But the Greek astronomers devised other and better methods for determining the positions of the heavenly bodies. Obelisks or dials were of use only with the Sun and Moon which cast shadows. To determine the position of a star, "sights" like those of a rifle were employed, and these were fixed to circles which were carefully divided, generally into 360 "degrees." As there are 365 days in a year, and as the Sun makes a complete circuit of the Zodiac in this time, it moves very nearly a degree in a day. The twelve Signs of the Zodiac are therefore each 30° in length, and each {44} takes on the average a double-hour to rise or set. While the Sun and Moon are each about half a degree in diameter, _i.e._ about one-sixtieth of the length of a Sign, and therefore take a double-minute to rise or set. Each degree of a circle is therefore divided into 60 minutes, and each minute may be divided into 60 seconds. As the Sun or Moon are each about half a degree, or, more exactly, 32 minutes in diameter, it is clear that, so long as astronomical observations were made by the unaided sight, a minute of arc (written 1') was the smallest division of the circle that could be used. A cord or wire can indeed be detected when seen projected against a moderately bright background if its thickness is a second of arc (written 1")--a sixtieth of a minute--but the wire is merely perceived, not properly defined. Tycho Brahe had achieved the utmost that could be done by the naked eye, and it was the certainty that he could not have made a mistake in an observation in the place of the planet Mars amounting to as much as 8 minutes of arc--that is to say, of a quarter the apparent diameter of the Moon--that made Kepler finally give up all attempts to explain the planetary movements on the doctrine of circular orbits and to try movements in an ellipse. But a contemporary of Kepler, as gifted as he was himself, but in a different direction, was the means of increasing the observing power of the astronomer. GALILEO GALILEI (1564-1642), of a noble Florentine family, was appointed Lecturer in Mathematics at the University of Pisa. Here he soon distinguished himself by his originality of thought, and the ingenuity and decisiveness of his experiments. Up to that time it had been taught that of {45} two bodies the heavier would fall to the ground more quickly than the lighter. Galileo let fall a 100-lb. weight and a 1-lb. weight from the top of the Leaning Tower, and both weights reached the pavement together. By this and other ingenious experiments he laid a firm foundation for the science of mechanics, and he discovered the laws of motion which Newton afterwards formulated. He heard that an instrument had been invented in Holland which seemed to bring distant objects nearer, and, having himself a considerable knowledge of optics, it was not long before he made himself a little telescope. He fixed two