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 180^{o}, one drawn on a circle will have angles whose total is greater than 180^{o}, and one drawn on a pseudosphere will have a total less than 180^{o}.

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.