ARCHES AND ARCHED RIBS 175 



of one joint pressure, as explained in Article 139 ; or, in general, it 

 is necessary to have given three conditions which the equilibrium 

 polygon must satisfy, such, for instance, as three points through 

 wliich it is required to pass. Since it is impossible to determine these 

 three unknowns by the principles of mechanics, the theory of the 

 arch has long been a subject of controversy among engineers and 

 mathematicians. 



Among the various theories of the arch which have been proposed 

 from time to time, the first and most important of the older theories 

 is called the principle of least resistance. This theory was introduced by 

 the English engineer, Moseley, in 1837, and later became famous on the 

 Continent through a German translation of Moseley 's work by Scheffler. 



In building an arch the material is assembled upon a wooden frame- 

 work called a center ; when the arch is complete this center is removed 

 and the arch becomes self-supporting, as explained in Article 137. 

 Now suppose that instead of removing the center suddenly, it is 

 gradually lowered so that the arch becomes self-supporting by degrees. 

 In this case the horizontal pressure or thrust at the crown gradually 

 increases until the center has been completely removed, when it has 

 its least possible value. This hypothesis of least crown thrust con- 

 sistent with stability is Moseley's principle of least resistance. 



In rnii>tru-tin.u r an equilibrium polygon the horizontal force, or 

 pole distance, is least when the height of the polygon is a maximum. 

 Therefore, in order to apply the principle of least resistance, the equi- 

 librium polygon must pass through the highest point of the extrados 

 at the crown and the lowest points of the intrados at the abutments. 

 Since this would cause tensile stress at both the crown and abut- 

 ments, the criterion for stability against overturning makes it neces- 

 sary in applying the theory to move the center and ends of the 

 equilibrium polygon, or linear arch, until it falls within the middle 

 third of the arch ring. There is nothing in the principle of least 

 resistance, however, to warrant this change in the position of the 

 equilibrium polygon, and consequently the theory is inconsistent with 

 its application. 



Culinann tried to overcome this objection to Moseley's theory by 

 considering the compressibility of the mortar between the joints. At 

 the points of greatest pressure the mortar will be compressed more 



