ost advanced Greek treatises on mathematics were translated in the 16th century, and advances made beyond the ancients included the solution of cubic equations and the innovative astronomy of Nicolaus Copernicus, Tycho Brahe, and Johannes Kepler. By the end of the 16th century, Galileo had taken the crucial step of applying mathematical models to the subject matter of physics. Geography was transformed by new empirical knowledge derived from explorations beyond Europe and from the first translations of the ancient works of Ptolemy and Strabo.In the field of technology, the invention of printing in the 15th century began to revolutionize the dissemination of knowledge. Printing increased the quantity of books, helped eliminate errors, furnished scholars identical texts with which to work, and turned intellectual endeavor into a collaborative rather than a solitary activity. The use of gunpowder transformed warfare between 1450 and 1550. Artillery proved devastatingly effective against the stonewalls of castles and towns. The medieval army, led by cavalry and supported by bowmen, was gradually replaced by one made up of foot soldiers carrying portable firearms and masses of troops with pikes; such armies were the first standing armies of Europe. Renaissance churchmen, particularly in the higher echelons, patterned their behavior after the mores and ethics of lay society. The activities of popes, cardinals, and bishops were scarcely distinguishable from those of secular merchants and political figures. At the same time, Christianity remained a vital and essential element of Renaissance culture. Preachers, such as San Bernardino of Siena, and theologians and prelates, such as Sant'Antonino of Florence, attracted large audiences and were revered. Moreover, many humanists were concerned with theological questions and applied the new philological and historical scholarship to the study and interpretation of the early church fathers. The human...
