DR WALLACE ON A FUNCTIONAL EQUATION. 635 



I am next, from the same source, but with the aid of the proposition that 

 has just now been demonstrated, to establish the theory of curves of equihbra- 

 tion, a practical application of mathematics which is of essential importance in 

 the construction of bridges, 



II. ARCH OF EQUILIBRATION. 



18. The construction of a bridge of considerable length, such as those across 

 the Thames at London, or that over the Menai Strait at Bangor, is one of the 

 noblest achievements of human power, whether we consider its conception or 

 execution. It has long exercised the ingenuity of mechanicians in devising arches 

 which shall unite the properties of stability with elegance of form. 



19. There are two chief theories regarding the proper form of an arch, 

 both resting on the principles of Geometry and Statics. One of these, the more 

 ancient, is the wedge theory. In this, the arch is formed of a series of stone- 

 wedges, which ought to be so adapted to each other, in regard to weight and po- 

 sition, that they shall have no tendency to move in any direction ; the pressures 

 throughout the arch being either counteracted by equal opposite pressures, or else 

 exerted against fixed points of support. 



The French mathematician La Hire explained this theory in a treatise on 

 mechanics, printed in 1695. It was followed by other French engineers and ma- 

 thematicians ; as by Parent, in the Memoirs of the French Academy for 1704 ; by 

 Couplet, in the same work for 1729; by Bouguer and Bossut, in 1774 and 1776 ; 

 and again by this last writer, in the third volume of his Cours cle Mathematiques ; 

 and in this country by the late Mr Atwood, who published a Dissertation on the 

 Construction and Properties of Arches in 1801 ; to this a Supplement was given in 

 1804. 



20. The second theory of an arch is that deduced from the properties of the 

 curve, formed by a cord or chain hanging ft-eely in a vertical plane from two fixed 

 points, which, because of the way in which it is formed, is called the Catenaria 

 or Catenary. This curve was fii-st noticed by Galileo, who, however, did not 

 precisely comprehend its nature, for, in his second dialogue on motion, he says 

 that it is a parabola ; but again, in his fourth dialogue, he says that, to a certain 

 extent in the lower part of the curve, it differs very little from a parabola. I 

 notice this because it has been said that he believed it to be exactly a parabola.* 

 The discovery of the true nature of the curve was hardly within the power of the 

 mathematical science of Galileo's time. The method of fluxions of Newton, 

 and the differential calculus of Leibnitz, however, enabled mathematicians to 

 surmount this, and many other difficulties in statics. 



* Leslie's Geometrical Analysis. The Catenary. 



