematical deductions and rules were based on terrestrial data, their essence went beyond the confines of the Earth, unlike the other, earthbound branches of science. However, the cosmic character (in the above sense) of mathematical thought was not realized even by its very representatives and creators (barring some philosophical speculations). Most important: the cosmic aspect was never explicitly stated in mathematics. This science never studied the quantitative properties of an actually given cosmic infinity, that is, it did not give any representation of the universe defined as an infinite entity.
Of course, the concept of infinity gradually made its way into mathematics. The problem of the infinitesimal was posed as far back as the time of the ancient Greeks. However, the first real progress in this direction occurred when the differential and integral calculus were created. Significantly enough, it was not until the concept of an infinitely great quantity was formulated (in the 17th century) that the first step toward the mathematical study of the universe was made.
The science of prehistorical time was thus above all a terrestrial science, even taking into consideration some individual exceptions (which did not go very far at any rate). Engels had good grounds for writing in his "Dialectics of Nature" that: "All of our official physics, chemistry, and biology are exclusively g e o c e n t r i c, designed solely for the Earth." In the natural science of modern times, however, new, cosmic trends sprouted forth from the terrestrial groundwork, notably, as was to be expected, in astronomy and mathematics. The achievement of Copernicus, continued by Giordano Bruno and Galilei, announced the definite cleavage of astronomical science from its terrestrial bondage. And though the change in outlook involved only the solar system, while everything that lay beyond was still regarded as a sphere of motionless stars, the Copernican system truly constituted a revolution, imparting the first serious blow to the geocentrism which had pervaded natural science up till then.
The discovery of spectral analysis led to other advances toward the proper knowledge of the universe. Until then astronomy had been chiefly based on classical mechanics'; from that point onward, the whole complex of astronomical science underwent a thoroughgoing refashioning on the basis of astrophysics. This complex was thus substantially supplemented by newly
developed astrophysical or related disciplines.
It may be said that the introduction of astrophysical methods and concepts (of a distinctly non-terrestrial character) brought about a rebirth of astro nomy. This new, essentially astrophysical astronomy differs from the old, classical one mainly in its having renounced the intrinsically terrestrial standpoint in the study of the cosmos.
Progress in the astronomical sciences has since proceeded at an accelerated pace, especially in astrophysics, which has come to be increasingly relied upon in the investigation of the regions around the Sun and in the study of galactic and extragalactic space. A new stage was reached with the advent and rapid development of radioastronomy (together with its offshoot, radar astronomy). This technique has been a powerful tool for probing into cosmic processes and making new discoveries, and has considerably expanded the portion of the universe accessible to observation.
Thus, since the time of Copernicus, astronomy has firmly set out on the way to becoming a truly cosmic science, with its sights free from terrestrial fog. The end of the 19th century and the first half of the 20th century have witnessed spectacular advances, both qualitative and quantitative, in the astronomical sciences. The line of development traced above constitutes the first fundamental trend toward the cosmization of natural science.
Following astronomy came mathematics. Aside from a few preceding efforts of little significance, the turn toward the cosmos was signaled by the creation of Lobachevskian geometry. Its cosmic character was perceived (at least in part) by the author himself. The strange content and results of the new geometry resulted from a definite cause. The mathematics involved dealt with properties of space that become manifest only over extensive portions of the universe and are "not visible" on a terrestrial scale.
Of course, the non-Euclidean geometry of Lobachevskii just laid out a new domain of mathematical interest. This initial venture was soonfollowed • J.D. Bernal: "... The greatest triumph of science of the 17th century was doubtlessly the completion of a general system of mechanics, capable of explaining the motion of the stars in terms of the behavior of matter observed on the Earth" (emphasis mine— E.F.).