PROGRESS OF MATHEMATICS AND MECHANICS 249 



In Statics Galileo had somewhat more from the ancients to 

 build upon. To him we owe the formulation of the law of virtual 

 velocities applying dynamical ideas to problems of statics. If 

 two forces are in equilibrium they are proportional to the cor- 

 responding paths, or : What one by any machine gains in power 

 is lost in distance. The parallelogram or triangle of forces in 

 equilibrium however escapes him, and his ideas about impulse 

 though remarkably in advance of his time were not fully worked 

 out. 



He investigates strength of materials under tension and fracture, 

 with reference to practical applications in construction. He draws 

 just inferences in regard to the relation between strength and 

 size of plants and animals as well as machines, comparing for 

 example hollow bones and straws with solid bodies of similar mass. 

 He derives an important formula for the stiffness of a horizontal 

 beam supported at one end and regarded as a lever. He discusses 

 the curve formed by a cord suspended between two points, recog- 

 nizing that it is not a parabola. 



In Hydrostatics he reviews the known work of Archimedes and 

 corrects the error of the Aristotelians in regard to the dependence 

 of floating on specific gravity. He develops the modern theory 

 that the fundamental factor in the mechanics of fluids is that they 

 consist of freely moving particles yielding to the slightest force. 

 He makes effective application of the principle of virtual velocities 

 to fluids. At the close of his third conversation he expresses his 

 modest confidence in the great future of his new ideas. 



The theorems set forth in this brief discussion will, when they 

 come into the hands of other investigators, continually lead to wonder- 

 ful new knowledge. It is conceivable that in such a manner a worthy 

 treatment may be gradually extended to all the realms of nature 



a prediction magnificently fulfilled in succeeding genera- 

 tions. 



Among other branches of physics in which Galileo accomplished 

 work of value may be mentioned the expansion by heat the 

 beginnings of thermometry, experiments on the acoustics of 



