Sometimes blogs start when some kind of reoccurring theme pops up over several days. In this case the theme was (loosely) drawing things using parametric functions. I saw the image above which is the Logo for the MIT Lincoln Library. It reminded me of something called Bowditch (or Lissajous) curves which were a common amusement I would use to introduce my students to parametric equations after graphing calculators. (more about these later) And I mused that someday I would have to look up the history of mechanical methods of producing parametric functions.

Then within a short period of time I read that AndrĂ© Cassagnes, the French inventor of the Etch A Sketch, had died near Paris on January 16, 2013, at the age of 86. If you haven't heard of the Etch A Sketch, (right) it was a mechanical toy that was used to draw on a screen with an internal stylus that was moved right or left by one twist knob, and up or down by the other, sort of a mechanical x=f(t) and y=f(t).

It reminded me of my earlier intention a few days earlier, and so I decided to begin filling out my knowledge about that history.

From my own notes I knew that Nathaniel Bowditch, an under-appreciated American self-taught mathematician had drawn curves like this. He first drew these parametric curves in 1815 with a compound pendulum."

Like most others, in school I had learned about them as Lissajous figures, images we drew on oscilloscopes using signal generators for the two inputs. But then,shortly after I first read about Bowditch I happened to be in Tokyo Visiting the Edo Museum for an exhibit named __Worlds Revealed - The Dawn of Japanese and American Exchange__. Like others, I had always had the misconception that Commodore Perry opened trade with Japan in 1853, so I was surprised to find that a number of American ships from Salem, Massachusetts, sailing under Dutch charters had traded with the Japanese as early as 1800. The company was called the East India Marine Society, and in 1802 the First Secretary was Bowditch. On exhibit was a much more popular mathematical creation of Bowditch; his book, __The New American Practical Navigator__, that Bowditch, and the Marine Society had published in 1802. The book was a compilation of the most accurate measures of the period giving the positions of major astronomical objects at numerous longitude and latitude coordinates. The book was, literally, a mariner's bible until an accurate sea clock would become commonly available that allowed sailors to conquer the longitude problem. Bowditch's position and accomplishments seem even greater in light of the fact that he was almost totally self educated in mathematics.

Then, in August of 2008 I read a post by Milo Gardner on the almost unheard of Wilkes Expedition, which explored the western Americas and the Pacific, and Milo added that "... mathematicians during the early 1800's were assigned to working on Manifest Destiny issues and projects. On the Wilkes Expedition you'll find Bowditch as one of its navigators. An island in the Pacific is named for Bowditch, since it had not been on any US or European map prior to the expedition's visit." The island, I found out, is sometimes called Fakaofu, and is located in the Stork Archipelago in the South Pacific.

I decided to go back a little farther by looking for any historical references I could find for the history of mechanical curve drawing and hit a jackpot with an on-line article by Daina Taimina, of Cornell University titled Historical Mechanisms for Drawing Curves. It seems to be from the book,Hands on History: A Resource for Teaching Mathematics.

She stated that "Mechanical devices in ancient Greece for constructing different curves were invented mainly to solve three famous problems: doubling the cube, squaring the circle and trisecting the angle."

She went on to give several examples, "There can be found references that Meneachmus (~380-~320 B.C.) had a mechanical device to construct conics which he used to solve problem of doubling the cube. One method to solve problems of trisecting an angle and squaring the circle was to use quadratrix of Hippias (~460-~400 B.C) {*this was the first named curve other than circle and line – it is also the first example of a curve that is defined by means of motion and can not be constructed using only a straightedge and a compass*.}

Proclus (418-485) also mentions some Isidorus from Miletus who had an instrument for drawing a parabola.[ Dyck,p.58]. We can not say that those mechanical devices consisted purely of linkages, but it is

important to understand that Greek geometers were looking for and finding solutions to geometrical problems by mechanical means. These solutions mostly were needed for practical purposes."

From her description it would seem that none of these still existed in physical or drawn form.

While her focus was on the use of linkages to create mechanical movement and drawings, I was searching for something closer to the idea of a parametric curve.

Certainly the early trammel which dates to Proclus or Archimedes (*indeed it is sometimes called the trammel of Archimedes*) but again, it is more of a mechanical linkage than parametric. And no offense intended to my neighbors here in Kentucky, but the instrument is often sold as a novelty made of wood with a crank knob on the end of the trammel bar that traces out the ellipse, and is referred to as a "Kentucky do-nothing".

Students may have also been shown how to draw an ellipse by taking a loop of string looped around two thumb tacks. By holding a pencil pulled against the string to keep it taut, and sliding it around the two thumb tacks as you keep the string taut, the pencil will trace out an ellipse. The first written description of this method of construction an ellipse by means with string was by Abud ben Muhamad, in the 9th century.

Then I came across an article in Wikipedia about the harmonograph, a mechanical platform that employs one or more pendulums to create a geometric image. Interestingly, they give credit for the first harmonograph to Scottish mathematician Hugh Blackburn. Trouble is, Blackburn was born in 1823; almost a full decade after Bowditch had written about his use of such a device.

I am beginning to accept that Bowditch may have been the first person to create the parametric images which sometimes, and should more often, bare his name. If someone has an example of an earlier non-linkage apparatus that suggests parametric input to draw figures, I would love to be notified.

Jules Antoine Lissajous, for whom the figures are more often called, invented a different type of device to create the images. He used a beam of light bounced off a mirror attached to a vibrating tuning fork, which then reflected off a second mirror attached to another vibrating tuning fork which was perpendicularly orientated (usually of a different pitch, creating a specific harmonic interval), which was then reflected onto a wall, tracing the figure.With frequency produced by audible frequencies the curve traced out by the light appeared as a complete image due to visual persistence. Lissajous device is sometimes credited with inspiring the two pendulum device, but he too was born after Bowditch had written of his device. None of this should be seen to diminish Lissajous mathematical stature. His experiments with waves, his novel method of creating the waves, and his dramatic lectures and demonstrations, including one at the Royal Society in London, exposed them to a much wider audience. These lectures were so impressive that he was awarded the Lacaze Prize in 1873 for his optical observation of vibration and, in particular, "for his beautiful experiments". Almost certainly he was completely unaware of Bowditch's work.

When I introduced these to my students I often used one similar to the Lincoln Library Logo at top and I called it the Chinese finger cuff curve (*I am still waiting for the rest of the mathematical world to adopt this term, fall into line people*) As I neared retirement it seemed that many of the students had never heard of finger cuffs, but there were always a few who knew of them, and often at least one student who would produce one from home over the next few days.

If you want to create you own, you can find on-line parametric graphers and even an ipad app for a harmonograph.

Several nice examples, with their equations, are given at this Wikipedia link. Enjoy

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