B 11 1 D G K. 



low it, nor injure the bridge by splintering away 

 its corners. 1 his thickness is made .!.; email as pos- 

 sible, that the bridge may not be unnecessarily clcva- 



id the roadway is preserved nearly liori/ 

 The other courses of archstones too, do not often 

 differ much in thickness from that at the crown. 

 But although these things are pretty constant, then- 

 is a considerable degree of latitude in filling up the 

 space between the back of the arch and the road- 

 way. It may be done with substances varying in den- 

 sity, from the lightest charcoal or pumice, open shi- 

 ver or chalk, to closely rammed clay, or even solid 

 masonry ; and it is not uncommon to make, in various 

 ways, open spaces in the masonry of the spandrel, co- 

 vering them above, so as still to support the roadway. 

 It will therefore be proper for us to enquire, what 

 is the density requisite over every section of an arch, 

 where the thickness of the crown is given, the road- 

 way horizontal, the arch of uniform thickness, and 

 the angles of abutment of the several sections constant, 

 that is, all drawn from the same centre ; or, what is 

 the same thing, let us suppose the structure built up 

 to the horizontal roadway with parallel sides, and 

 then enquire, what is the proportion between the 

 pressure borne by each section, in this way and the 

 pressure of equilibrium ; we shall thereby discover 



iiio in which the density of the backing must, 

 it needful, In- iliiiiini .hcd ; .mil the quantity of ex- 



'in necessary towan'. , of the an 



that tin- advantages of equilibration may be preserved, 

 even in this state of things. 



Before we give a more rigid determination, 

 should wish to shew the practical builder, that thr 

 solution of this problem m,iy be easily approximated 

 to, by the help of the trigonometrical tallies. For 

 we may suppose the matter of the archstones to b 

 the same in specific gr.w : 'y with that which lies 

 above it; and as there can be no impropriety in con- 

 sidering the arch as polygonal, from joint to joint, 

 our mean versed sine is only half the sum of those 

 at the two joints. The supposition is not strictly 

 accurate, but it is sufficiently near : greater strictness 

 would only serve to render the calculation more com- 

 plicated, without making it more useful. 



The following Table exhibits, in the first line, the 

 supposed sections of the polygonal vault, taken 5 

 asunder. The second line is the angles of abutment, 

 or inclination of each joint to the vertical. The 

 third, the ratio of weight in each section, taking as 

 the standard 8721, or the length of the arch of 5 in 

 lOOOths of radius ; of course this line is merely the 

 differences of the natural tangents taken at every 5. 



Key 



3 



4 



6 



10 



11 



12 



22* 271 



47 J 



524 



571' 



872 880 



900 



936 | 989 | 1UQ3 | 116-t | 1302 | 14-89 | 1749 | 2119 | 266 1 



The excess of each section over the key section will therefore be 



430 | 



28 



117 



191 



292 



But we must consider, that a part of the thickness at the crown is roadway, and this not, like the 

 arch-stones, kept of uniform magnitude over each section, but must be diminished in the ratio of the 

 differences of the sines of the abutments, or horizontal bases. These differences are 



872 



869 



859 



842 



819 | 790 



1* 



668 



C17 



561 



499 



Now suppose the roadway, and other constant superstructure, to be at the crown of equal weight with 

 the arch-stones; a supposition not far from the truth, and from which any small variation is not of 

 great consequence ; the weight of arch-stone and roadway will be a mean beween lines 3 and 5, or 

 872 | 871 | 866 r857~|~8T6'T~83 1 | 8lT~| 793 |~770 | 745 | 716 | 686 |~ 



And the excess of weight necessary on each section, will be 



79 



14.3 



232 



'MO 



509 



719 | IQOt | 14-03 | 1978 



Which is only about a fifth more than the preceding. And this must be provided for in the spandrel, 

 or over the flank of the arch ; the solidity of which will be found, by multiplying the numbers in 

 line 5 as bases, into the mean height of the matter at each section. These mean heights, or versed sines, 



.00024. | .00381 1 .01519] .03-1-07 ^06037 1 .09369 1 .1331)7 1 .18085 |. 23306 | .29289 | .35721 | .1261^ |_ 

 when the roadway is horizontal. And the effect of line 5, into line 8, which will express the end of 

 the prismatic section over each arch-stone, will be _ 



'~' ~~~'~~~ JJ200.3J 



.2 



3.1- 



129.1 



In which observe, that the unit of height is radius, the bases being expressed in the same notation as in 

 expressing the weights. If a number in this line be divided by its corresponding number in line 7, it 

 will express the thickness at the crown, in terms of the radius, which equilibrates the matter over the 

 corresponding section, between the arch and^horizontal roadway ; when that stuff is filled solid, and of 

 equal density with the arch, of course these thicknesses are 



i- ' . '--i . __ i .. . r .. . _ i .', 7TT '" i " ' , ... . .- . i ' 



.385 | .363 | .345 | .319 | .288 | .254 | .217 | .ISO | .1*3 | 



.39 



.108 



The use of this will be understood by the follow- 

 ing example. Suppose the thickness at crown 4 of 

 the radius of the circle, or ,142857> which answers 



VOL. IV. PAUT II. 



very nearly to the llth key. We see by this last 



table, that the flanks of the arch over that key, or 



at 50 on each side of the arch, must be filled solid 



3u 



