PROGRESS OF MATHEMATICS AND MECHANICS 251 



individual method consisted namely in always conforming to the limits 

 of scientific investigation, and confining his attention to seizing the 

 phenomena sharply in their progress and in their relation with allied 

 processes, without wandering into a fruitless search after the ultimate 

 bases of the phenomena. Dannemann. 



Such a limitation has been of the highest value for the renewal of 

 natural science as it followed at the beginning of the seventeenth 

 century. 



Galileo was not chiefly interested in mathematics, but he em- 

 phasizes the dependence of other sciences upon it. 



True philosophy expounds nature to us; but she can be under- 

 stood only by him who has learned the speech and symbols in 

 which she speaks to us. This speech is mathematics, and its sym- 

 bols are mathematical figures. Philosophy is written in this greatest 

 book, which continually stands open here to the eyes of all, but can- 

 not be understood unless one first learns the language and characters 

 in which it is written. This language is mathematics and the 

 characters are triangles, circles and other mathematical figures. 



He gives an acute discussion of infinite, infinitesimal and con- 

 tinuous quantities leading up to the conclusion "that the attri- 

 butes 'larger,' 'smaller/ and 'equal' have no place either in 

 comparing infinite quantities with each other or in comparing in- 

 finite with finite quantities." Again "the finite parts of a con- 

 tinuum are neither finite nor infinite but correspond to every as- 

 signed number." 



In commenting on Galileo's achievements, Lagrange the great 

 mathematician of the eighteenth century says: 



These discoveries did not bring to him while living as much 

 celebrity as those which he had made in the heavens ; but to-day his 

 work in mechanics forms the most solid and the most real part of the 

 glory of this great man. The discovery of Jupiter's satellites, of the 

 phases of Venus, and the Sun-spots, etc., required only a telescope and 

 assiduity ; but it required an extraordinary genius to unravel the laws 

 of nature in phenomena which one has always under the eye, but the 

 explanation of which, nevertheless, had always baffled the researches 

 of philosophers. 



