Tractrix - Claude Perrault is not a well known mathematical name, in fact there is no record of his ever proving any mathematical theorem. Trained as a doctor, he did eventually gain some respect as an architect and an anatomist before his unusual death as a result of an infection he acquired while dissecting a camel. He is mostly remembered today for being the brother of the author of "Cinderella" and "Puss-in-Boots".
Yet he did make one interesting contribution to mathematics. Here is the story as told by the esteemed math historian V. F. Rickey in an article in Learn from the Masters, published by the M.A.A. " When Liebniz was inventing the calculus in Paris in 1676, Claude Perrault (1613-1688) placed his watch in the middle of the table and pulled the end of its watchchain along the edge of the table. He asked: What is the shape of the curve traced by the watch?" The path is shown in the image at right.
Perrault's question would today be called a differential equation, and the name of the curve is the tractrix, which is drawn from the Latin root tractus, that which is pulled or drawn along. Liebniz claimed to have found a solution, but no record of his solution has been found in his existing work. The first known solution is credited to Christiaan Huygens who gave the solution in a 1693 letter to a friend.
The German name for the same curve is hundkurve, the hound curve. The name makes sense if you imagine the watch as a dog following along the leash as the master walks away.
For students with ability in differential equations who wish to solve the equation, the problem may be expressed as , where a is the length of the watch chain. For those who just want to graph the function, it is given by
A tractrix can also be found by taking the involute of a catenary. Imagine a horizontal bar held at the vertex of the catenary and the point of contact marked as P. When the bar is rolled against the cantenary without slipping, the path of P will be a tractrix. If a tractrix is rotated around its axis, the shape formed is called a pseudosphere. You can view an image of the pseudosphere from the famous curves page at St. Andrews University in Scotland. A pseudosphere is a model of part of a hyperbolic (non-Euclidean) space. It was first proposed as a model for non-Euclidean geometry by Beltrami in 1868. The pseudosphere is a surface with constant negative curvature, as opposed to a sphere which has constant positive curvature. A triangle drawn on a plane has angles that total 180o, one drawn on a circle will have angles whose total is greater than 180o, and one drawn on a pseudosphere will have a total less than 180o.
A tractrix is also known to be orthogonal (perpendicular) to all the set of circles with a common radius along a straight line. A good image of this, and several other of the properties mentioned above can also be found at the webpage of Xah Lee.