Monday 3 October 2022

# 5 from old math term notes: Catenary


A catenary curve is the shape that a perfectly uniform rope would form when suspended between two points. The word is from the Latin catena for chain. The name was applied by Christen Huygens while studying the form of suspended chains. Galileo thought the shape would be a parabola. As can be seen from the image below, near the vertex a parabola and a catenary look very similar. When x is slightly greater than three however, the catenary begins to rapidly outgrow the value of the parabola. 
The two shapes are related by another relationship. If a parabola is rolled along a straight line, the focus of the parabola will move along a catenary curve. In the figure the parabola y=x2 + 1 is on the inside and the catanary whose equation is y= (ex+e-x)/2, is on the outside. 

For students of American History, it may be interesting that the first use of "catenary", rather than the longer, more formal "catenaria", may have been in a letter from Thomas Jefferson to Thomas Paine. Jeff Miller's wonderful web-site on the first use of mathematical words has 

In a letter to Thomas Jefferson dated Sept. 15, 1788, Thomas Paine, discussing the design of a bridge, used the term catenarian arch: 
Whether I shall set off a catenarian Arch or an Arch of a Circle I have not yet determined, but I mean to set off both and take my choice. There is one objection against a Catenarian Arch, which is, that the Iron tubes being all cast in one form will not exactly fit every part of it. An Arch of a Circle may be sett off to any extent by calculating the Ordinates, at equal distances on the diameter. In this case, the Radius will always be the Hypothenuse, the portion of the diameter be the Base, and the Ordinate the perpendicular or the Ordinate may be found by Trigonometry in which the Base, the Hypothenuse and right angle will be always given.
In a reply to Paine dated Dec. 23, 1788, Thomas Jefferson used the word catenary: 
You hesitate between the catenary, and portion of a circle. I have lately received from Italy a treatise on the equilibrium of arches by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it, but I find that the conclusions of his demonstrations are that 'every part of the Catenary is in perfect equilibrium.'
The earliest citation for catenary in the OED2 is from the above letter.

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