412 On the differ e7it Theories of Arches, Vaults, &c. 



vertex B of the key-stoiie a line BO be drawn at right an- 

 gles to BG, and the radii answering to the beds of the voiis- 

 soirs PQli be prolonged until they intersect BO, all the 

 voussoirs will be in equilibration, if their absolute weight 

 be expressed bv HK, KL, LI\1, Sec. the distances between 

 the points of intersection. 



'{"he tiistorian of the Academy of Sciences at Paris reports, 

 1704, that i\J. Parent determined accordinu; to the same 

 principles, hv points, the form which the extrados of a semi- 

 circular vault and pier ought to assume. This solution has 

 not been seen by the writer of this paper, nor docs he kno.v 

 whether it has been printed. 



In 1712, M. de la Hire inserted a paper in the Mtm. 

 AccuL Par. in v\ hich he considers the problem of the re- 

 8i>tance of abutment piers umn the suppositions which. 

 have been before hinted at. He gives a mode of construct- 

 ing an ahutmeut pier, which is as follows. Let pv (fig. 1) 

 be a quadrant of a circle, bisected in I.Jo the extrados, and 

 fc perpendicular to the horizon, c live centre of the circle, 

 5/?. the base of the pier. Parailel \.o fc draw //;, cutting 

 the base line in b. Through L draw qd perpendicular to 

 fc, cutting vc in s, iVJake Ix and 1% equal to the square 

 root of the superficies of the part of the vault Imfv (whicli 



bv prop, of ivi. Couplet = -—~ X ^y^i'cr passing though' 



I, and fr and iv being tangents to the intrados and extrados). 

 Draw zi and its parallel xa, cutting Lb in a. Draw /•=, and 



cq perpendicular to it, cutting qd\n q. Ivlake Id =s ~ + 



sq — sb, and having ntade Ih = It/, with the radius hd and 

 centre d, descrii)e the arc hi, cutting lb in ?'. Make s/t = 

 {/ — li/, the wulth of the abutment pier required. 



In tiie volumes 17-26, 1727 and 17-S, Acad. Sci. Par, 

 M. Couplet published a paper on embankment walls, and 

 was thence induced in 17'-20 and 1730 to publish two iiie- 

 nioirs on the equilibration, principally of circular vaults, 

 and the thickness of their, abutment piers : in the first he 

 considers the voussoirs, as in all fornter inquiries ihcy had 

 been considered, as perfectly lubricous: he kivestigales 

 from the funicular polvgon the vault of uniform thickness: 

 he also considers the charge uf a vault on a centre of wood, 

 in reference to the paper of M. Pilot in vol. 1 726. In the 

 second memoir he takes into the consideration, cohesion 

 and fricljon. iM. Bouguer observes, that we maybe sure, 

 that if the voussoirs will maiiitain their places when per- 

 fectly s'.ippervj they must when subject to cohesion and 



friction ; 



