5'36 DR WALLACE ON A FUNCTIONAL EQUATION. 



21. James Bernouilli, in the year 1690, proposed in the Leipsic Acts " to 

 find the nature of the curve formed by a rope which hangs freely suspended be- 

 tween two fixed points." This problem was resolved by Huygens, Leibnitz, and 

 John Bernouilli. In England, David Gregory gave a complete solution of Ber- 

 NOuiLLi's problem.* In his memoir, he says, " The catena, placed in an inverted 

 position, maintains its figure, and does not fall downwards, so makes a thin arc 

 or fornix ;" and he afterwards adds, " The catenaria are the only true arches or 

 for7iices, and an arch of any other figure is sustained for this reason only, because 



a catenaria is included in its thickness." In this assertion Gregory went too 

 far ; for, as John Bernouilli truly said, an arch may have the form of a circle or 

 an ellipse, or indeed any curve whatever, and be perfectly secure, by adapting 

 the mass which it supports to its form. 



22. Writers on the Method of Fluxions have exemplified the use of that cal- 

 culus by applying it to the catenary, and arches of equilibration. Thus, Emerson 

 gave their theory in various parts of his works ; and Dr Charles Hutton has also 

 explained it in his Treatise on Bridges. He there also adverts to the wedge 

 theory, which had been delivered before by Atwood. 



Dr J. RoBisoN, formerly Professor of Natural Philosophy in our University, 

 adopted the theory of equilibrated curves, in a valuable article on Arches which 

 he contributed to the Supplement to the fourth edition of the Encyclopaedia Bri- 

 tannica.f Since that essay was published, a period of nearly forty years has 

 elapsed, and in this time Bridges of Suspension have come much into use. To 

 these the simple catenary, which is inapplicable to stone bridges, finds an im- 

 portant application. On this subject the late Da vies Gilbert, Esq., Presi- 

 dent of the Royal Society of London, gave a memoir, which is published in its 

 Transactions for 1826. This contains tables of the co-ordinates and arch of a 

 catenary ; the numbers extend to eight decimal places, supposing the parameter 

 to be an unit. Such tables, if correct, must be highly useful to engineers in the 

 construction of bridges ; it so happens, however, that, instead of the numbers 

 being true to eight decimal places, they are only exact in general to about five. 

 If the last three figures of each be rejected, the remaining figures will be nearly 

 correct. 



23. The roadway along a bridge should be nearly a horizontal straight line. 

 An exact catenarian arch, with such a roadway, would require to be of great 



* Philosophical Transactions, No. 231, (vol. i. p. 39 of Lowthorp's Abridgment), Gregory's 

 Memoir, which was in Latin, was translated and published in Miscellanea Curiosa, edited (I believe) 

 by Dr Derham. 



■ t His articles in that edition of the Encyclopaedia and its Supplement, were, in 1822, collected 

 and published in 4 vols. 8vo. The article on Arches is republished in the seventh edition of the Ency- 

 clopsedia, to which I added a short supplement on Equilibrated Curves. 



