Hi again Derek, 

    Here are some more detailed ideas for your thesis prospectus.  Feel
    free to "recycle" any of this that you find useful. 

    First, I don't think it's reasonable to focus on just one of the 
    construction problems, since they have often been treated as a "package" and 
    their various solutions are closely related on several levels.  We will want
    to treat all three of the most famous construction problems:
 
    (1)  "squaring the circle":  Construct a square with the same area as
          a given circle

    (2)  "duplicating the cube":  Construct a cube with volume twice that
          of a given cube

    (3)  "trisecting a general angle":  Construct an angle exactly 1/3 a 
          given angle (the construction should work for any given angle, not
          just special ones like right angles)

    T.L. Heath (a very famous British historian of Greek mathematics from
    the end of the 19th and the beginning of the 20th century) says 
    "It is evident that the Greek geometers came very early to the conclusion
    that the three problems in question ... required for their solution either
    higher curves than circles or constructions more mechanical in character
    than the use of the ruler and compasses in the sense of Euclid's Postulates
    1 - 3". ([3], p. 219)  In other words, they weren't able to solve these using 
    just straightedge and compass constructions in a very restricted sense, but 
    they developed methods for solving these using other given curves, or more 
    general constructions that involve taking what we would call limits ("neusis 
    constructions" or "verging constructions").  Eventually all three construction 
    problems were proved to be unsolvable using just straightedge and compass, 
    but only in the 19th century, and then using ideas from field theory (a 
    branch of abstract algebra).  The connection between the algebra and the 
    geometry comes, of course, from Descartes' ideas of introducing coordinates
    to describe locations of points in the plane (something the Greeks did not 
    do).

    I think a great thesis would be to 

    (a) Investigate the Greeks' ideas about methodology of geometric constructions.
        They were the really the first to consider questions of this sort: How 
        you solve particular types of problems and what methods are allowed?  Main 
        Question:  Where did this train of thought come from?  Can we trace the history?
  
    (b) Study and present (in modern language and in your own words) some of these 
        neusis or "mechanical" constructions, especially the use of the "quadratrix" 
        curve of Hippias and the spirals of Archimedes to square the circle, the 
        neusis construction of Nicomedes for trisection problem, and the cissoid 
        of Diocles for the duplication of the cube.   (One interesting side-angle here
        is that all of these can be presented very effectively using the geometry
        construction/visualization software packages like GeoGebra or Geometer's 
        Sketchpad that are often used in secondary math education now.)  Main Question:
        Is Heath's statement quoted above really justified?  Or was it just that 
        they did not know how to derive solutions with simpler methods and "settled"
        for any solution they could find?  Can we find evidence in the original 
        sources?

    (c) Learn and present the ideas in field theory that led to the eventual
        proof of impossibility of the three constructions using only the strict
        straight-edge and compass constructions.  

    Some first sources we will want to consult -- not in any particular order (and 
    not alphabetized, either!)

  
    [1] Knorr, R.W.  The Ancient Tradition of Geometric Problems, Dover reprint, 2012.
        
        He views the Greek geometric enterprise almost like a modern mathematical 
        research program and argues that the construction problems were a motivating
        force for the whole development of Greek geometry.  Is that a valid comparison?  
        Is he being anachronistic?
 
    [2a] Eves, H.  An Introduction to the History of Mathematics, 5th ed., Saunders, 1983.

    [2b] Boyer, C. and Mertzbach, U.  A History of Mathematics, 3rd ed.,  Wiley, 2011.

        (Both for general background and overview only -- we'll see that their 
        discussions of this topic are somewhat superficial.)

    [3] Heath, Sir T.L. A History of Greek Mathematics, vols. 1 and 2, Dover reprint, 1981.
        
        This has never really been superseded as an overview of the surviving Greek texts.
        Chapter VII is an indispensable summary of the Greek efforts to solve the
        three construction problems. 

    [4] Heath, Sir T.L., ed. The Works of Archimedes, Dover reprint, 2002.
 
    [5] Klein, J. Greek Mathematical Thought and the Origin of Algebra, Dover reprint, 1992.
        
        For background on the geometry-algebra connection.

    [6] Proclus, A Commentary on the First Book of Euclid's Elements, translated by G. Morrow, 
        2nd ed., Princeton University Press, 1992.

        Proclus lived about 700 years after the time of Euclid, but his commentary is one
        of the main surviving texts that discuss how the Greeks thought about their 
        mathematics.  

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 J Delattre and R Bkouche, Why ruler and compass?, in History of Mathematics : 
 History of Problems (Paris, 1997), 89-113.