Dudley, Underwood, *A Budget of Trisections*, Springer Verlag, 1987.

This delightful book begins with a historical overview of angle trisections, using devices other than a straightedge and compass, such as Archimedes' angle trisection using a compass and a straightedge with two marks. He then describes the personalities of some would-be angle trisectors, then details dozens of angle trisection attempts.

Eves, Howard, *A Survey of Geometry*, Allyn and Bacon, 1963.

This encyclopedic text is chock full of fascinating tidbits. In particular, Section 4.4 (pp. 198-204) discuss the Mohr-Mascheroni construction theorem which states that our constructions could be carried out with a compass alone and no straightedge. Section 4.5 (pp.204-210) detail the Poncelet-Steiner theorem, which shows that a straightedge along with one given circle and its center is sufficient to carry out any desired Euclidean construction. Section 4.6 (pp. 210-217) discusses other construction results, and mentions Lemoine's 1907

geometrography,a counting scheme for determining the complexity of a construction. Lemoine generally counts three operations where Hartshorne's construction counts one "step." We of course have followed Hartshorne's simpler counting method.

Hartshorne, Robin, *Companion to Euclid*, American Mathematical Society,
Berkeley Mathematics Lecture Notes, Volume 9, 1997

Hartshorne, Robin, *Geometry: Euclid and Beyond*, Springer, Undergraduate
Texts in Mathematics, 2000.

The 2000 title is an update of the 1997 version. Pages 20-22 in the newer version discuss the number of steps Euclid used for a proof versus the number needed for a mere construction. The homework following this section contains several constructions with an average or "par" estimate of how many steps an experienced geometer might use. Page 25, problem 2.14, asks for both of the trisection points of a segment and rates it "par 6". Of course, we are only asking for the left trisection point, so our equivalent is "par 5." Fascinatingly, in the 1997 version, the same problem 2.14, only now on page 23, says par=9. Either his brightest students improved a lot in three years, or the original textbook had a typo.

Heath, Sir Thomas, *The Thirteen Books of Euclid's Elements with introduction and
commentary*, Dover Publications, 1956.

The bisection construction pictured is very close to that in Euclid's first proposition. Euclid actually does not prove the bisection construction until Proposition 10. (The relevant Propositions 1 and 10 are pages 241 and 267-268.) The bisection construction we have pictured is said to be due to Appolonius. Euclid clearly is more interested in the logical development of the proofs rather than sharing our interest in the shortest constructions.

Lemoine, Emile, Geometrographie, ou Art des Constructions Geometriques, Scientia,
Phys.-Math. no. 18, Paris, February 1902.

Lemoine invented a method to measure the complexity of geometric constructions. His method has four parameters; one gives the required number of lines, a second the number of circles, while two others count the moves needed to place the ruler and the compass. Trisection is not explicitly mentioned in this monograph, although pages 34-36 give a more general construction, a corollary of which is essentially our third trisection method (see Reusch and Ringenberg below).

Martin, George, Geometric Constructions, Springer, 1998.

This nice undergraduate geometry textbook explicitly develops the Mohr-Mascheroni and the Poncelet-Steiner constructions. We use his convention for the "ruler points", pages 69-82, for the Poncelet-Steiner type of constructions.

Nelson, Roger, *Proofs Without Words*, Mathematical Association of America,
1993.

Our first trisection is taken from page 13, attributed to Scott Coble.

Reusch, J., Planimetrische Konstruktionen in Geometrographischer Ausfuhrung, Druck and Verlag von B. G. Teubner, Leipzig and Berlin, 1904.

Reusch expands on Lemoine's monograph with many diagrams and even more explicit analysis of the basic geometric constructions. Pages 17-20 explicitly deal with trisections; he gives three different constructions. The one he calls classical takes 4 lines and 6 circles; his second is the our third one a la Hartshorne that we are calling classical. Reusch's third construction uses four circles and a line (another "par" 5 construction). Here are scans of the relevant pages: V, VI, 17, 18, 19 ,20

Ringenberg, Lawrence, *Informal Geometry*, John Wiley and Sons, 1967.

This standard text contains the typical method of trisecting a segment. Here is his version of the classical construction, page 139. In our third construction, we use two circles to construct his point C and draw the line

A C and then a third circle constructs C2. We do not need to draw C3 nor B C3 since the geometry is such that our line C2 D is already parallel to B C3. Thus, this slick version of the classical construction takes only three circles and two additional lines.

Our second and fourth trisection constructions do not seem to appear on the web or in standard modern texts, nor are our Mohr-Mascheroni and the Poncelet-Steiner trisection constructions explicitly shown anywhere.

The third construction given, using three circles and two lines, is well known.
Here is the typical diagram that is used to trisect the segment AB (due to Dr. Math at the
Math Forum.) In our third
trisection, the two initial circles construct the point C in Dr. Math's and Ringenberg's
classical construction.

/ D o. / `. / `. C o. `. / `. `. / `. `. A o--------+---------+--------o B `. `. / `. `. / `. `o E `. / `. / `o F /

Here are some trisection constructions that appeared in a Google search.

http://forum.swarthmore.edu/dr.math/problems/cbr.8.13.99.html

http://www.math.niu.edu/~rusin/known-math/98/trisect

http://www.cut-the-knot.com/arithmetic/rational.html#const

http://rick.dgrc.crc.ca/trisection/

The following web page shows the Quadratrix of Hippias, which connects trisecting segments
and trisecting angles (see also Dudley's *A Budget of Trisectors*, pages 5-7):

http://www.perseus.tufts.edu/GreekScience/Students/Tim/Trisection.page.html

Dr Math also discusses angle trisection, http://mathforum.com/dr.math/faq/faq.impossible.construct.html.

Return to Main Trisection Page

Villanova Home Page Math Dept Home Page 25 April 2001