Chap. XXV.

ON STONE BRIDGES*

1487

The forms which are here described for the arches of bridges are given without anyreference to their equilibrium, a subject which has been found one of the most difficult themathematician has had to consider, and perhaps of little value to the practical engineer, ifit could be rightly solved ; for the latter must ever be guided by those rules which expe-rience dictates ; if he follow implicitly in the path theory points out, he may fail, fromobserving the nice and too often close deductions made in her calculations : when, however,theory is borne out by practice, we are warranted in submitting to all that is propounded.Emerson’s Fluxions and Hutton’s Principles of Bridges may be consulted by the readerwho is desirous of becoming acquainted with these delicate but important applications ofmathematical science. The practical method of hanging up a semicircular polygon, withweights at the joints of each link or side, will afford the student data for calculating theform that should be given to the extradossing of an arch of any figure, as the weights whichwould pull it into any given curve would, if placed upon an arch, maintain it in the sameposition, or cause it to be in a state of equilibrium.

Offlat Arches described with more than three Arcs of a Circle. — Different methods havebeen employed to determine the positions of the centre, and the length of the radii of thearcs of which these curves should be composed; the following was that adopted forthe arches of the bridge at Neuilly .

After having fixed the radius F B of thefirst arc at the springing, on the pro-longation of the lesser diameter C D, adistance is taken, C E, which is arbi-trarily made triple C F, and which mightbear any other proportion to that line:having then divided C Fi into five equalparts, C F into five parts, which are toeach other in the proportion of the num-bers 1, 2, 3, 4, and 5, join the points ofdivision by the lines LF,MG,XH,OI,

E K ; take for centres of the differentarcs which compose the arch the pointsE, P, Q, R, S, F, which are found at therespective intersection of these lines.

In the curve described in this manner,the relation of the height C D to thespan A B depends on the data from•which we started, that is to say, on thelength C F, and of its proportion to CE.

But when a flat arch is to be described,the two diameters are generally fixedbeforehand, so that after having con-structed a curve by the preceding method,that curve must be so modified that theheight C D becomes precisely equal tothat which is assumed. Call a half thespan of the arch required, and b itsheight, x the dimension C F, and y thatof C E. Suppose, moreover, that the primitive and arbitrary values which shall have beengiven to C F and C E be represented by n and m ; and that by the development of theportion of the polygon EPQ.RSFB, a length =s results, whilst that in the flat archdescribed on the diameters a and b this length will be = z. We have the relation,

z + a — x = y + b ;

and if we suppose the figure that will be constructed on the lines x and y to be similar tothe figure E C F, constructed on the lines C F = n, and C E = ?«, we shall have

Fig. 2706.

y= —, z = •*

n n

substituting these values in the preceding equation and taking that of x, we find

n (a —6)

which is the value to be given to CF, in order that the opening and the height of thearch may be precisely equal to the lines represented by a and b.

This method is applied to the flat arch, composed of any number of arcs whatever: bymeans ot* the same centres, parallel curves may be described, in which the proportions ofthe diameters shall vary; and if the ratio between C F and CE be correctly determined,curves will be obtained, which, although having the same axis, will assume different forms,