Van Schooten's Ruler Constructions - Note to Teachers

C. Edward Sandifer

There are a number of ways to use this material in a geometry class.  Here is a plan that could be used.  We give more activities than most teachers would want to use.  Teachers who want to devote less time to this material may cut activities as they see fit.

I.  Read the biographical and historical Introduction.

                 A. Briefly describe the contributions of one of the four most important mathematicians mentioned here.  For this, the MacTutor History of Mathematics Archive is one useful source.

                                 1.  Vieté

                                 2.  Descartes

                                 3.  Huygens

                                 4.  Newton

                 B. Try an exercise with a trig table of the style van Schooten used.  Here are trigonometry exercise with a table based on a radius of 200.

                 C. If they are very good at analytic geometry and know about focus and directrix, you might ask your students to figure out how van Schooten’s two mechanisms work to draw an ellipse or a parabola.  Van Schooten's parabola drawer and ellipse drawer are shown at work in Figures 2, 6, and 8 of the module "Curve Drawing Then and Now" in the Convergence article, "Historical Activities for Calculus."

II.  The first two ruler constructions (Problems I and II)

                 A.  Go over the detailed first solution to the first problem, bisecting an angle.  Note how the third postulate works.

                 B.  Mention that circles still exist in van Schooten’s geometry.  It’s just that you can’t construct them with the available tools.  In much the same way, parabolas and ellipses exist in Euclid’s geometry, but you can’t construct them with a ruler and compass.

                 C.  Ask the students to prove that the first solution actually works.

                 D.  Ask the students to go over van Schooten’s other solutions to Problem I and his solutions to Problem II.

                 E.  Discuss why people are interested in solving the same problem more than one way.

                 F.  Discuss why van Schooten may have been interested in solving geometry problems without a compass.

III.  The other eight problems

Let your students try to solve these problems on their own.  Expect them to be rather challenging.  Require that they prove that their solutions are correct.  Unless they can prove correctness, the solutions are probably not correct.

It is probably too much ask your students to solve all eight problems at once.  You might want to use them more slowly, two or three a week for three or four weeks.

Some problems are harder than others.  Problems IV and VIII are kind of tricky.

Problem V depends on solving Problems III and IV first, but then it is very easy.

Problem VI depends on solving problems IV and V first, and is fairly easy, though not as easy as Problem V.

Problem VII depends on solving Problems II and VI first.  Then it is fairly easy.

Problem VIII depends on solving Problem II and V.  This problem is rather difficult.

Problem IX depends on Problem III.  This problem is rather easy, if you already know how to do it using Euclidean techniques.



The other eight problems

Trigonometry exercises