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 There were other “Elements” books circulating in antiquity by Hippocrates, Leo, and Theudius, but Euclid superseded them all and none of the other books have fully survived into the modern day.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.

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.

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.

On Shirley Jackson’s “The Lottery”

“Lottery in June, corn be heavy soon” -fictional proverb

The year was 1948. The New Yorker Magazine was celebrating its 23rd anniversary when it published a disturbing little story called “The Lottery.” The story was to cause decades of controversy. At the time, The New Yorker apparently did not distinguish between works of fact or fiction and, as a consequence, they received more angry letters than any other publication. “The Lottery” was written by Shirley Jackson, a troubled writer of horror stories who hailed from the Bay Area before settling in the town of North Bennington, Vermont to start a family. North Bennington is a tiny town in southern Vermont and it bears a great deal of resemblance to the town in “The Lottery.”

The story begins in a beautiful, bucolic summer scene:

“The morning of June 27th was clear and sunny, with the fresh warmth of a full-summer day; the flowers were blossoming profusely and the grass was richly green. The people of the village began to gather in the square, between the post office and the bank, around ten o’clock; in some towns there were so many people that the lottery took two days and had to be started on June 26th, but in this village, where there were only about three hundred people, the whole lottery took only about two hours, so it could begin at ten o’clock in the morning and still be through in time to allow the villagers to get home for noon dinner.”

However, the horrifying nature of “The Lottery” occurs precisely because it takes place in a comfortable, safe, familiar setting: summertime in small-town America, in a village not much larger than 300 people. The story distorts this feeling of security and monochrome innocence. By the end, the truth about the lottery is revealed and a gruesome or macabre sense of dread overpowers the reader. In a crude plot twist, the townsfolk gather and draw lots (i.e. a “lottery”) but unlike in a typical lottery, nobody wants to win this contest. Like a heartless pagan ritual, the townsfolk of “The Lottery” gather each summer to select one person to be stoned to death, perhaps to purge the town of evil. The story ends as a woman known as ‘Mrs. Hutchinson’ is selected to be killed. The frenzied townsfolk quickly gather piles of stones while Mrs. Hutchinson screams, “It isn’t fair, it isn’t right!” The story concludes just as she is about to be stoned to death.

Upon its controversial publication, “The Lottery” spawned an onslaught of public criticism. Almost immediately, The New Yorker received hundreds of angry letters decrying its portrayal of a cult-like small town America. To add some context “The Lottery” was published shortly after the end of World War II, on the heels of the Cold War and the accompanying cultural anxieties related to impending nuclear warfare. In a world where nuclear warfare was an ever-present possibility, a return to innocence in Middle America captured the Zeitgeist however “The Lottery” shockingly localized some of these anxieties and redirected these fears inward toward the safest places in American culture.

Jackson, Shirley. “The Lottery,” The New Yorker, June 26, 1948.

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.

A Classical Hero in the Modern World: A Reading of Macbeth

A common reading of Shakespeare’s notorious “Scottish Play” is that Macbeth was written in tribute to the King of England and Shakespeare’s royal patron, King James I. Being a relatively new king on the throne of England, James was fascinated with two chief themes which are rife throughout Macbeth: witchcraft and regicide. James was a prolific writer and he wrote a book on the subject of witchcraft entitled Daemonologie. Thus both witchcraft and demonology were topics which James vehemently accepted as true. And on the topic of regicide, the infamous ‘Gunpowder Plot’ of 1605 was fresh in the minds of all Englishmen, as tensions between Protestants and Catholics continued to breed civil strife. Shakespeare’s Macbeth explores both themes of witchcraft and demonology in important ways with an eye toward political philosophy.

Macbeth takes place in 11th century Scotland, a pseudo-Homeric world filled with ruling Thanes who govern various regions beneath the reign of an appointed king. Geographically, the Scottish world in Macbeth is torn between a Hobbesian state of nature to the north (where the invading Irish Celts reside) and an orderly Christian kingdom to the south (in England). In addition, Norway serves an ever-present threat as it forges an alliance with a traitorous Scotsman, the Thane of Cawdor. In the pre-modern era, the fearsome Norsemen of Norway and their Viking longboats were the terror of Europe, especially under the leadership of Sweyn “Forkbeard” (who is mentioned in Macbeth as the current king of Norway). With these key geographic and political regions in mind, the Scottish world of Macbeth comes to light as a borderland, both physically and metaphorically. Much like Hamlet and Othello, Macbeth takes place in between competing ideas of civilization (especially pagan and Christian) as both value systems clash between visions of the future. For reference, Shakespeare uses this idea of a moral and geographic borderland to craft his seminal tragedies —Macbeth, Hamlet, and Othello. In Hamlet, the geography of the play is wedged between Norway to the north, Denmark in the center, and an orderly Christian Europe to the south. The character of Hamlet is a Christian prince assigned to complete a Pagan task of revenge. In Othello the geography of Cyprus is caught between Christian Venice and Islamic Turkey. In the play, the character of Othello is a Turk who becomes a Christian but is forced to end his own life as a Turk. In Macbeth, the northern lands are the barbaric worlds of the Irish Celts, in contrast to England in the south –England is the monarchy toward which Scotland is striving. In all three of these plays, the setting is modern (in contrast to Shakespeare’s Roman Plays) and the central tension of the play exposes a deep cleavage between classical antiquity and modern Christianity. In other words, Macbeth and its counterpart plays of Hamlet and Othello explore and test the prevailing sense of Renaissance optimism which was the prevailing wisdom in Shakespeare’s day, that modern Christian culture can successfully be harmonized with the virtues of classical antiquity.

An etching of Macbeth and Banquo meeting the Three Witches from the Holinshed Chronicles which Shakespeare used as chief inspiration

The tone of Macbeth is ominous, the mood is eerie, perhaps even evil. Three witches (or ‘weird sisters’) foretell of a dark prophecy in which what is “fair is foul, foul is fair.” In other words, the moral order is set to be upturned in Scotland. The character of Macbeth appears to us out of the fog of war as a classical hero, not unlike Achilles or Heracles. He ‘disdains Fortune’ as a fierce soldier. We first encounter him being honored as “noble Macbeth” and a “worthy gentleman” as well as “brave Macbeth” for his brutal killings on the battlefield (he is praised for slicing the rebel, Macdonwald, Thane of Cawdor in half and placing his head on a pike). Macbeth is surely a great war hero for Scotland, however by the end of the tragedy, Macbeth is no longer praised by his countrymen, and instead he is derided as “the dead butcher” with “his demon-like queen” (Act V, scene viii). How does Macbeth degenerate from a classical hero at the start of the play into a tyrannical villain by the end? The answer lies in Macbeth’s evolving beliefs throughout the play, particularly his own supernatural beliefs which delude him into committing a most heinous regicide.

Throughout the early parts of the play, Macbeth is contrasted with Scotland’s saintly and pious King Duncan, a most gentle and meek king. Duncan is the opposite of a warrior like Macbeth or even a soldier-king like Henry V. In performances of Macbeth, Duncan is often clad in white like a priest. Amidst the backdrop of a brutal two-front war, against the Celts and the invading Vikings from Norway, Duncan is almost wholly absent from the battlefield, even as his own son Malcolm is captured by the enemy and rescued by Macbeth. Under Duncan’s reign, Scotland has become excessively “gospeled.” Indeed, when Duncan finally arrives on the battlefield after the end of the fighting, he can hardly even recognize one of his own “bloody” captains. Duncan might best be compared with his counterpart to the south, Edward “The Confessor,” an equally delicate and weak king of England. Aside from being a feeble leader, Duncan’s second transgression is in naming his son, Malcolm, as his successor. At this time in Scotland, kingship was based on an elective monarchy rather than primogeniture. The king was merely an appointed leader, the first among equals. In naming his son as the future king, Duncan looks southward for emulation, to the example of England and its hereditary monarchy as a solution to the problem of political successorship. However in highlighting this parallel between England and Scotland, Shakespeare also illuminates Scotland’s distinctness from England as a uniquely democratic monarchy. The selection of Scotland as the setting is doubly important when considering the play’s first performance was likely delivered before the court of a Scottish king who sat on the English throne. Additionally, James I believed himself to be a descendent of Banquo (and therefore also of Banquo’s son Fleance who narrowly survives assassination in the play).

At the same time that Duncan’s kingship seems to be at its weakest point in the play, a dark prophecy begins to creep into the mind of Macbeth. Three ‘weird’ sisters appear (the word “weird” here comes from the Anglo-Saxon word “wyrd” meaning fate or destiny). Also called ‘witches,’ these weird sisters deliver a mysterious riddle that suggests Macbeth will become Thane of Cawdor (at present, he is only the Thane of Glamis). Their prophecy also states that Macbeth will become king of Scotland but that Banquo’s seed will actually spawn the line of future kings (i.e. Shakespeare offers a nod to King James I in this line of succession). At any rate, Macbeth contemplates these strange supernatural prophecies and, by surprise, he is soon appointed Thane of Cawdor in partial fulfillment of the prophecy. As a result, he quickly begins to lose faith in his own free will, and starts trusting otherworldly prophecies. Instead of making his own luck, Macbeth believes himself slave to the supernatural –“nothing is, but what is not.” Gradually, he is transformed from a soldier with limitless potential (‘disdaining Fortune’), into a hostage of Fate (“come what come may”). He also comes to believe in the idea of tyranny (in the modern sense, rather than the ancient notion of tyrannos), and his idea of tyranny informs his own practice as a tyrant (i.e. he becomes a murderer of families and children). In other words, when Macbeth begins to accept an absolutist supernatural ‘be-all and end-all’ power that controls his own fate, he begins to mirror that very absolutism within his own dominion. After committing his fateful act of regicide against Duncan, which is spurred on by his Clytemnestra-esque wife Lady Macbeth, we begin to see Macbeth’s inner struggle. The warrior’s conflict turns inward. He becomes king and the Thanes abandon him. We are given glimpses of his guilt over a string of seemingly endless savage murders (particularly his assassination of Banquo and the slaughter of Macduff’s whole family). The result is akin to the Furies which plague Orestes in Aeschylus’s Oresteia, the cycle of revenge which continues unabated in Macbeth. As king, Macbeth sees no end in sight to the vast numbers of people who must be killed in order for his own kingship to endure. And if there is the possibility of an absolute supernatural force that supersedes the strength of a warrior, then his being-in-time in the present-moment becomes irrelevant. Macbeth begins obsessing over the future (rather than the past or present) in the hopes of discovering supernaturally revealed signs which may prove the witch’s riddles true.

Despite being a changing world, filled with a moral system in conflict with itself, there are still limits to politics and kingship in Macbeth. Political philosophy remains enduring amid this conflict, as does the persistence of Nature. The subversion, or perhaps perversion, of Nature is addressed in the uncomfortable relationship between Macbeth and Lady Macbeth. Both spouses desire to be manly, however their notion of manliness (as in the Greek word aner rather than anthropos) differs from one another and it is devoid of any notion of justice. Macbeth is the warrior archetype not unlike a ‘guardian’ from Plato’s Republic, but contra Polemarchus’s notion of justice as explored in Plato’s Republic, Macbeth suddenly decides to turn his sword inward against his own kingdom. Why? In part, it is caused by his belief in supernatural revelations, but also his rejection of Nature which leads to his own downfall. His conception of manliness, bravery, and courage was once associated with violence against the enemies of Scotland, however the boundary between friends and enemies has become blurred for Macbeth and he ‘dares do all that may become a man.’ His decision to become treasonous is in part spurred on by questions of his manhood, as well as his belief in otherworldly prophecy. The ‘best of men’ according to Macbeth is someone who forcibly claims ownership over whatever he wants, following his base desires, and in so doing, his friends become enemies. In short, ‘what is fair becomes foul.’ Perfect tyranny is the telos toward which Macbeth is striving. Similarly, Lady Macbeth wishes to be ‘unsexed’ and made into an uncaring, cold-hearted woman. She questions Macbeth’s manhood, as if he is not strong enough to kill Duncan, accusing him of being “…too full o’the milk of human kindness.” She pushes Macbeth to “look like th’innocent flower, but be the serpent under’t.” There is something decidedly unnatural about this cruelty displayed by Macbeth and his Lady. Along these lines, they have no children, though apparently Lady Macbeth has previously “given suck” to a baby (we are not offered any explanation as to what happened to this baby) and their marriage is apparently a calculated political partnership in advancement of their own ends. Lady Macbeth rejects her nature as a woman, and she reimagines their marriage as the truest test of courage: to murder a king and claim the throne. After they begin the killing of all those who stand in their way, both Macbeth and Lady Macbeth face what we moderns might call severe mental illness or ‘brain-sickness’ because “unnatural deeds breed unnatural troubles.” At one point, Macbeth interrupts a meal filled with guests because he is haunted by the ghost of Banquo, and Lady Macbeth sleepwalks through the castle with “a great perturbation in nature” while furiously rubbing her hands together as if washing away the invisible blood (the idea of “blood” and “bloodiness” is mentioned over 40 times in the play). As with many people in the modern world, characters like Macbeth and his wife spend a great deal of time lost in their own heads, deep in thought, contemplating ideas of the absolute, the eternal, the infinite (as in Macbeth’s famously nihilistic soliloquy “tomorrow, and tomorrow, and tomorrow…” -which occurs immediately following the apparent suicide of Lady Macbeth). Macbeth and his Lady seek a perfect rule without the blemish of enemies or even half-friends. However in Shakespeare, Nature shows us that this glimpse perfection is actually unnatural –there must be nuance, blemish, and aberration. Tragedy strikes whenever modern human beings attempt, with great difficulty, to force Nature into a kind of divine perfection via purgation of impurity (i.e. those who call upon humans to ‘be ye therefore perfect’). Hence, when the protagonist faces his inevitable downfall, Shakespeare aspires to mirror Aristotle’s idea of tragic catharsis as described in the Poetics.

Lady Macbeth sleepwalking by Johann Heinrich Füssli (1781-1784)

Macbeth is a play that explores the nature of tyranny in the modern world. Is it possible for a tyrant to take power within the boundaries of modernity? Contra the optimism of Renaissance England, Shakespeare suggests that a tyrant like Macbeth is indeed a very dangerous possibility. As a pagan war hero dressed in the cloak of a Christian or modern king, Macbeth appears to us like Achilles –only with a conscience. As time passes, Macbeth justifies killing children and families, including Macduff’s whole family, though he cannot rest easy with this decision. Meanwhile, Macduff is called a “traitor” by one of the murderers sent to slaughter his family; and the king’s sons, Malcolm and Donalbain, are blamed for the death of the king –thus, the leaders of Scotland are so ‘gospeled’ that they have become incapable of spotting a true tyrant like Macbeth.

Perhaps in Macbeth, Shakespeare offers several points of caution to England’s new king, James I –lessons about the nature and limits of kingship, which includes a particular advocacy of Aristotle’s golden mean between serving as a meek king like Duncan, and a cruel tyrant like Macbeth (ironically the gentlest and most pious king runs the risk of inviting overthrow by the harshest and most savage tyrant). Shakespeare also offers a cautionary tale against the dangers of excessive belief in supernatural prophecies. Again and again in Shakespeare, Nature is shown to have limits that curb human desires, but characters like Macbeth decide to place their faith in supernatural whims. In the case of Macbeth, he embraces fateful prophecies that hold him hostage to an unfolding destiny, one which he believes he must act in accordance with. Time itself becomes merely a self-fulfilling divine revelation for Macbeth –he grows obsessed with the future, looking only for “Tomorrow and tomorrow and tomorrow.” Consider the moment Macbeth persuades his wife of the witches’ prophecy (“thy letters have transported me beyond this ignorant present, and I feel now the future in the instant”). By the end of the play, Macbeth believes a new prophecy that ‘none of woman born’ can harm him, and thus he views himself as an invincible superman, protected by unexplained otherworldly whims, at least according to his interpretation of the witches’ riddle. However, the invading soldiers descend on his castle clad in the branches of trees from Birnam forest (thus fulfilling another part of the prophecy) and Macbeth learns that his enemy, Macduff the Thane of Fife, was never technically born of a woman. Instead he was “untimely ripp’d” from his mother’s womb (i.e. he was born via a caesarean section). So Macbeth meets his fateful end according to the witch’s prophecy after all –he is slaughtered and decapitated offstage by Macduff who, himself, was never truly born of woman (note: very few characters are actually killed onstage in Macbeth, exceptions include Banquo as well as Macduff’s family. Both are killed indirectly at the behest of Macbeth).

At the end of Macbeth, Scotland is cured of its particular disease and it hangs on the promise of a new king: Duncan’s heir, Malcolm, a non-Christian who stands in sharp contrast to his pious father, Duncan (Malcolm gives thanks to the “grace of Grace” rather than the “Grace of God” and promises to rule in “measure, time, and place”). Earlier in the play, while in exile Malcolm hesitated at the prospect of becoming king (“a good and virtuous nature may recoil in an imperial charge”). Malcolm confessed to Macduff his uncontrollable sexual desires (“your wives, your daughters, your matrons, and your maids, could not fill up the cistern of my lust”) and he also confessed to having a deep hunger to rob the nobles of their wealth. Malcolm worries that his personal vices are worse than the rule of a tyrant, like Macbeth, because he takes no stock in virtues like Justice, Mercy, Courage, and so on. Macduff cries out that Malcolm is not fit to live, much less to govern, but in response Malcolm quickly covers over his thoughts with a praise of God and a series of lies to reassure Macduff, though it is difficult to “reconcile” what Malcolm has just uttered. This little interlude is deeply revealing about the character of Malcolm in contrast to his father, and perhaps foreboding about the future of Scotland. At any rate, when Malcolm becomes king he renames his thanes as “earls” to mirror the orderly monarchy of England to the south, and he calls his exiled friends abroad to come home. Macduff kills Macbeth in much the same way Macbeth once killed the rebel Macdonwald in Act I –Macbeth is slaughtered offstage and his head is brandished by Macduff. The disease Scotland is cured of is Macbeth’s uniquely modern form of tyranny –a belief in absolutism, a tyranny modeled on the idea of an all-controlling and unblemished Fate or ‘destiny’ or divine will. Macbeth believed he could become omniscient like a god, and thus he had degenerated into the worst of all evils. The danger of an all-perfect, all-good divinity is that it inspires the greatest of all evils on earth. Therefore, in order for a king to be successful in the modern world, without stooping to the moral depravity of Macbeth, he must find an Aristotelian golden mean. He must be both gentle and pious like Duncan, as well as prideful and disdaining of Fortune like Macbeth. In other words the city (polis) depends upon a certain degree of evil, such as the callousness of a classical soldier (like Macbeth cutting another man in half, rather than merely ‘turning the other cheek’). However, when the guardians of the city like Macbeth are turned inward, they will unjustly assassinate the king and unleash a far more cruel tyranny. By the end of the play, Macbeth’s fortunes are terrifyingly reversed, not unlike Oedipus, and the witches are proven correct, though not in the way Macbeth had expected. Extreme forms of political rule are characterizes by either weak and ineffectual leaders or else vicious and cruel tyrants. The introduction of Christianity into the modern world (in contrast to Shakespeare’s Roman plays) entirely upends classical notions of political life, however it does not destroy the enduring political questions as investigated by the ancients. Instead, it exposes something deeper and more authentic about ourselves which remains worthy of exploration.

Macbeth is a horrifying tragedy because it reveals deep fault-lines within our ethical standards of judgment. It exposes a stark conflict between two differing conceptions of the good (this conflict is the prototype of tragedy according to Hegel). At times, we celebrate aggressive impulses and admire a person for his sheer strength and power, like Macbeth and his ability to triumph in combat over others. The great monument to this attitude in Western culture is Homer’s portrait of Achilles in The Iliad. At other times, we assert the need to tame aggressive impulses and brand them as evil, or at least the most significant impediment to achieving social order. A memorable example of this attitude is the portrait of Jesus in the New Testament. In Macbeth, Shakespeare exposes the opposition between these two ethical viewpoints, one classical and the other Christian, and in doing so, he offers a meditation on the very concept of modern manhood in the play. As in the dialogue between Malcolm and Macduff, we see that the question “What is it to be a man?” which sits at the heart of Macbeth, offers two different answers in response —one pagan and the other Christian—both of which run throughout the play in tragic tension with one another. With the introduction of dark prophecies, Macbeth is tormented by doubts of his manliness. At the same time, he feels the pull of Christianity, and the virtue of meekness, which is also held in high regard in his country. Hence, when he commits his crimes, it is not with a clear conscience. He is indeed horrified by his own misdeeds, haunted before and after committing them as he witnesses frightening images, exposing his own guilt and criminality. If Macbeth were not torn in opposing directions, his life would be much simpler. If he were fully Christian, he would never commit these crimes. If he were fully pagan, he would hardly be so tormented by his deeds and would instead proceed without hesitation. But the Macbeth of Shakespeare is torn between two conceptions of what it is to be a man –and this conflict makes him a truly tragic figure. Tragedy does not provide us with simple moral lessons, such as “pride goes before a fall.” Unlike melodramatic works, which simply appeal to our conventional moral beliefs, tragedy is unsettling; it disturbs us and unnerves us by revealing that our ordinary moral platitudes do not necessarily complete nor adequately explain the full range of human possibilities. Understandably, we do not take joy in reflecting on these problematic aspects of the human condition, aspects which Shakespeare exposes in his tragedies, but nevertheless they show us a terrifying glimpse of something true about our nature.

For this reading I used the essential Arden 3rd Edition of Shakespeare’s Macbeth as well as the writings and lectures of Paul Cantor as well as Timothy Burns’s Shakespeare’s Political Wisdom (2013).