Videos

The Istanbul Museum for the History of Science and Technology in Islam prepared a video about the perfect compass.

A tradition of curve tracing

My favorite 17th century Dutch mathematician Frans van Schooten (1615-1660) wrote a book on mechanical devices to trace hyperbola, ellipses, and parabola. Part of this website is devoted to his book "De Organica Conicarum Sectionum ...". Each of his devices can draw only one specific conic section, for example an ellipse. The perfect compass is a much more general instrument. It can draw all conic sections. Its roots go back to 10th century Arabic sources. In Greek mathematics, one might expect perfect compasses too. Euclid mentions cutting cones at a right-angle, Apollonius mentions all kind of cones and sections, but there are no traces of a drawing device. We have to dig deeper to find physical evidence. Later on in the European Renaissance there is a revival of the perfect compass. The quest for continuous drawing curves served one goal: prove that continuous curves do exist. In the old times, mathematics was much closer to physical experiences. Curves were related to trajectories of physical objects in a controlled experiment.

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Formulas

I start with an example to underline the importance of graphing parabolas. Suppose you want to double the content of a cube. Trying to solve the riddle with an integer number of small cubes does not work. You need a multiplication factor p between the base cube and the double volume cube. So p³=2 and p is not rational.
It is best to start with a compass that is at 45°. It produces a parabola. Suppose we want to draw the parabola 2x=y². All we need is a perfect compass whose length is AB=√2.
In order to solve the problem of doubling the cube, we need a second parabola y=x². Now we need a second perfect compass whose length is AB=½√2. To avoid having multiple perfect compasses, we could try a universal perfect compass. Of course, we can use the previous perfect compass whose length is AB=√2, but now we have to find the proper angle, which is less than 45°. For a parabola, I have a formula in which the required angle is expressed in the properties of the parabola.
Things gets complicated for an ellipse. Given the angles, it is easy to calculate the properties of the ellipse.
However, the other way round, is much more difficult. Again, I have a formula.
If you want to draw an ellipse, the perfect compass will do, but there are many more alternatives. My favourite 17th century mathematician Frans van Schooten wrote a book on all kind of devices to produce conic sections, The gardener ellipse is so easy. Frans van Schooten on the gardeners ellipse

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GeoGebra

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Literature

Blåsjö, V. (2022). Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry. Found Sci 27, 587-708
DOI
Hogendijk, J.P. (1995). Ibn al-Haytham's Completion of the Conics. New York: Springer, Sources in the History of Mathematics and Physical Sciences no. 7
Hogendijk, J.P. (1998). L'étude des sections coniques dans la tradition Arabe, in: Actes du 3ème colloque Maghrébin sur les Mathématiques Arabes, Tipaza. Alger: École Normale Supérieure, pp. 147-158
Milici, P. (2020). A machine for conics and oblique trajectories
Hal
Raynaud, D. (2007). Le tracé continu des sections coniques à la Renaissance, in: Arabic Sciences and Philosophy , volume 17 , pp. 299 - 345, Cambridge: Cambridge University Press
DOI
Rashed, R. (2003). Al-Quhi et Al-Sijzi: Sur Le Compas Parfait et le Tracé Continu des Sections Coniques, in: Arabic Sciences and Philosophy, volume 13, pp. 9-43, Cambridge: Cambridge University Press
DOI
Rashed, R. (2005). Geometry and Dioptrics in Classical Islam, London: al-Furqan, Islamic Heritage Foundation
Rose, P.L. (1970). Renaissance Italian methods of drawing the ellipse and related curves in: Physis, volume 12, pp 371-404, Milano: Museo Nazionale
Sijzi, (10th century), Risala fi wasf al-Qutu al-Makhrutiyya,
Leiden University Library
Vittori, T. (2009). The perfect compass: conics, movement and mathematics around the 10th century in: History and epistemology in mathematics education, Proceedings of the 6th European Summer University, pp.539-548. Vienna: Verlag Holzhausen
Hal
Woepcke, F. (1874). Trois traités arabes sur le compas parfait, publiés et traduits par François Woepcke, in: Notices et extraits des manuscrits de la Bibliothèque impériale (Paris) 22/1874/1?175 (reprint in: F. Woepcke, Études sur les mathématiques arabo-islamiques. Nachdruck von Schriften aus den Jahren 1842-1874, Frankfurt, 1986, vol. 2, pp. 560-734 and in: Islamic Mathematics and Astronomy, vol. 66, Frankfurt 1998, pp. 33-209

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Presentation

This contribution focuses on the benefits and the shortfalls of the perfect compass. It raises doubts on the practical use of a universal perfect compass. It shows that the perfect compass is well-suited for mathematical education, but not for graphically solving equations. In theory, a universal perfect compass can draw two parabolas and its intersection point, but it turns out that one is better off with two dedicated tailor made compasses because of the computational effort to set up that universal device correctly.

Presentation PPT

website Annual Meeting

Drawing instrument

I made a low-cost perfect compass from scrap material.