194 A SHORT HISTORY OF SCIENCE 



is "to determine a triangle when the difference of two sides, the 

 perpendicular on the base, and the difference between the seg- 

 ments into which the base is divided are given : i.e. a b, a sin B, 

 a cos B b cos A are known ; to find, a, b, c, A, B, C." An- 

 other is to construct from four given lines a quadrilateral which 

 can be inscribed in a circle. 



CONDITIONS NECESSARY FOR PROGRESS. The genius of 

 Hipparchus and Ptolemy had brought Greek astronomy to its 

 culmination. Higher it could not rise until three conditions 

 should be fulfilled, even though here and there the heliocentric 

 hypothesis might be adopted through an unsupported inspiration 

 of individuals. First, there must be better astronomical instru- 

 ments and more accurate observations, extended over long periods. 

 Second, there must be improved methods of mathematical com- 

 putation for the reduction and interpretation of these observations. 

 Third, there must be substantial progress towards clear thinking 

 as to the fundamental facts and laws of motion. These conditions 

 were met one after another during the sixteenth and seventeenth 

 centuries by an extraordinary series of men of genius, among whom 

 the chief were Copernicus, Tycho Brahe, Kepler, Galileo, and 

 Newton. Their work constitutes a great part of the history of 

 science during these two centuries and one of the most won- 

 derful chapters of all time. 



Of these five, Copernicus and Kepler were predominantly inter- 

 ested on the mathematical and theoretical side, Tycho Brahe was 

 a great observer, Galileo combined experimental and observa- 

 tional skill with a new appreciation of physical laws, while Newton, 

 building on the foundation laid by all the others, made a magnifi- 

 cent synthesis of their results into a rational and consistent mathe- 

 matical theory of the solar system. These five represent Poland, 

 South Germany, Denmark, Italy, and England. Scientific progress 

 is no longer localized or dependent on princely patronage. It has 

 now become international. 



NICOLAUS COPERNICUS (1473-1543) was born in the remote 

 little city of Thorn on the Vistula, and having relatives in the 

 Church, prepared himself for an ecclesiastical career. This led 



