t 



k 



On the different Theories of Arches, Vaults, &c. 411 



become the source of the various theories which have been 

 invented to determine the equilibration of the arch. 

 While the corporation of the Free Masons as builders was 

 expiring, Dr. Robert Hooke, under the mask of a cipher, 

 gave what he calls "The true mathematical and mechanical 

 form of all manner of arches for building, with the true 

 hutment necessary to each of them, a problem which no 

 architectonic writer halh ever yet attempted, much less per- 

 formed." The cijiher deciphered is, " ut pendei continuum 

 flexile sic stabil contiguum rigidum inversum." Dr. Hooke 

 applies the principle of the catenaria not only to all man- 

 ner of arches, but their abutments also. About this time 

 the problem of the catenaria was investigated by James and 

 John Bernoulli, Huyghens, Leibnitz, and Dr. David Gre- 

 gory : the latter in his paper in the Philosophical Transac- 

 tions deduces the relation of an arch and its abutment walls 

 from the catenaria. He, like Hooke, claims the priority 

 of the invention (see his answer to the animadversions on 

 his paper on the catenaria). Dr. D. Gregory's deduction 

 translated is : " None but the catenaria is the figure of a 

 true and legitimate arch or fornix ; and when an arch of 

 another figure is supported, it is because in its thickness 

 some catenaria is included ; neither would it be sustained, 

 if it were very thin and coiuposcd of lubricous parrs." From 

 a preceding corollary he says " it may be collected, by what 

 force an arch or buttress presses a wall outwardly, to which 

 it is applied. P'or this is the same with that part of the 

 force sustaining the chain; which draws according to a 

 liorizontal direction. For the force which in a chain draws 

 inwards, in an arch equal to the chain drives outwards." 



James Bernoulli, afer the manner of Gregory to whom 

 he refers, gave two solutions of the problem " De curvatura 

 fornicis, cujus partes se mutuo proprio pondere sufFulciunt 

 sine opere caementi." They were published in his post- 

 humous works (see Opera Jac. Dcr, Gen. 1744, page 1119). 

 The application of the theory of ihe catenaria to the abut- 

 ment, by these mathematicians, seems to favour the theory 

 by which an arch, standine on a pier, should be considered 

 an arch Of greater altitude, or to contain in its thickness a 

 catenaria. 



M. De la Hire, in his Trait c de ]Meca?:iqne, 1695, pub- 

 lished from the theory of the wedge, the proportion accord- 

 ing to which thcvoussoir ought to be augmemed, from the 

 key-stone to the i|r)post, in a semicircular vault. His theo- 

 rem is this. Lei ABC (lig. 1, Fl. XL) be a semicircular 

 arch toinposed of many equal voussoirs^ and if from the 



vertex 



