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The civilization of ancient Greece remains a beacon for its advancements in literature, architecture, philosophy, and mathematics.  Indeed, the advancements of Greek geometers provided a foundation for the entire mathematical enterprise for the next two millennia, from the Classical period, through the Middle Ages, and into modernity in both the Islamic world and Europe.  This progress is manifested not only by the content of the Greek works---their brilliance is evident---but also in how the ancients went about their research.  They opened schools, championed reason and rationality, and were intrigued by the physical and natural complexities of the world.  Such is the context for one of the greatest mathematical endeavors history has ever witnessed. 

Historians have divided the Greek mathematical enterprise into two distinct periods.  The first of which, the Classical Period, dates from 600 to 300 BCE, roughly the three centuries from the demonstrative works of Thales to the extraordinary \textit{Elements} of Euclid.\footnote{Eves, p.84.}  During this time, Athens flourished as the cultural hub, culminating in the establishment of Plato's Academy around 387 BCE.  The Hellenistic, or Alexandrian, Period ranges from 300 BCE to roughly 415 CE, beginning with the establishment of the University of Alexandria, of which Euclid served as the first mathematical chair.\footnote{Eves, p.112.}.  Throughout the entirety of the epoch, there remained three clear features of Greek mathematics: the importance of deductive reasoning, the emphasis on abstract mathematics, and the prominence of geometry to solve problems.\footnote{Hollingdale, p.113}  Nowhere are these characteristics more apparent than the three construction problems that dominated the mathematical field in antiquity.  These three problems are described as follows: 


1. Duplication of the cube: constructing a cube having twice the volume of a given cube. 

2. Trisection of an angle: constructing an angle with one-third the measure as that of a given angle.

3. Quadrature of the circle: constructing a square with the same area as that of a given circle.\footnote{Eves, p.89.}
	

The duplication of the cube, tradition believes, likely began from pronouncement of the citizens of Delos around 430 BCE.  A plague had stricken Athens, and a delegation of concerned citizens sought the oracle of Apollo to determine how the plague could be alleviated.\footnote{Burton, p.119.}  The oracle responded that the altar to Apollo---a cubical shrine---should be doubled in volume.  Ignorant of geometry, the obedient Athenians merely doubled the edges of the altar, increasing the volume eightfold.\footnote{Boyer, p.64.}  Apollo then cursed the city by strengthening the plague.  Plato claimed that this divine problem was conceived not to double the size of an altar, but to make the Greeks aware of their ``neglect of mathematics and their contempt of geometry.''\footnote{Burton, p.119.}  The mathematician and historian Howard Eves gives a similar tale, where King Minos commands that the cubical tomb of his son be doubled.\footnote{Eves, p.90.}  The former description appears to have been more common, as the duplication of the cube has long been referred to as the Delian problem.  

The origins of the other two construction problems are not associated with such romantic tales.  Historians believe that the trisection of an angle likely arose as a natural progression to the multi-section of a given angle, after both an angle and a segment's bisection was proved by Eulcid.\footnote{Eves, p.92}.  Or perhaps, some argue, the trisection of an angle stemmed from attempts to construct a regular nine-sided polygon, after the construction of a regular pentagon had been discovered.\footnote{Heath, p.235.}  The origin of the quadrature of the circle is attributed to Anaxagoras, who attempted the problem while passing the time in prison.\footnote{Boyer, p.64.}  Collectively, these three problems would dominate the field of geometry not just for the ancient Greeks, but for many mathematicians over the following centuries.  

In solving these problems, however, the Greeks also initiated a deeper study of mathematical methodology by asking for solutions using particular tools. In fact, many commentators of Greek mathematics assert that the Greeks restricted themselves to the use of only two tools: a straightedge and a compass, though the origin of such a restriction is still greatly debated.  Knorr, for example, writes that scholars agree that Euclid ``would become the one responsible for the formal project of effecting geometric constructions within the restriction of employing compass and straightedge alone.''\footnote{Knorr, p.15.  Knorr goes on to say that while the restriction to these tools stems from Euclid's postulates, it is Oenopides of Chios, a fifth century BCE geometer, who ``set about the task of regularizing and classifying problems according to the means of construction adopted,'' (p.16).}   Similarly, Eves asserts that the postulates of Euclid's \textit{Elements} restrict geometers to the use of the straightedge and compass.\footnote{Eves, p.90.}  These tools, therefore, are often called Euclidean tools, named after the geometer who begins his \textit{Elements} with three postulates on how lines and circles were to be constructed.

Other scholars assert that it is Plato who is responsible for imposing the restriction to the Euclidean tools.  In his work \textit{Quaestiones Convivales}, Plutarch notes Plato condemned Greek mathematicians who endeavored to ``bring down'' geometry to ``mechanical operations; for by this means all that was good in geometry would be lost and corrupted.''\footnote{citation?}  James Gow, a nineteenth-century historian of Greek mathematics, therefore states that ``to Plato we owe the strict limitation of geometrical instruments to the ruler and compass.''\footnote{Gow, p.181.}  David Burton, author of \textit{The History of Mathematics}, agrees, stating ``tradition has it that Plato insisted that the task be performed with straightedge and compass only.''\footnote{Burton, p.116.}  

No matter the origin of this restriction, it is important to specify how these tools were to be used in problems of geometric constructions.  With a straightedge, geometers could draw a straight line through any two points.  Similarly, with a compass they could draw a circle with any point as its center and passing through any other point.  Transferring distance was prohibited---thus the straightedge could not be marked or graduated and the compass must be ``regarded as collapsing as soon as either point is lifted off the paper.''\footnote{Burton, p.116.}  In the fourth century BCE, nearly seven hundred years after Euclid, the mathematician Pappus comments on the classification of geometric problem solving, stating that ``there are three kinds of problems in geometry.''\footnote{Knorr, p.341}  The first are called planar problems, so named because they can be solved by straight lines and circles and therefore their construction has its origin in the plane.  There are also linear geometric problems, which are generated from more complex lines, such as ``spirals and quadratrices and conchoids and cissoids.''\footnote{Ibid, p.342.}  Finally, Pappus classifies problems as solid if their solution involves the use of curves defined using solid figures, such as plane sections of a cone.  

Pappus' influence on geometry and the legitimacy of constructions extended into the early modern period, over one thousand years after his death.  Indeed, the Latin translation of his \textit{Collection} in 1588 by Commandino would provide the spark for many mathematicians to take up these concerns.\footnote{Bos, p.37.}  The mathematician and historian Henk J.M. Bos describes how Christopher Clavius, Johannes Kepler, Johannes Molther, Francois Vi\`ete, and Ren\'e Descartes each addressed the matter of demarcating between legitimate and illegitimate geometrical constructions and classifying the different geometrical procedures.  Each, in turn, offered his own solutions to these three problems of antiquity, extending the field of mathematical knowledge.  Perhaps most significantly, Descartes made advancements in the field of algebra that would ultimately provide the bridge for Classical geometry and more modern applied and analytical mathematics.  

Moreover, although they were unable to solve the three construction problems using just the Euclidean tools, ancient Greek geometers did produce remarkable solutions using other methods.  Hippias of Elis, for example, developed the quadratrix curve in the fifth century BCE that would later be used to both square the circle and trisect the angle.\footnote{Boyer, p.69.}  Archimedes, the third century BCE geometer and inventor, developed a spiral that could produce solutions to the angle trisection and the squaring of the circle.\footnote{Ibid, p.126.}  Around 350 BCE, the Greek mathematician Menaechmus employed certain conic sections to duplicate the cube, while nearly a century later Nicomedes achieved the same result using conchoidal curves.\footnote{Burton, p.123.}  Some geometers even invented their own apparatuses to construct these curves, stopping at no limits to provide solutions to these problems, and advancing the mathematical field beyond what their predecessors had established.    

While mechanical solutions to the three construction problems flourished throughout antiquity and the early modern period, solutions using just the Euclidean tools were not proved to be impossible until the nineteenth century.  In 1837, for example, the French mathematician Pierre Wantzel detailed the proofs of impossibility of trisecting an angle and duplicating the cube using just a straightedge and compass.\footnote{Burton, p.121.}  Rooted in these proofs is the concept of reducing geometric problems to algebraic equations, first set forth by early sixteenth and seventeenth century mathematicians, particularly Descartes.  These two problems reduce to cubic equations that do not yield rational roots and do not have a degree of a power of 2, and hence their construction using the Euclidean tools is impossible.\footnote{Ibid, p.121.}  The quadrature of the circle was proved impossible in an 1882 paper by Ferdinand von Lindemann.  He proved that $\pi$ is a transcendental number that cannot be the root of an algebraic equation, hence it follows that a circle cannot be squared using just a straightedge and compass.\footnote{Boyer, p.573.}   

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