PRINCIPLES OF MECHANICS. 539 



and rupture end, and the compression and crushing begin : a point 

 which has been called the neutral axis. This was pointed out by 

 Mariotte ; and the notion, once suggested, was so manifestly true that 

 it was adopted by mathematicians in general. James Bernoulli, 5 in 

 1705, investigated the strength of beams on this view ; and several 

 eminent mathematicians pursued the subject ; as Varignon, Parent, 

 and Bulfinger ; and at a later period, Dr. Robison in our own country. 



But along with the fracture of beams, the mathematicians consid- 

 ered also another subject, the flexure of beams, which they undergo 

 before they break, in virtue of their elasticity. What is the elastic 

 curve ? — the curve into which an elastic line forms itself under the 

 pressure of a weight — is a problem which had been proposed by Gal- 

 ileo, and was fully solved, as a mathematical problem, by Euler and 

 others. 



But beams in practice are not mere lines : they are solids. And 

 their resistance to flexure, and the amount of it, depends upon the re- 

 sistance of their internal parts to extension and compression, and is 

 different for different substances. To measure these differences, Dr. 

 Thomas Young introduced the notion of the Modulus of Elasticity : 3 

 meaning thereby a column of the substance of the same diameter, such 

 as would by its weight produce a compression equal to the whole 

 length of the beam, the rate of compression being supposed to continue 

 the same throughout. Thus if a rod of any kind, 100 inches long, 

 were compressed 1 inch by a weight 1000 pounds, the weight of its 

 modulus of elasticity would be 100,000 pounds. This notion assumes 

 Hooke's law that the extension of a substance is as its tension ; and 

 extends this law to compression also. 



There is this great advantage in introducing the definition of the 

 Modulus of Elasticity, — that it applies equally to the flexure of a sub- 

 stance and to the minute vibrations which propagate sound, and the 

 like. And the notion was applied so as to lead to curious and impor- 

 tant results with regard to the power of beams to resist flexure, not 

 only when loaded transversely, but when pressed in the direction of 

 their length, and in any oblique direction. 



But in the fracture of beams, the resistance to extension and to com- 

 pression are not practically equal ; and it was necessary to determine 



a Opera, ii. p. 976. 



s Lecture xiii. The height of the modulus is the same for the same substance, 

 whatever its breadth and thickness may be ; for atmospheric air it is about five 

 miles, and for steel nearly 1500 miles. 



