In this page we study an instrument for drawing ellipses. It is called "Ellipsograph" or "Trammel of Archimedes".
The word 'trammel' in this context has a meaning of restrain, something impeding activity, a constraint that restrics freedom. The origin of this word is related with a net to catch fish.
We can see a classical representation of this tool in Cundy and Rollet's book 'Mathematical Models'.
Classical representation of this tool in Cundy and Rollet's book 'Mathematical Models'.
"Four triangular cheek-pieces are bolted firmly to a base-plate which rests on the paper. They form the walls of two slots at right angles in which the sliders A and B can run. PAB is a slotted arm which can be screwed to the sliders at A and B in such a way that it is free to rotate on the slider, but not to slip along the slot AB. The sliders must be longer than the width of the slots at O to ensure free travel across the opening.(...) By adjusting the screws, ellipses of different sizes can be drawn by a tracing point at P." (Cundy and Rollet, p. 240)
We can say that the two sliders are 'trammelled' or confined by the two slots and the rod.
We want to prove that point P draws an ellipse.
Some properties of ellipses have been studied in other pages of matemáticasVisuales.
To calculate coordinates of point P we are going to use this notation and trigonometry:
Then the X coordinate of point P is:
And the Y coordinate is:
Then, the coordinates of point P are:
These coordinates verify the implicit formula of the ellipse:
The two semi-axis of the ellipse are a and b, it is to say, the distances between the end of the rod and the two pivoted points:
Changing the distances between points we get different ellipses:
This idea was used to build devices for drawing ellipses (called ellipsographs).
This instrument is in the Museo Nacional de Ciencia y Tecnología in Madrid:
Ellipsograph in Museo Nacional de Ciencia y Tecnología in Madrid.
Each point in the rod draws an ellipse, as you can see in the next interactive application:
You can get ellipses even if the point P is between the two sliding points:
A circle is a particular case of ellipse:
Tom Apostol and Mamikon Mnatsakanian, 'New Horizons in Geometry' (Chapter 9. Trammels, ), Mathematical Association of America, 2012.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961.
Hilbert and Cohn-Vossen, Geometry and the Imagination. Chelsea Publishing Company. pag.278.
Robert C. Yates, 'A Handbook on curves and their properties', J.W.Edwards-Ann Arbor, 1947.
Elliptic Trammel in The National Musuem of American History and a comment about ellipsographs.
J.L. Coolidge, The Mathematics of great Amateurs. Second Edition. Claredon Press. Oxford. Jan de Witt's proof in page 124.
Historical Mechanisms for Drawing Curves by Daina Taimina. With De Witt's proof why the trammel describes an ellipse.