1488

THEORY AND PRACTICE OF ENGINEERING.

Book IT.

and afford more or less passage for the water. The tediousness of the preceding method is itsonly objection ; but in the case in which the arch would he lowered J, it is useless tocompose the Hat arch of so large a number of arcs of a circle; five will generally be founda sufficient number, and the process be much simplified.

Suppose, for example, (fig. 2706.) thatthe length A F and D E of the radii of thearc at the springing and the summit bedetermined, and represented by r R : call pthe radius of the intermediate arc, whichmay be determined by the condition of itsbeing a mean proportional between r andR: then we shall have

p =- ^Rr.

From the point F as a centre, and with aradius equal to p — r, describe an arc;and from the point E, with a radius equalto II — p, a second, which shall cut thefirst in G ; the point G will be the centreof an arc uniting that at the springingand that of the summit.

If a flat arch described from sevencentres is required, we must determinetwo mean proportionals, p and p 7 , betweenthe radii of the extreme arcs r and R,which would give

p = 4K R r*, and p'= ^R a r:from the point F as a centre, (fig. 2707.),and with the radius equal to p — r, de-scribe an arc which should contain the centreof the arcs,’the radius of which is p ; andfrom the point E, and with the radiusEII=R —p / , a second arc, which shouldthe radius. To fix on each of these arcs the respective position of the two centres, drawa line II G, the length of which must be equal to p' — p, between the two arcs ; but as theposition of this line is not determined by this single condition, we must imagine it to beso fixed, that the length of the arcs of which the arch is composed decreases uniformlyfrom the summit to the springing. This method may be extended to flat arches composedof a greater number of arcs, but it is useless to use more than five.

In constructing moulds for the arches of bridges, it is not possible, except for the partnear the springing, to use beam compasses for tracing the arcs of which they are composed ;the engineer commences, therefore, by fixing the extremities of each of these arcs, of whichthe data have been calculated beforehand; and they are described by means of two rulesfirmly united, so as to form an angle the supplement of which is equal to half of thearc. This angle is made to move so that the sides always pass through the extreme pointsof the arc : the summit gives the intermediate points.

Flat Arches not formed by Arcs of a Circle. — The difficulty of exactly tracing theprojected curve, when composed of several arcs of a circle, has given rise to differentmethods for describing the flat arch.

Carpenters generally use, for the purpose of unitingtwo sides of an angle A E D a curve, the form ofwhich may bo obtained by dividing the two sides ofthe angle into the same number of equal parts, andby joining the points of division by lines regardedas tangents to the curve, and which, supposing theminfinitely near, determine each of these points bytheir successive intersections; in performing thisoperation for an angle, a portion of the curve equalto the first wull be obtained, which will completethe description of the arch ADC.

It has long been remarked, that the curve tracedby the preceding method was a portion of a para-bola, the summit of which is situated between thepoints A and D ; it gives more water-way than aflat arch composed of three arcs of a circle, or asemicircle which could be constructed on the same

E 12 *3 456789 10 D

Fig. 2708.

Fig. 2707.

contain the centre of the arc of which p/ is