SCIENCE OF THE SIXTEENTH CENTURY. 91 



first became Tycho's assistant; but it was the observations of the 

 Danish astronomer which furnished the data for the endless calcula- 

 tions made by Kepler in successively testing one theory after another 

 until he had hit upon one according with the facts. He found that 

 it was impossible to represent the observed positions of Mars by any 

 combinations of uniform circular motions, and after testing by calcu- 

 lation every theory which presented itself to his mind, he was led at 

 length to the supposition that the orbit of Mars was an ellipse, having 

 the sun at one of its foci. The same laws, he soon found, applied to 

 the other planets, and hence the first of the three celebrated theorems 

 called Kepler's Laws : The planets move in elliptical orbits round the 

 sun, which is placed at one of the foci. The second law which he de- 

 duced in the course of this investigation is famous among astronomers 

 as " the Law of Equal Areas." Its purport 

 may easily be understood by any reader with 

 the aid of the diagram Fig. 37. Let the 

 ellipse represent the orbit of a planet Mars, 

 for example ; and let s be the place of the 

 sun in one of the foci of the ellipse. Let M X 

 be the place of the planet at a given instant, 

 and M 2 its place after any given interval of 

 time, as an hour, a day, or a month ; and, 

 further, let M 3 be the planet's place in its FIG. 37. 



orbit after another equal interval of time ; so 



that at the end of the second hour, day, month, etc., from M 2 the planet 

 is passing the position M 3 . Suppose that a straight line always passing 

 from the centre of the sun to the centre of the planet is carried round 

 by the movement of the planet, like the cord tied to a stone whirled 

 in a circle about the hand. Kepler discovered that in the case of 

 Mars, and of each of the other planets, this line (the radius vector} 

 swept out equal areas in equal times. Thus in our diagram the area 

 of the space enclosed by the lines M 2 s, s M 3 , and the arc of the ellipse 

 intercepted between them, would exactly equal the similar area en- 

 closed by M 2 s, s M 2 , and M X M 2 . 



Kepler was conducted to brilliant results by a method very different 

 from that by which others have advanced to great discoveries. His 

 reasonings were often vague, and they frequently rested on no other 

 foundations than arbitrary assumptions. But Kepler had much mathe- 

 matical knowledge, and his perseverence in calculations which would: 

 appal most men was something extraordinary. He would assume an 

 hypothesis at random almost, and work out its consequence by labo- 

 rious calculations. The results thus obtained he then carefully com- 

 pared with actual observation : if they contradicted it, he abandoned 

 the hypothesis, and tried another the first that presented itself to his 

 fertile imagination repeating his calculation on the fresh supposition, 

 however formidable the labour this might involve, and again rejecting 



