Meanwhile, in July of 1499, the French army had reached Lombardy. Ludovico was now in a state of desperation. He tried to appeal to the people of Milan, explaining that their heavy taxes had been due to the constant threats from abroad. But, however hard he tried to arouse their sense of loyalty to him, the public of Milan turned a deaf ear. They had not forgotten how Ludovico had allied himself with Charles VIII—a foreign king! Ludovico now had to put his trust in his army commander, Galeazzo da Sanseverino, despite warnings that this was a man of doubtful loyalty. Moreover, to make matters worse, Louis XII had succeeded in forming an alliance against Ludovico; and, among his allies was a powerful cardinal, son of Pope Alexander VI—the notorious Cesare Borgia. From a note on a page of designs for supplying and heating a bath we know that Leonardo continued his quiet life, only vaguely disturbed by the political upheaval taking place around him. His note reads, “On the first day of August 1499 I wrote here of movement and weight.” He had made many experiments and calculations concerning the movement and weight of objects. He had drawn, for example, the flight of an arrow to describe motion through air and although he wrote no specific formula, he marked the three stages of its trajectory—the initial push, the slowing and the steeper downward path as the arrow’s momentum was overcome by the resistance of the air. He also defined the law of movement on an inclined plane and he arrived at the root principle of Newton’s law of gravitation when he wrote, “every weight tends to fall toward the center by the shortest way.” A diagram of this period is probably the first scientific graph. Leonardo had experimented with two balls dropped from a height. First he dropped them together and then one after the other. In attempting to solve the mathematical problems presented by these falling bodies he drew a graph of vertical and horizontal lines. The times it took for the balls to fall were marked on the horizontal lines and the distances on the vertical lines—thus, he could trace their relationship. But this peaceful time of productive work was running out for Leonardo. Ludovico’s commander, Galeazzo, had yielded the fortress of Alessandria to the French at the first battle. Ludovico himself had sent his sons and his treasure to his brother, Cardinal Ascanio, in Germany. When he saw that his cause was lost, he turned the Sforza castle over to Bernardino da Corte, a trusted commander, making certain that it was fully supplied with arms and food. Then in sorrow, Ludovico Sforza, Duke of Milan, left his city for the last time as ruler of Lombardy. The gates of Milan were opened to the French in October of 1499, and Bernardino da Corte surrendered the Sforza castle.