On The Puzzling History of Euclid’s Fifth Postulate

At the outset of Euclid’s Elements he offers twenty-three definitions, five postulates, and five common notions (sometimes translated as “axioms”). Of the five postulates, the fifth is the most troubling. It is known as the Parallel Postulate. The word postulate can be roughly translated to mean “request,” “question,” or “hypothesis” (postulat in Latin means “asked”).

The Parallel Postulate is translated from Greek as follows:

“That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”

In the picture above, two lines are intersected and their inner angles are less than two right angles, and therefore both lines will meet if extended indefinitely. It defines parallelism. It also deals with concepts of divergence and convergence -the key implication in both concepts is a certain degree of motion.

Why does the Parallel Postulate not occur as a Proposition in Euclid’s Elements? Why is it not, for example, a demonstrated reductio? In some ways the Parallel Postulate begs to have a proof of its claims (note: Euclid’s Proposition 27 in Book I).

The problem with the Parallel Postulate is that a “proof” or demonstration has not been sufficiently made and thus it requires a visual demonstration to understand its claim. The Parallel Postulate requires significant use of the human imagination. We imagine someone drawing two straight and parallel lines and then another line ‘falling’ on the two unparallel lines at an angle that is not perpendicular in a way that both lines will eventually meet. The key term in the Parallel Postulate is indefinitely. In all likelihood, Euclid used the term indefinitely to encourage us to consider a hypothetical exercise in drawing two parallel lines onward without actually doing so -does Euclid’s surface also continue indefinitely? And if so, is the surface indefinitely flat? The crucial distinguishing factor is that Euclid invites us to draw these two parallel lines on a what we might call a mental plane, perhaps not even an existing surface.

Many mathematicians have attempted to prove the Parallel Postulate but to little avail. Ptolemy thought he had proved the Parallel Postulate but Proclus found an error in his proof, and Proclus could not prove the Parallel Postulate either. However, mathematicians have also explored what might happen if the Parallel Postulate was untrue: names like Ibn Al-Haytham, the great poet-mathematician Omar Khayyam, Nasir al-din Al-Tusi, the famous polymath Muhammad ibn Musa al-Khwarizmi, Giovanni Saccheri, Adelard, Descartes, Janos Bolyai, Newton, Leibniz, Carl Gauss, and Nikolai Lobachevsky. The modern shift from the Middle Ages onward was an exploration into the possibilities of geometry. They found that the negation of Euclid’s Parallel Postulate gives rise to new forms of geometry (non-Euclidean geometries). The key distinction relies on the surface upon which the parallel lines are constructed -is it indefinitely flat or is there a curvature to the surface, as is the case with objects in the physical world. In the real world, elliptical geometry better describes shapes that have being, while other forms of geometry are orderly but puzzling, like M.C. Escher’s artwork which displays hyperbolic geometry. Einstein used non-Euclidean geometry to describe the ways in which the space-time continuum becomes warped in the presence of matter in his General Theory of Relativity. This means that the curvature of space implies that straight and parallel lines will, in fact, meet at one point if extended indefinitely.

Thus what began as an ancient quest in search of geometric and Platonic perfection (Pythagoras, Euclid, Proclus etc) was transformed into a project to better understand and map the world around us (Descartes, Newton, Lobachevsky, Einstein etc). The investigation of the true “earth measurement” continues. Defenders of Euclidian geometry argue that Euclid never intended for his Elements to resemble anything existing in the world around us. They say his geometry is pure abstraction in search of perfection. However this poses certain problems because Euclid’s geometry is in fact not pure abstraction. It relies upon a certain understanding and demonstration of postulates in the physical world -i.e. not in some fabled celestial or divine realm. Therefore the question of whether indefinitely parallel lines will ever meet remains a vexing theoretical quandary, and indeed it is worth entertaining the modern position of doubting Euclid and exploring where new geometries take us.


For this reading I used the wonderful translation of Euclid’s Elements by Thomas L. Heath for Green Lion Press. Mr. Heath was a Cambridge scholar who translated Euclid directly from the original Greek in the early 20th century.

On the Definitions, Postulates, and Common Notions of Euclid’s Elements

Euclid’s Elements (“Stoikheîon”) is the foundational text of classical, axiomatic, and deductive geometry (“earth-measurement”). The Elements is composed of thirteen books, each filled with propositions that beautifully unfold a theory of number, shape, proportion, and measurability. The Elements was the essential geomtery textbook for nearly 2,000 years thanks to the preservation efforts of the Byzantines, Arabs, and English. Sadly, the Elements fell out of favor for students in the 20th century and very few, if any, students attempt to summit the extraordinary heights of Euclid in our modern era. The Elements has been cited by every major mathematical and scientific figure including Copernicus, Galileo, Kepler, Newton, Hobbes, Descartes, Spinoza, Whitehead, Russell, Einstein, and so on.

We know almost nothing about Euclid. The only two things we infer about his life, as referenced by ancient sources (primarily Diogenes Laërtius), is that he lived after Plato (died 347 BC) and before Archimedes (287 BC). He worked or perhaps founded a school in Alexandria, Egypt. Thomas L. Heath surmises that Euclid was originally schooled in Athens under the geometric pupils of Plato (in many ways we can see echoes of Plato found in Euclid’s Elements -recall the mathematical instruction of the boy in Plato’s Meno). Take note of a common mistake: Euclid, the author of the Elements, is distinct from Euclid of Megara who appears in Plato’s Theaetetus.

Euclid appears briefly in Archimedes’s On the Sphere and the Cylinder and also in Apollonius’s Conics. There were other “Elements” books circulating in antiquity by Hippocrates, Leo, and Theudius, but Euclid superseded them all and none of the other books have fully survived into the modern day.

Euclid begins his Elements not with a series of “problems” or “equations” like many math modern textbooks but rather with a list of foundational metaphysical claims: Definitions, Postulates, and Common Notions. The Definitions appear first and a general descent occurs. The Postulates follow the Definitions, and lastly we are offered a list of Common Notions. Things that are common occur last in order of importance.

Definitions
The Definitions are 23 statements (they were later numbered by 16th century editors after the advent of the printing press). The Definitions proceed from small elements to constructions of shapes. They are brief declarations that we can imagine as a response to Socratic questions, “what is…?” The Definitions do not permit a modern conception of the infinite. The first Definition is of a point -an irreducible and indivisible element (“A point is that which has no part”). A point gives us a sense place, perspective, and grounding. A point grants permission to draw a line (“breadthless length”) between two points. Where do we draw these elements? On a surface (“that which has length and breadth only”). A surface is presumed to be flat, unlike modern formulations of elliptical and non-linear geometry (i.e. Lobachevsky). This is evidenced by the final Definition of parallel lines (“straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction”). The assumption is that a) the straight could be produced indefinitely in a hypothetical situation and b) the straight lines are produced on an indefinitely flat plane/surface. This is distinct from modern conceptions of rounded or spherical surfaces upon which to conduct geometric demonstrations. We imagine an ancient geometer demonstrating Euclid’s Definitions in the dirt or on a chalk board.

As the Definitions descend we begin with foundational elements like points and lines (Definitions 1-7), then with Definitions pertaining to proportions between foundational elements like angles (Definitions 8-13), and then Definitions concerning shapes or figures (A figure is defined in Definition 14, Definitions 13-18 concern circles, and Definitions 19-23 concern rectilinear figures). It is worth noting that a plane surface does not appear first in the list of Definitions. Instead human activity (i.e. creating a point and a line) takes precedence over the plane surface. Perhaps Euclid’s Elements was not intended to be translated from the conceptual to the physical world (“earth-measurement”). Perhaps it is meant to be an exploration of the Platonic eidos.

Postulates
While the Definitions are firm and unquestionable, the Postulates are a series of “requests” or “demands” placed upon the reader. They are a demonstration of the authority or authorship of Euclid. The Postulates do not necessarily deductively follow from the Definitions, rather they are five rules offered by Euclid.

The five Postulates begin with three active requests: first that it is possible to “draw” a straight line between any two points; second that it possible to “produce” a finite straight line; and third that it is possible to “describe” a circle with any center and distance. The descent of the Postulates begins with three active possibilities: ‘drawing’ lines between points in practice and ‘producing’ lines as well as ‘describing’ circles in concept.

The fourth Postulate concerns the equality of all right angles (in other words, there are no modern notions of gradation), and the fifth and final Postulate concerns lines that pass through parallel lines at an angle which will meet if produced indefinitely, and that the intersecting lines will meet at interior angles that are less than two right angles.

Common Notions
The Common Notions are the most democratic of Euclid’s metaphysical claims. They are ideas everyone understands -common to everyone. They are visual, whereas the Definitions and Postulates are more conceptual and analytical. There are five Common Notions: the first four Common Notions concern equality, and the fifth defines the “whole” as greater than the parts (i.e. a triangle is not superseded by its lines or points -it is a whole triangle).

Unlike Aristotle who often begins his books with commonly held opinions and then proceeds into nuanced discussions of greater depth which ultimately yield a higher perspective, Euclid begins his Elements in Platonic fashion -answering Socratic questions as if posed to a geometer -“What is a point?” “What is a line?” “What is a plane surface?” “What is a figure?” Thus, Euclid’s book is as much an examination of the human mind as it is a lesson in mathematics.


For this reading I used the wonderful translation of Euclid’s Elements by Thomas L. Heath for Green Lion Press. Mr. Heath was a Cambridge scholar who translated Euclid directly from the original Greek in the early 20th century.

“Crawling Between Earth and Heaven:” A Reading of Hamlet

Hamlet is a deeply puzzling and unsettling play. It remains elusive, alluring, cryptic, macabre, and mysterious: it is Shakespeare’s drama par excellence (contrary to T.S. Eliot’s unimpressive criticism of the play). On the surface, Hamlet is a classic revenge story that mirrors the theatrical works of classical antiquity. However, upon closer investigation Hamlet draws on uniquely modern qualities that are divergent from the works of Aeschylus or Sophocles. In the play, almost as if awakening from a dream, Prince Hamlet is haunted by the ghost of his father, a war hero who is also named Hamlet. The elder Hamlet’s ghost is apparently trapped in Christian Purgatory yet he is also familiar with the river Lethe and other classical markers of the underworld (in James Joyce’s Ulysses, Leopold Bloom offers the theory that the ghost of Hamlet’s father is actually intended to represent Shakespeare, himself). The ghost claims that while he was poisoned while resting in his orchard and then usurped by his luxurious and incestuous brother, Claudius. We are dropped into the plot of Hamlet approximately two months after the death of the elder Hamlet. In that short timespan Claudius (Prince Hamlet’s Uncle) has been crowned king of Denmark and he has wedded the widowed queen, Gertrude (Prince Hamlet’s mother). Prince Hamlet has returned home from his education in Wittenberg (Northern, Germany). He is forbidden to return to school. As directed by his father’s ghost, Hamlet’s task is to avenge his father’s death and claim the rightful crown of Denmark.

Horatio, Hamletm and The Ghost by Hnery Fuseli (1789)

Hamlet’s 13th century world is a cosmopolitan hub linking Denmark with France, Germany, Norway, Poland, and England. The geography of the play informs some of its central themes. Hamlet’s homeland of Denmark is a borderland between opposing values (similar to the metaphorical borderland of Scotland in Macbeth or Cyprus in Othello). North of Denmark lies Norway – a heroic nation where King Hamlet won a victory over the elder Fortinbras (the elder Hamlet killed the elder Fortinbras and now Norway is governed by the elder Fortinbras’s feeble brother). Norway is an old world that is comparable to Homer’s Achaea. South of Denmark on the Continent lies Paris, France and Wittenberg, Germany. Paris is a city of fashion -it is where Laertes desires to go (Polonius advises him to be honest with himself before leaving: “to thine own self be true”). Paris is a city of fancy and pleasure, where young men learn the skill of fencing, while in Norway men learn to use the battle-axe. Wittenberg is strongly associated with Martin Luther and the Protestant Reformation. It is the embodiment of the modern world, a world of students and scholarship. Wittenberg is linked with Rosencrantz and Guildenstern -the sycophantic young scholars and friends of Hamlet. Norway is the fringe, it is the past; while Paris and Wittenberg represent the future. Meanwhile, Denmark is caught between these two polarities: the old world and the new world. The frustrated and restless youth of Denmark feel compelled to go elsewhere, away from Denmark, particularly to mainland Europe. This cosmopolitanism of the Renaissance world stands in opposition to classical notions of commitment to one single polis (i.e. patriotism). Lastly we learn of England -the place where Hamlet is sent to die (the unnamed king of England owes Claudius a favor). However, Hamlet escapes while en route to England and instead Rosencrantz and Guildenstern are killed in England. Ironically, Shakespeare associates England with death in Hamlet.

Why would Shakespeare choose Denmark as the primary setting of Hamlet instead of England? Is there something uniquely illustrative about Denmark? The Danes were once the fearsome raiders of medieval England. Their viking longboats were the terror of river towns. However, Denmark as featured in Hamlet is no longer the expansionist regime of bygone years. Now, it struggles to be sure of itself and its own kingship is easily usurped. The contemporaneous English context of Hamlet also informs the setting of the play. Prior to the rule of King James I in England, Queen Elizabeth’s court was also elderly and cautious, neither aggressive nor expansionist -England was not yet a great power like France, the Hapsburgs of Spain, or the Ottoman Empire. This same tension between aging leadership and active young men is apparent in the play. The court of Denmark is elderly and there is some discussion about whether Denmark should pursue a policy of diplomacy, expansionism, or war. The central problem of an indecisive and elderly ruling class is: what to do with thumotic and rivalrous young men? How should leaders direct the energetic youth outward toward public good? The goal is to prevent young men from becoming destructive -to prevent them from seeking private revenge and personal gain. At any rate, King James I’s mother (Mary Queen of Scots) was married to the murderer of her husband (he was murdered in his garden much like the elder Hamlet in the play). Also James I’s wife was Anne of Denmark and Norway (Denmark and Norway were united as one single country at this time). James and Anne were married in Oslo and honeymooned at Elsinore Castle (the primary setting of Hamlet). Shakespeare is thus directing the court of James I to pay attention to particular dilemmas in Hamlet that echo the challenges facing 16th and 17th century England (as well as universal questions pertaining to political philosophy).

In the same way that Denmark represents a confluence of opposing worlds, Hamlet is also the locus for a variety of tensions: classical and modern, pagan and Christian, Orthodox and Reform. In light of these tensions, Prince Hamlet is a troubling hero. He represents an exploration into (and perhaps an inversion of) the classical heroic archetype. In Hamlet, the outward battle has turned inward. Hamlet is a prince who is called to exact revenge on a usurper (his uncle Claudius) and claim the rightful throne of Denmark, but in a modern context he finds the classical idea of revenge deeply troubling. In a decidedly Christian world, where souls are immortal, what profit is it to kill Claudius if his soul will simply be sent to heaven or perhaps purgatory? Should Hamlet simply ‘turn the other cheek’ to his nemesis? Can Denmark even be redeemed? In Act III scene iii, Hamlet finds an open moment to kill Claudius. He sneaks into Claudius’s room but just before committing the deed he stops himself because Claudius is praying. Claudius is on his knees asking for divine forgiveness of his sins, and thus by killing Claudius, Hamlet would merely send Claudius’s soul straight to heaven – and in this way the modern “conscience does make cowards of us all.” Instead, Hamlet must take into account vengeance on Claudius’s soul. This is an entirely distinct concept from the vengeance Achilles exacts on Hector in Homer’s Iliad where vengeance on Hector’s body is all Achilles desires. The immortal soul, in Shakespeare, emerges as a modern concept. The great difficulty, however, is in seeing the true intent of one’s soul without ‘shuffling off this mortal coil.’ Hence, why there is so much sneaking around, hiding, and scheming in Hamlet. There is a desire to reveal what is deeply hidden (i.e. “catch the conscience of the King” when characters like the aging counsellor, Polonius, hide behind a curtain not unlike Gyges in Herodotus’s Histories). Thus, Hamlet feigns madness in order to truly conceal his own personal intentions.

At any rate, despite his inner quandary Hamlet is not a wimp (contra Goethe’s assertion). Hamlet kills nearly half the characters in the play whether directly or indirectly. He has thumos -he is a political man. Similar to the political situation in Macbeth, in Hamlet a group of leaders led by Claudius plans to appoint Hamlet as the inheritor of the throne of Denmark but Hamlet is unsatisfied with this arrangement (“Sir, I lack advancement”). His standards of a political hero are classical, and thus he looks to Norway for courage because his home state of Denmark has lately become “rotten” and ruled by an aging gerontocracy (notice how many errors of judgment and false predictions Polonius makes in the play).

Hamlet is not a one-dimensional classical figure. He is a popular prince among the people of Denmark (hence why Claudius cannot simply have him executed). He is an unusually thoughtful hero, capable of examining things from multiple perspectives. He is not a single-minded avenger like Laertes (son of the old counsellor Polonius -note: Laertes was also the name of Odysseus’s father) and yet Hamlet is also distinct from his good friend, Horatio, who calls himself “more an antique Roman than a Dane” (his name is a combination of two Latin words ratio “reason” and orator or “speaker”). In many ways, Horatio is the most crucially important character in the play. He stands alongside Hamlet during the most critical scenes in the play and he is also one of the few characters to survive to the end of the play (he tries to commit suicide but Hamlet begs him not to). Hamlet’s character is somehow split between these two figures: Laertes and Horatio. Hamlet’s tragedy is that he cannot simply embrace one or the other. He is the tragedy of modern or Renaissance man -a man who tries to be too many things all at once and becomes trapped in paralysis. The crux of Hamlet is an interrogation of the possibility of a philosopher king in a Christian/modern context (i.e. a synthesis of a theoretical and practical kingship). Shakespeare seems to offer an example of a modern heroic warrior king in Henry V, yet even King Henry V faces the problem of the church -an established religion of peace which competes for authority with the expansionist state. The impossibility of Plato’s modern philosopher king becomes a tragic impossibility in Shakespeare. There is hope in Horatio, a man with Greco-Roman virtues in the modern world who doubts the fanciful claims of modern theology (“…there are more things in heaven and earth, Horatio, than are dreamt of in your philosophy”). However, Horatio is not destined for kingship.

The great hope of the Renaissance was to synthesize two antithetical value systems: Christian and Classical. In an effort to examine this tension, Shakespeare offers us Hamlet -a very modern European, who is dropped into a Norse feudal saga with a twist: Hamlet is tasked with a supernatural quest for vengeance and requital. Hamlet admires the classical virtues of his father, yet he is also restrained by what he has learned in Wittenberg (not least of which concerns the Protestant Reformation). He is constrained and haunted by notions of a life after death. Meanwhile, Denmark has embraced a secular court. There are no bishops or archbishops like those found in Shakespeare’s History plays wherein the church emerges as a troublesome political force contra the warrior kings of England. While in Hamlet the state of Denmark is not beholden to the church, Hamlet explicitly agrees with the theological doctrine that the world is fallen and sinful. This causes him to have a problem with politics. Everywhere he looks he sees corruption. He believes women are inherently corrupt -a product of ‘original sin’ and he calls Ophelia a harlot and commands her to “get thee to a nunnery!” Women are either chaste nuns or licentious whores according to Hamlet. In this way, virtue for Hamlet is akin to absolute sexual purity. Indeed, Hamlet possesses a certain type of absolutism that is not present in Shakespeare’s Roman plays. This view of virtue also causes Hamlet to despise everything around him, including his own country. Hamlet is a modern thinking man and he longs for the monastic or ascetic ideal (he embraces a Christian interpretation of a Roman appropriation of Greek things). The irony of Hamlet is that he is a modern Christian man but he is also aware of the tension between ancients and moderns, and he embraces certain Homeric virtues as well. In fact, Homeric epithets are found throughout Hamlet (around the time of Hamlet’s first performance George Chapman had just completed the first English translations of Homer’s Iliad and Odyssey, circa the 1590s). At any rate, Hamlet finds it disgraceful that the task is left to him to restore order to his chaotic and fallen world (in The Birth of Tragedy Nietzsche likens Hamlet to the Dionysian man). Hamlet’s disposition is somewhat philosophical -he resents the custom of his own people (particularly Claudius’s heavy drinking) and he despises the whole of humanity itself:

“What a piece of work is a man!
How noble in reason, how infinite in
faculty! In form and moving how
express and admirable! In action how
like an angel, in apprehension how
like a god! The beauty of the world. The
paragon of animals. And yet, to me,
what is this quintessence of dust? Man
delights not me: no nor woman neither…”
(Act II scene ii -Hamlet speaking to Rosencrantz and Guildenstern)

In his Interpretation of Dreams, Sigmund Freud links Shakespeare’s Hamlet with Sophocles’s Oedipus cycle, however in many ways the better contrast is between Shakespeare and Aeschylus. While Oedipus becomes tragically aware of his horrifying circumstance at the end of Oedipus Tyrannus, Hamlet becomes aware of his situation at the beginning of the play. In a certain light, Hamlet is better understood as Shakespeare’s response to Aeschylus’s Oresteia. In Aeschylus, we find a tale about the tragic cycle of revenge. Orestes returns home from exile to avenge his father, Agamemnon, who has been murdered and usurped by Clytemnestra (Orestes’s mother) and her lover Aegisthus. Whereas in Aeschylus’s Oresteia we are offered a deus ex machina in the form of Athena delivering a trial to release Orestes from his guilt, in Hamlet no such divine trial arrives to save the hero.

Hamlet spends a great deal of time contemplating his future actions, even though his actions will depend on a veiled illusion whether it be supernatural or not (Samuel Taylor Coleridge dismisses Hamlet as a flawed man who simply thinks too much, however this is far too simplistic an assessment). Hamlet is a man of thought who is contemplating actions (i.e. he embodies the conflict between thought and action), but the difficulty in the play is that it incorporates a vision of classical heroism (i.e. a man of action) coupled with a Christian critique of that heroism (a man of thought). His inner dilemma leads to stasis and eventually nihilism -it becomes an ontological question: ‘to be or not to be.’ Is it better to be and simply suffer at the hands of his enemies? Or else take up arms against a sea of troubles and surely die? But what is death if life merely continues on and on into a future world? Either way Hamlet will end up in heaven, purgatory, or hell. In contemplating the macabre, Hamlet notes that all men must die and disintegrate into mere “dust” -even the skull of Sir Yorick, the late court jester. Despite being a man of “infinite jest” Yorick’s finitude is likened to the meaningless sophistic trappings of Osric, a current Danish courtier. All of life becomes tragic for Hamlet (he notes the beauty of something anatomical like a skull). Beauty is ugly and life is empty for Hamlet.

Ophelia by John Everett Millais (1852

Hamlet also explores the problem of suicide. Does Ophelia commit suicide? In Act V scene i the two gravediggers debate the merits of a Christian burial for Ophelia. They decide that if she had intentionally ended her own life, then she does not deserve a Christian burial (the key question is whether she knowingly drowned herself -the possession of knowledge is a fluid problem throughout the play). In contrast in the classical world, suicide was sometimes the honorable thing to do (i.e. the suicide of Ajax; see Sophocles’s Ajax). However in the modern Christian world, suicide is a sin. Since there is no honor in suicide Hamlet must find another way out of his problems. The beginning of his “to be or not to be” soliloquy is a praise of the easiness and simplicity of the Pagan world. It begins with the idea of death in the ancient world, with echoes of Socrates and Cicero. The end of life in the ancient world meant the end of everything: ‘the end of heart-ache and the thousand natural shocks the Flesh is heir to’ -but things change in the modern world. Life truly begins after death and we perchance to dream in “the undiscovered country.” The Christian notion of an afterlife is described as a ‘calamity’ by Hamlet. If life never truly ends, what is the point of the politics of revenge?

“To be, or not to be, that is the question:
Whether ’tis nobler in the mind to suffer
The slings and arrows of outrageous fortune,
Or to take Arms against a Sea of troubles,
And by opposing end them: to die, to sleep;
No more; and by a sleep, to say we end
The heart-ache, and the thousand natural shocks
That Flesh is heir to? ‘Tis a consummation
Devoutly to be wished. To die, to sleep,
To sleep, perchance to Dream; aye, there’s the rub,
For in that sleep of death, what dreams may come,
When we have shuffled off this mortal coil,
Must give us pause. There’s the respect
That makes Calamity of so long life:
For who would bear the Whips and Scorns of time,

The Oppressor’s wrong, the proud man’s Contumely,
The pangs of dispised Love, the Law’s delay,
The insolence of Office, and the spurns
That patient merit of th’unworthy takes,
When he himself might his Quietus make
With a bare Bodkin? Who would Fardels bear,
To grunt and sweat under a weary life,
But that the dread of something after death,
The undiscovered country, from whose bourn
No traveller returns, puzzles the will,
And makes us rather bear those ills we have,
Than fly to others that we know not of?
Thus conscience does make cowards of us all,
And thus the native hue of Resolution
Is sicklied o’er, with the pale cast of Thought,
And enterprises of great pitch and moment,
With this regard their Currents turn awry,
And lose the name of Action. Soft you now,
The fair Ophelia? Nymph, in thy Orisons
Be all my sins remember’d”
(Act III scene i)

In this way Hamlet is a play about modern reformulations of heroism. The politics of Hamlet are guided by supernatural revelation coupled with a particular conception of the afterlife. Everything in Hamlet’s world is haunted by a vision of the next world. It makes the world appear to be irredeemably corrupt. Ultimately, Hamlet chooses to “take up arms against a sea of troubles” in an effort to end them, knowing that he marches toward his own death -there is not even redemption through art for Hamlet. His play “The MouseTrap,” which is a version of The Murder of Gonzaga, is not the dramatic conclusion of the play. It merely reveals Claudius’s guilt. Instead Hamlet must end his troubles by fighting to the death in a farcical parody of a duel (fencing is a more delicate way to settle disputes than, say, a proper sword-fight). No characters die by means of violence at the conclusion. Instead they die by trickery (i.e. poisoning). It is not a hero’s death and Denmark is not redeemed -perhaps Hamlet saw this fatal end coming.

Hamlet and Horatio in the Graveyard by Eugène Delacroix (1839)

At the conclusion, Hamlet’s Denmark is contrasted with Fortinbras’s Norway. Both young men have uncles who have claimed the kingship of their respected countries. For Hamlet the battle has turned inward on his own country, whereas Fortinbras extends outward conquering the worthless lands of Poland (against his enfeebled uncle’s demands). The Norwegian army has been allowed full rights to march across Danish lands toward Poland and the army of Fortinbras arrives at Elsinore just as Hamlet dies. With his dying breath Hamlet offers his “vote” for Fortinbras, his rival, to take the throne of Denmark (the message is conveyed by Horatio to Fortinbras). As is often the case in Shakespeare, the final speech in the play is delivered by the highest ranking person: in this case it is Fortinbras -a foreign man who will be the future king of Denmark (hopefully the accession will solve Denmark’s ‘rottenness’). Hamlet is given a hero’s burial by the Norwegian army, though few readers of Shakespeare will suggest that Hamlet is truly a noble hero -he is the tragedy of modern man, paralyzed while “crawling between earth and heaven.”


For this reading I used the impressive Arden edition of Shakespeare’s Hamlet along with Paul Cantor’s excellent lectures.

Introduction to Hamlet

The century was coming to an end. The rapidly approaching 1600s marked the end of the long reign of Queen Elizabeth I, and consequently the twilight of the Tudors. William Shakespeare, an unknown man without a formal education from the rural town of Stratford, had exploded onto the London theatrical scene, quickly becoming a star actor and playwright virtuoso. Back in his hometown of Stratford, his ailing father, John Shakespeare, who once was a prominent landowner and magistrate, had squandered the family’s status and riches over a scandal that surely dimmed William’s hopes of ever becoming a proper gentleman. After his fall from grace, John Shakespeare remained under lock and key at home, rarely leaving even to attend church and drinking himself to the grave. He died in September 1601. Meanwhile, William Shakespeare left his wife, Anne, and fled to London where he quickly rose alongside the growing popularity of the English theatre. Anne was left alone to raise the three Shakespeare children in Stratford (including an elder daughter, Susanna, and fraternal twins: Hamnet and Judith). There has been much speculation about Shakespeare’s relationship with his family, why did he abandon his family? His last will and testament seems to indicate a distant and perhaps unpleasant relationship with his wife that likely lasted for much of his life.

The writing of Hamlet coincided roughly with the death of Shakespeare’s father, but also, and more importantly, it coincided with the untimely death of Shakespeare’s only son, Hamnet, who died in 1596 perhaps of the bubonic plague. Hamnet was eleven years old when he died -he was another victim of the high infant mortality rates in England at the time (about one in every three children died before the age of ten). Hamnet and his twin sister, Judith, were both named after friends and neighbors of the Shakespeares in Stratford. After the deaths in the family, Shakespeare found himself with neither a son nor a father, and his relationship with his wife appears to have collapsed. On top of that, the mood of England was dark and conspiratorial. The plague continued to spread through the country. The end of the House of Tudor meant new questions about political stability -would England return to the extraordinary violence and chaos of the Wars of the Roses? The legacy of the Reformation continued to pit ‘radical’ Protestants against ‘Papist’ Catholics -often violently. Puritanical religious fanatics would soon ban theatrical performances and many other ‘worldly entertainments’ that were labeled sinful. The infamous Gunpowder Plot was soon to strike at the heart of the monarchy, amidst other conspiracies. At the time, London had grown into a crowded and growing city where odd superstitions proliferated. The people were quick to believe odd demonic theories and they held fast to grievances and resentments (in some ways the era was not too dissimilar from our own). Within this convoluted milieu, Shakespeare began writing Hamlet. And while Hamlet surely delves into his own personal, as well as political, and philosophical questions, Shakespeare took great labors to disguise himself within his own works, not unlike his predecessor Plato and his successor Nietzsche. Unlike the insatiable Christopher Marlowe, who wound up dead on a bar room floor, or Ben Jonson, who explicitly devoted a public poem in honor of his own deceased son, Shakespeare preferred a much more subtle, mysterious, and esoteric path.

Shakespeare’s personal intentions aside, Hamlet is the essential Renaissance play. It remains comprehensive in its scope -novel and imaginative yet melancholy and ominous- its horizon is both elusive and expansive by exploring universal themes. Details about the origins of the play remain something of a mystery. Unlike Macbeth, the sources for Hamlet are likely varied -from the revenge story of Brutus (the fabled founder of Rome) to a 13th century Norse saga/Danish history (“Geno Danorum”) by Saxo Grammaticus which tells the story of “Amleth,” a vengeful prince. Some literary scholars have suggested there was an earlier ‘Hamlet’ revenge-play by Thomas Kyd called the “Ur-Hamlet.” No copies of this rumored play have survived.

Hamlet was likely first performed at either the end of the 1590s or else at the very start of the 1600s at the Globe Theatre (actually the earliest documented evidence of the play’s performance is from an amusing oceanic voyage by the ship’s crew as they docked off the coast of Africa). In addition, there are three distinct versions of Hamlet that survive today (the First Quarto, the Second Quarto, and the First Folio). Each version contains lines that are not present in the other versions. Hamlet was almost certainly written with Richard Burbage in mind. Burbage was the the leading tragic actor in the Lord Chamberlain’s Men. The Globe Theatre apparently opened shortly before the first performance of Hamlet, the first performance at the Globe was likely Julius Caesar.

Hamlet is Shakespeare’s longest and most complex play.