And yet, Leonardo refused to remain idle—he had to work. The need for mirrors in the vast halls and rooms of the papal palace was great. Leonardo turned his mechanical skill to redesigning and improving methods of making them, and even inventing his own machines for the grinding of the glass. Also, for Giuliano, who dabbled in alchemy and magic, he made distorting mirrors and burning lenses. In addition, Leonardo invented a machine which could be run hydraulically for producing long strips of copper of equal width for use in soldering the mirrors. But, with the making of these mirrors, Leonardo began to run into trouble with his German assistant, Georg. The boy was a loafer; he spoke little Italian and took every opportunity to spend his days with his countrymen in the Swiss guard. Leonardo tried to alter the situation by suggesting that the boy have his meals with him at his worktable, thus giving Georg a better chance to learn the language. This however did not appeal to him. Then, because Leonardo’s inventions were so extraordinary, he began to give away the secrets of their mechanisms to Johannes the mirror-maker, another German, who had been replaced by Leonardo in the favors of Giuliano. This naturally made Johannes jealous of Leonardo. Georg gossiped, too, and told stories about the old, eccentric man who lived like a miser in the midst of all the luxury and who drew crazy circles on pages of paper. These “crazy circles” were geometric exercises that had fascinated Leonardo from the time he had wandered across Italy with Fra Luca Pacioli. Pacioli’s book _De Divina Proportione_, containing sixty illustrations from designs of Leonardo, had been published in Venice in 1509. Leonardo intended to entitle these geometric exercises _De Ludo Geometrico_. In geometry a lune is a crescent-shaped figure bounded by two intersecting arcs of circles on a plane or a sphere. Leonardo drew pages of these lunes and then proceeded to transform their curvilinear figures into squares of equal area. He also reviewed Archimedes’ method of squaring a circle and developed it into a variety of ways for cubing spheres and cylinders. He returned as well to formulating theories of friction. He wrote in his notes, “the tallest wheel is the easiest to pull”—for example, a big wheel turning at the same speed as a smaller one has less friction to overcome because it makes less revolutions. His experiments in friction predated men like Amontons and Coulomb by two and three centuries. He established a formula for the building arch which he described as “a strength caused by two weaknesses”—if one half of an arch is removed, the other half collapses. They support and give strength to each other. In addition, Leonardo determined, before Galileo, the center of gravity of any pyramid and of a tetrahedral, or four-sided body.