To take, for example, the immortal voyage of Christopher Columbus. His purpose in setting out into the west was to discover a new way to India. The Venetians and Genoese practically possessed the overland route across the Isthmus of Suez and down the Red Sea. Vasco da Gama had opened out the route eastward round the Cape. Firmly convinced that the world was a globe, Columbus saw that a third route was possible, namely, one nearly due west; and when, therefore, he reached the Bahamas, after traversing some 66° of longitude, he believed that he was in the islands of the China Sea, some 230° from Spain. Those who followed him still laboured under the same impression, and when they reached the mainland of America, believed that they were close to the shores of India, which was still distant from them by half the circumference of the globe. Little by little the intrepid sailors of the sixteenth century forced their way to a true knowledge of the size of the globe, and of the relative position of the great continents. But this knowledge was only attained after many disasters and terrible miseries; and though a new kind of navigation was established--the navigation of the open ocean, far away from any possible landmark, a navigation as different as could be conceived from the old method of coasting--yet it remained terribly risky and uncertain throughout the sixteenth century. Therefore many mathematicians endeavoured to solve the problem of determining the position of a ship when at sea. Their suggestions, however, remained entirely fruitless at the time, though in several instances they struck upon principles which are being employed at the present day. The first country to profit by the discovery of America was Spain, and hence Spain was the first to feel keenly the pinch of the problem. In 1598, therefore, Philip III. offered a prize of 100,000 crowns to any one who would devise a method by which a captain of a vessel could determine his position when out of sight of land. Holland, which had recently started on its national existence, and which was challenging the colonial empire of Spain, followed very shortly after with the offer of a reward of 30,000 florins. Not very long after the offer of these rewards, a master mind did work out a simple method for determining the longitude, a method theoretically complete, though practically it proved inapplicable. This was Galileo, who, with his newly invented telescope, had discovered that Jupiter was attended by four satellites.