Jen-chung Chuan

Deparment of Mathematics

National Tsing Hua University

Hsinchu, Taiwan 300

e-mail: jcchuan@math.nthu.edu.tw

Mascheroni dedicated one of his books Geometria del compasso (1797) to Napoleon in verse in which he proved that all Euclidean constructions can be made with compasses alone, so a straight edge in not needed. This theorem was (unknown to Mascheroni) proved in 1672 by a little known Danish mathematician Georg Mohr. In the setting of dynamic geometry, the Mohr-Mascheroni constructions ask for specific procedures in which the figures are constructed using the compasses alone. In what follows we are to concentrate the constructions of

- the conics: hyperbola, parabola and ellipse.
- the epicycloids (the cardioid and the nephroid), hypocycloids (the deltoid and the astroid) and their osculating circles.
- the Lemniscate of Bernoulli.
- the Bowditch curve.

The dynamic geometry environment provided by CabriJava is essential in our exploration.

**Hyperbola **

Total number of intermediate circles: 8

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/hyperbola-with-compass.html

Principle: hyperbolas are the inversions of the lemniscates. [Lockwood;
p. 116]

**Parabola **

Total number of intermediate circles: 8

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/parabola-with-compass.html

Principle: parabolas are the inversions of the cardioids. [Lockwood;
p. 180]

**Ellipse **

Total number of intermediate circles: 8

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/ellipse-with-8circles.html

Principle:

1. Center of the reference circle, the inverse and the point itself are collinear.

2. x = a cos t, y = b sin t.

**Cardioid **

Total number of intermediate circles: 4

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/cardioid-from-circle.html

Principle: x = 2 cos t - cos(2t), y = 2 sin t - sin(2t).

**Cardioid and its Osculating Circle **

Total number of intermediate circles: 10

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/osc-cardioid-compass.html

Principle: the points [cos t,sin t], [cos(2t), sin(2t)] separate the point [2 cos t -
cos(2t), 2 sin t - sin(2t)] and the center of curvature harmonically.

**Nephroid and its Osculating Circle**

Total number of intermediate circles: 11

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/osc-nephroid-compass.html

Principle: the points [cos t,sin t], [cos(3t), sin(3t)] separate the point [3 cos t -
cos(3t), 3 sin t - sin(3t)] and the center of curvature harmonically.

**Deltoid and its Osculating Circle **

Total number of intermediate circles: 11

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/osc-deltoid-compass.html

Principle: the points [cos t,sin t], [cos(-2t), sin(-2t)] separate the point [cos (2t) - 2
cos(t), sin (2t) +2 sin(t)] and the center of curvature harmonically.

**Recovering the Center of a Circle**

Total number of intermediate circles: 6

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/center.html

Principle: Inversion.

**Fermat Point**

Total number of intermediate circles: 16

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-fermat.html

Principle: Simititude.

**Peaucellier's Linkage**

Total number of intermediate circles: 5

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/peau.html

Principle: Inversion.

**Intersection of a Line and a
Circle**

Total number of intermediate circles: 4

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-line-circle-intersections.html

Principle: Symmetry.

**Envelope Formaing Deltoid and
3-Cusped-Epicycloid **

Total number of intermediate circles: 11

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-epi-hypo.html

Principle: Symmetry.

**Regular Pentagon**

Total number of intermediate circles: 12

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/5-gon-compass.html

Principle: Inversion.

**Linkage Drawing the Ellipse**

Total number of intermediate circles: 11

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/ellipse-linkage-with-compass.html

Principle: Inversion.

**Square Constructed from One Diagonal**

Total number of intermediate circles: 6

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-sq-diag.html

Principle: Symmetry.

**Square Constructed from One Side**

Total number of intermediate circles: 6

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-sq-side.html

Principle: Symmetry and translation.

**Dividing a Cirlce into Four Equal Parts**

Total number of intermediate circles: 6

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-sq.html

Principle: Symmetry and translation.

**Arc Bisection**

Total number of intermediate circles: 7

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/bisect-arc.html

Principle: Symmetry.

**Bowditch Curve**

Total number of intermediate circles: 13

Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/bowditch-with-compass.html

Principle: Coordinates.

- H. M. Cundy and A. P. Rollett, Mathematical Models, Oxford University Press,1961.
- Heinrich Dorrie, 100 great problems of elementary mathematics; their history and solution, New York, Dover, 1965.
- Howard Eves, A Survey of Geometry, Boston, Allyn and Bacon, 1963-65.
- Hilda P. Hudson, Ruler and Compasses, reprinted by Chelsea in the collection "Squaring the Circle"
- A.B. Kempe, How to draw a straight line; a lecture on linkage, reprinted by Chelsea in the collection "Squaring the Circle"
- A.N. Kostovskii, Geometrical Constructions Using Compasses Only, Popular lectures in mathematics series,v. 4, Translated from the Russian by Halina Moss. Translation editor: Ian N. Sneddon, New York,Blaisdell, 1961.
- E. H. Lockwood, A Book of Curves, Cambridge University Press, 1961.
- Robert C. Yates, A Handbook on Curves and Their Properties, Ann Arbor, J. W. Edwards, 1947.