The activity incident to these great movements made Florence and Venice renowned for their wealth, while it also gave the Hanseatic League command of the trade of the north. With the growth of prosperity came increased leisure for intellectual development, resulting in the Italian Renaissance and the European Revival of Learning. The crusades also influenced development still more directly by opening lines of communication with the east, whereby the learning that had lain dormant in the Byzantine empire became current in Europe.
Toward the close of the fifteenth century the discovery of America, closely followed by the circumnavigation of the world, gave dominance once more to the influence of the sea. The effect of such a strong suggestion of boundless and unknown possibilities, intensified by the element of hazard and daring, became at once an important factor in development, stimulating ambition, creating moral fiber and inspiring a passion for freedom. With the opening of the sixteenth century the narrow and vague ideas characteristic of scholasticism began to give place to clear and strong thinking. As the church had been the center and source of medieval authority, the struggle for freedom naturally centered around this institution. Beginning with the reform of certain abuses, the spirit of the reformation ended by repudiating the entire authority of the church, epitomized by the action of Luther in nailing his ninety-five theses to the door of the church in Wittemberg, thus undermining the whole system of tradition and inaugurating a new principle of action based on individuality.
The relation of this mental attitude to the development of culture was nowhere more evident than in the trend taken by mathematics. Everywhere old methods were questioned and new ones substituted. The first great advance naturally occurred in Germany and Italy. In the former the time-honored system of Ptolemaic astronomy gave place to the Copernican theory, and notable advances were also made in other branches of mathematics, especially algebra and trigonometry. The intervention of the Thirty Years' War, followed by the Prussian war, stayed German development for a time, but with the return of peace the German spirit again manifested itself in the critical attitude toward science and religion which found expression in mathematics in the function theory, and in philosophy and religion in agnosticism.
In France where the invigorating effects of climate and race were less marked, the sixteenth century was characterized by such acts of religious intolerance as the massacres of Vassy and St. Bartholomew, leaving no energy for scientific pursuits. The ascension of Henry IV. to the throne, however, followed by the Edict of Nantes which terminated the religious strife, produced an immediate effect, the Age of Richelieu being remarkable for scientific and cultural progress. Great literature was produced and in mathematics the period was made illus-
