‘Superfluous Death’ and The Mathematics of Revenge
[a] ‘Oh…Millions of deaths’ complains the Duke as he expires in Middleton’s The Revenger’s Tragedy (3.5.186), thus fulfilling Vindice’s fantasy of ‘constant vengeance’ (3.5.109). John Kerrigan and others have long recognised the excessively reciprocal nature of revenge on the early modern stage. This article seeks to quantify that process, or rather to survey the quantities that recur with surprising frequency in the genre of revenge tragedy.
[b] In Antonio’s Revenge, Pandulpho wishes to prolong his enemy’s death ‘till he hath died and died/ Ten thousand deaths in agony of heart’ (5.5.78), while Hamlet’s Laertes curses whoever has made his sister mad: ‘O, treble woe/ Fall ten times double on that cursed head’ (5.1.235). On the one hand this can be linked to the competitive intertextuality of the genre itself, where each author tries to outdo his predecessor – the logical conclusion of Renaissance emulatio. But might it also point to a deeper psychology of revenge, that struggles to equate life with life, and refuses to accept parity?
[c] In King Lear Shakespeare demonstrates how love cannot be quantified, through the motif of Lear’s diminishing train of knights: ‘Thy fifty yet doth double five and twenty,/ And thou art twice her love’ (2.2.448). Revenge tragedy is similarly interested in arithmetic, where the variable is not love but vengeance.
 In its simplest form, the equation of revenge is A injures B, and so B injures A. If the injury happens to be fatal, then B can no longer retaliate, and so must (often from beyond the grave) recruit C to take his or her place. Kerrigan says of this need for a third party that ‘[t]he displacement of revenge from one character to another creates a structure of obligation which modifies the economy of vengeance’ (1996: 3). The suggestive phrase ‘economy of vengeance’ implies that human lives are in some sense quantifiable, as codified in the so-called lex talionis — an eye for an eye. Similarly, Linda Woodbridge has argued persuasively that ‘revenge speaks a language of debt and obligation’ (2010: 84). As England experimented with Italian double-entry ledgers of debit and credit, revenge tragedies too can be seen to ‘balance the books’, meticulously matching injury for injury, death for death. While Woodbridge is right to identify the genre’s ‘numeric idiom’ (2010: 83), I wish to focus on a particular instance of numeracy in revenge tragedies that cannot be reduced to the logic of book-keeping. Revenge by its very nature exceeds the double entry system, upsetting any desire for ‘balance’ through its refusal to allow one debt/death to cancel another. In what follows, I seek to quantify what is at stake when characters on the early modern stage engage in the mathematics of revenge.
 From Titus Andronicus’s complaint that he is ‘as woeful as Virginius was,/ And have a thousand times more cause than he’ (5.3.49) to Laertes’s curse, ‘O, treble woe/ Fall ten times double on that cursed head’ (5.1.235), stage revengers are drawn to expressing themselves mathematically. This may be attributable to the fact that a body count is such an integral part of how these plays are plotted; reciprocity requires a tally:
As dear to me was my Horatio,
As yours, or yours, or yours, my lord, to you.
My guiltless son was by Lorenzo slain,
And by Lorenzo and that Balthazar
Am I at last revenged thoroughly.
(The Spanish Tragedy, 4.4.167)
Where revenge differs from mathematics is that while most totals can be reached by both addition and subtraction, the sum total of revenge can only ever go one way: upwards. A death toll, like any other toll, is accumulative. In the early modern theatre it both necessitates and creates a constant supply of stage revengers. In Hamlet, Laertes may demand of Claudius ‘Give me my father’ (4.5.116), but the redundancy of his request highlights the impossibility of any real reciprocity in revenge. In exchange for Polonius, Laertes will only receive the death of Hamlet, a perverted form of gift-exchange. While subtraction may not be an option for dramatists of revenge — characters can only come back to life momentarily in their ghostly, minus, form — division and multiplication do have an important part to play. That is, by dividing the revenge up in some manner, it becomes possible to multiply the ‘vengeances’ exacted. In The Tragedy of Hoffman, when the victims turn revengers, it is natural for them to envisage the potency of their vengeance in numerical terms: ‘And ’twould more vex him…/…/…than a hundred deaths’ (5.1.283). It is in the revengers’ attempts to make a single life suffer multiple deaths, rather than in the sheer number of dead bodies at the play’s climax, that the genre shows its abiding concern with the quantification of revenge. Furthermore, in undermining the parity of one character’s life for another’s, the genre reflects on its own laws of composition: lex talionis, an eye for an eye.
 This essay will focus on John Marston’s Antonio’s Revenge and William Shakespeare’s Hamlet, both revenge plays from the turn of the seventeenth century that are so intertwined that the order of their composition still cannot be definitively established (Neill and Jackson 1998). The dialogue between these two is well documented (Mercer 1987), but what has not been noted previously is the equations that unite the revengers in both. Paying attention to the mathematics of revenge sheds new light on the relentless intertextuality of the genre, and also makes sense of certain mathematical problems embedded in the plays, such as why Hamlet kills Claudius twice. I touch on Thomas Middleton’s The Revenger’s Tragedy as a coda of sorts; its protagonist Vindice returns to the problem of quantifying revenge in such a way as to undermine the very process of quantification. But first, it is necessary to establish the connection between revenge, mathematics, and hyperbole, which are each intimately bound up with the self-referentiality that these plays are known for (Kerrigan 1996). It is not my intention to engage in numerology, or to construct a grand equation of revenge, but rather to put forward a theory (as opposed to a theorem) that accounts for the abiding connections between mathematics and revenge on the early modern stage.
Intertextuality by Numbers
 Rosalie Colie describes the early modern author as being ‘in conscious competition with the very best that tradition could offer him’:
[T]here was an insistence on outdoing and overgoing earlier achievements, each man newly creating out of and against his tradition. (Colie 1974: 5)
Revenge tragedy offers a perfect example of just such a tradition, making the excesses of each play essentially emulatory in nature. From the mythological Atreus who feels ‘For yet even this too little seems to me’ (Thyestes, 5.3.85), stage revengers have always sought to make their revenges stand out against their historical and literary background: ‘For worse than Philomel you used my daughter,/ And worse than Progne I will be revenged’ (Titus Andronicus, 5.2.194). The intermeshing of Classical precedent and violent comparison is key to Shakespeare’s method in Titus, as the protagonist ‘self-consciously strives to surpass his classical models’ (Miola 1992: 29).
 Hallett and Hallett rightly point out that ‘[r]evenge is itself an act of excess’ (1980: 11), but this does not account for the constant cross-referencing of a genre in competition with itself. As Piero puts it in Antonio’s Revenge: ‘Is’t to be equalled thinkst thou?’ (Antonio’s Revenge, 1.1.79). The quotation encapsulates the way in which the concept of revenge challenges the very notion of equality, while also betraying a certain insecurity on the part of the character. At the same time, these plays are filled with numbers and equations. Take for example Aaron’s boastful admission of guilt in Titus Andronicus:
Tut, I have done a thousand dreadful things
As willingly as one would kill a fly
And nothing grieves me heartily indeed
But that I cannot do ten thousand more.
In the course of four lines, Aaron’s crimes multiply from a thousand to ten thousand effortlessly, exemplifying the exponential use of numbers and figures permeating these plays. By the time of Middleton’s revenge tragedy to end all revenge tragedies, knowingly titled The Revenger’s Tragedy, revengers are not satisfied with thousands and tens of thousands. Rather, the Duke is subjected to ‘[m]illions of deaths’ (3.5.188) by the arch-revenger Vindice. Such a claim is consciously exaggerated, pointing as it does to the intersection of revenge, hyperbole, and numerical superiority.
 While hyperbole was only one of any number of rhetorical figures taught and practiced in the grammar schools of early modern England (Mack 2002), the figure is yoked to — and transformed by — a distinctive language of revenge, as with Aaron’s boast quoted above. The rhetorical figure of hyperbole was famously called the over-reacher by Puttenham in his Art of English Poesie (1589: 159, sig. Y2r), which aptly describes stage revengers such as Titus, Hoffman, and Vindice.
Chettle’s Hoffman shows himself so eager to be revenged that he hopes to ‘pass those of Thyestes, Tereus,/ Jocasta, or duke Jason’s jealous wife’ (1.3.21): personal revenge is measured against classical precedent. Titus Andronicus similarly models himself on Priam (1.1.83), Coriolanus (4.4.66), and Virginius (5.3.49), explicitly inviting comparison with those who have gone before. The eponymous Antonio of Marston’s play sounds a different note when he complains ‘Let none outwoe me: mine’s Herculean woe’ (2.3.142).
 When such comparisons are scattered so liberally through the revenge genre, it is clear that the impulse to compare goes deeper than any particular play. The motif exceeds the explanation offered by straightforward intertextuality, and in fact offers a critique of intertextuality per se. Having been trained to copiously imitate Classical authors in the schoolroom, revenge dramatists show the violent fruits of such a labour. Shakespeare’s Titus is the clearest example: it is Chiron and Demetrius’ familiarity with Ovid that teach them to lop off Lavinia’s hands as well as her tongue. Thus the literary competitiveness inherent in Renaissance emulatio leads to bloodshed, showing its kinship with revenge. As protagonists strive to be ‘peerless in revenge’ (Antonio’s Revenge, 3.5.29), dramatists draw attention to this through Classical reference and vengeful precedents, with Senecan tragedy acting as a touchstone of sorts: ‘scelera non ulcisceris, nisi vincis’, translated by Jasper Heywood as ‘Thou never dost enough revenge the wrong/ Except thou pass’ (2.1.20). The intertextuality of such an excessive genre is indicative of early modern dramatists’ efforts to compare and quantify vengeance itself. However, such violent hyperbole belies the revenger/author’s own fear that revenges cannot be ranked so easily. Therefore, rhetorical superlatives are combined with numerical superiority to ensure that one’s revenge is excessively notable and notably excessive. This is best illustrated by John Marston’s Antonio’s Revenge, first performed by the Children of the Chapel at the turn of the seventeenth century.
3:1 in Antonio’s Revenge, or the Problem of Piero
 Typical in its plot, Marston’s play provides its central character, Antonio, with a host of reasons for seeking the death of his duke, Piero: the latter has murdered Antonio’s father Andrugio, falsely accused Antonio’s fiancé Mellida of inchastity, and framed Antonio’s recently-murdered friend Feliche as Mellida’s lover. In the course of the play Mellida too succumbs to Piero’s plots, giving Antonio a plethora of reasons to be revenged. In a sense, this is where the revenger’s problem lies: Antonio must take separate revenges for father, fiancé and friend in order for balance to be restored. If he were only to cite Andrugio in the final scene, does this leave Mellida and Feliche unrevenged? The deaths of Andrugio, Feliche and Mellida are greater than Piero’s single life, so that Piero’s multiple homicides create a crisis of revenge that can be thought of symbolically as: 3 > 1. Furthermore, Marston gives his protagonist Antonio numerous accomplices, including Pandulpho, the father of Feliche, and Alberto, mutual friend of Antonio and Feliche. But if each of these would-be revengers is to achieve a ‘full’ revenge on Piero, then surely they would have to kill their enemy single-handedly. Indeed the title seems to suggest just such a proposition. How then can the sole enemy Piero be equally divided among a range of revengers, while still conforming to the quid pro quo of revenge tragedy?
 A solution begins to emerge after Antonio has been visited by the ghost of his father Andrugio, who reveals all to his son. Andrugio accuses Piero of murder and exhorts, seven times in the course of a twenty-line speech, his son to be revenged (3.1.32-51). Thenceforth Antonio’s mind is fully focused on revenge: ‘May I be cursed by my father’s ghost/…/If my heart beat on aught but vengeance’ (3.2.35). He is soon presented with the opportunity to take an indirect vengeance, when Piero’s young son Julio enters. Antonio’s slaying of the innocent Julio has been called ‘one of the worst excesses in revenge literature’ (Hallett and Hallett 1980: 170), but surely this is precisely the point. Antonio is then ‘forced into a gruesome rationalisation of his savagery’ (Mercer 1987: 78), as he attempts to make a spurious distinction between Julio’s ‘soul’ and Piero’s blood that runs in the boy’s veins:
|He is all Piero, father all. This blood,
This breast, this heart, Piero all,
Whom thus I mangle. Spirit of Julio,
Forget this was thy trunk. I live thy friend.
It is difficult to accept such specious reasoning from the revenger. Nor, I would argue, are we expected to. Antonio’s fantasy of slaying Piero in the person of his son Julio highlights the problem of Piero having only one life to lose. But due to the nature of Piero’s crimes, a single death is not sufficient. Thus Antonio seeks to prolong Piero’s life only to prolong his death, so to speak: ‘I’ll force him feed on life/ Till he shall loathe it’ (3.2.89).
 In the aftermath of the murder of Julio, Antonio appears almost indistinguishable from his enemy Piero. Just as Piero had earlier boasted of his ‘unpeered mischief’ (1.1.10), Antonio is urged by his father’s ghost to ‘be peerless in revenge’ (3.5.29), which is precisely what he has done. Marston creates visual correspondences between the two: Antonio comes onstage with ‘his arms bloody, [with] a torch and a poniard’ (SD 3.5.13) just as Piero’s first entrance was ‘smeared in blood, a poniard in one hand bloody, and a torch in the other’ (SD 1.1.1). But the play’s two final acts significantly modify this image of a protagonist dripping with blood and crying out for ‘vindicta’, as Marston retires the Senecan model of a single revenger bent on the destruction of his enemies, replacing it with a more inclusive form of retribution. In bringing together all those wronged by Piero in the course of the play, Marston must necessarily lessen the protagonist’s role in the final revenge. But as this is revenge tragedy, a dramatic genre as opposed to a real crime, the literal can give way to the literary; multiple homicides require multiple deaths.
 It is notable that while the group revenge against Piero begins as meticulously balanced, this gives way to an excessiveness close to sadism. The band of revengers exact a protracted and cruel revenge on Piero that includes taunting him with the dead body of his son Julio (5.5.52). Their justification comes from Piero’s own excesses, which they rehearse for the benefit of both Piero and the audience:
Antonio: My father found no pity in thy blood. Pandulpho: Remorse was banished, when thou slew’st my son. Maria: When thou empoisoned’st my loving lord,
Exiled was piety.
Antonio: Now, therefore, pity, piety, remorse
Be aliens to our thoughts: grim fiery-eyed rage
Possess us wholly.
This is followed by a further twenty lines of torture, insults and curses for Piero, including the curious stage direction ‘They offer to run all at Piero, and on a sudden stop’. This action, seemingly merciful in its import, is in fact further evidence of their sadism. It is designed to prolong Piero’s pain, as Pandulpho makes clear:
let him die, die, and still be dying.
And yet not die, till he hath died and died
Ten thousand deaths in agony of heart.
The multiplication of deaths for Piero suggested by Pandulpho’s grim lines goes beyond the ‘eye for an eye’ logic of lex talionis. It plays with the possibility of inflicting innumerable deaths, by prolonging the moment of revenge indefinitely.
 On the point of death, Marston structures the final blow to be highly stylised and communal:
|Antonio:||This for my father’s blood.|
|[He stabs Piero]|
|Pandulpho:||This for my son.|
|Alberto:||This for them all.|
The balanced lines here replicate the restoration of order which Piero’s death brings about. At the same time, the lineation cannot do justice to the simultaneity of revenge, as the printed text can only approximate the united revenge action being staged. Indeed more than one stage direction is needed here, to give equal weight to each revenger in the scene. To return to the initial problem of Piero’s multiple homicides being represented as 3 > 1, Marston solves this by making one Piero die in triplicate.
 In the aftermath of Piero’s death, the revengers do not try to hide their crime, but rather jostle with each other to take full credit for the murder. When asked by a Venetian senator, ‘Whose hand presents this gory spectacle?’ (5.6.1), they reply:
|Antonio:||I will not lose the glory of the deed.|
This competitiveness is allayed by Alberto’s ‘Tush, to say troth, ’twas all’ (5.6.11); again individuality gives way to inclusiveness: ‘Mine’ becomes ‘all’. This word is picked up by the senator who responds in kind with, ‘Blessed be you all, and may your honours live/ Religiously held sacred, even for ever and ever’ (5.6.12). Instead of having his revengers die for their part in Piero’s death (thus unbalancing the symmetry achieved), Marston exonerates Antonio and his accomplices for having successfully solved the problem of Piero. In a sense, the title of Antonio’s Revenge is misleading, in that it suggests a single protagonist. Marston multiplies not only the motives for revenge, but also the revengers whose duty it is to carry out vengeance. Faced with only one enemy, they must join forces in such a way that manifold revenges can be achieved.
40,000:0 in Hamlet, or ‘Forty thousand to love’
 As a play, Hamlet is most famous for its protagonist’s deep philosophical cogitations, yet there are also a surprising number of equations running through the text that merit closer analysis. These are tied up with Hamlet’s concern to quantify and measure the world around him, from the celestial (‘What a piece of work is man’, 2.2.269) to the sub-human (‘a slave that is not twentieth part the kith/ Of your precedent lord’, 3.4.95). This may account for Shakespeare’s above-average usage of the word ‘quantity’ in the play, an inherently ambivalent term that can mean both ‘a specific or definite amount’ and ‘an indefinite (usually large or considerable) amount’ (OED 1.a; 1.b). The word ‘quantity’ appears in only ten plays, and in just two of these is it used more than once. Timon of Athens, that most economic of plays, uses the term twice (5.1; 5.4). Yet in Hamlet the count rises to four, twice as many as in Timon. From Hamlet himself, we hear of the clown who will laugh only to ‘set on some quantity of barren spectators to laugh too’ (3.2.39). In the same scene, the Player Queen equates love and fear, seeing them as proportional to each other:
For women fear too much, even as they love,
And women’s fear and love hold quantity –
Either none, in neither aught, or in extremity.
This obscure passage struggles to equate fear and love, even suggesting that the two are somehow interchangeable. Shortly afterwards in the closet scene, Hamlet resorts to the word once again in questioning his mother’s powers of judgement, for going from Old Hamlet to Claudius, ‘this to this’ (3.4.69):
sense to ecstasy was ne’er so thrilled
But it reserved some quantity of choice
To serve in such a difference.
Finally, we have Hamlet’s impassioned declaration of love:
I loved Ophelia – forty thousand brothers
Could not with all their quantity of love
Make up my sum.
When it comes to comparing — or struggling to compare — like with like, then whatever is being compared must first be made quantifiable; hence the quantity of ‘quantities’ in Hamlet.
 The term ‘quantity’ is also strangely apposite to the paradox of revenge, where deaths are measured one against another; as Laertes bluntly puts it, ‘Give me my father’ (4.5.116). Throughout Hamlet, Shakespeare meticulously balances parallel revenge plots, inviting the audience to make comparisons. Claudius has killed Old Hamlet, and so the younger Hamlet must be revenged. Hamlet mistakenly kills Polonius, setting in motion Laertes’ own revenge plot. Characters within the text even appear to be aware of this equivalence, as when Hamlet tells Horatio that he is sorry for having offended Laertes: ‘For by the image of my cause I see/ The portraiture of his’ (5.2.77 Folio). However, it is this very similarity that gives rise to the play’s interrogation of equivalence per se, and the problem of quantification. If Claudius is as inferior to Old Hamlet as we are led to believe (‘not twentieth part the kith/ Of your precedent lord’ (3.4.95)), how can his death possibly ‘equal’ that of Old Hamlet’s? More to the point, are the two characters who share the name Hamlet in some sense equals? Both problems come together in Hamlet’s first soliloquy. ‘My father’s brother (but no more like my father/ Than I to Hercules)’ (1.2.152): in seeking for equivalence, Hamlet only finds difference. And there’s the rub. In the hall of mirrors that is Hamlet, Shakespeare makes his audience painfully aware of the difficulty in the very notion of comparison. In the first instance, I want to problematise critics’ use of Laertes as a ‘foil’ for Hamlet, by analysing Shakespeare’s use of comparisons in the graveyard scene. I then move on to the final revenge(s), to see how characters mete out vengeance in the wake of the play’s ambivalent equivalences.
 Ahead of the fencing match that will precipitate the end of Hamlet/Hamlet’s revenge, we witness verbal sparring of a different sort between Hamlet and Laertes. Ophelia is quite literally the grounds of the argument, in the graveyard scene that opens the final act. Insults and taunts quickly accumulate on both sides, and this is underlined by an incessant use of arithmetic. Consider Laertes’ curse on the person responsible for his sister’s madness: ‘O, treble woe/ Fall ten times double on that cursed head’ (5.1.235). 3 x (10 x 2) = 60. What are we to make of this bizarre equation of woe? It is accompanied by the hyperbolic gesture of leaping into the grave, ‘Till I have caught her once more in mine arms’ (5.1.239). Laertes’ larger than life demonstration of grief continues as he wishes to be buried until the grave mound reaches higher than ‘old Pelion or the skyish head/ Of blue Olympus’ (5.1.242). It is this last, with its combination of histrionic excess and classical emulatio, that prompt Hamlet to declare himself:
What is he whose grief
Bears such an emphasis, whose phrase of sorrow
Conjures the wandering stars and makes them stand
Like wonder-wounded hearers? This is I,
Hamlet the Dane.
That this battle is a rhetorical one is evident from Hamlet’s terms of reference: ‘emphasis’, ‘phrase of sorrow’, and shortly, ‘I will fight with him upon this theme’ (5.1.255). Ophelia is no longer the ground of the argument, but rather argumentation is: ‘Nay, an thou’lt mouth,/ I’ll rant as well as thou’ (5.1.272). At this point Hamlet picks up on Laertes’ paltry quantifications, as if to degrade his mathematics through numerical superiority:
I loved Ophelia – forty thousand brothers
Could not with all their quantity of love
Make up my sum.
The presence of numbers in this highly charged scene, while unexpected at first, ties in with larger networks of comparative and competitive language. Just as making Ossa like a wart can be achieved only through moving ‘Millions of acres’ (5.1.270), Hamlet’s choice of forty thousand is as arbitrary as it is meaningless. Clearly this is as much a linguistic combat as it is a genuine outpouring of emotion. And as in any combat, someone must keep score. While in the fencing match of 5.2 that task falls to Osric, here it is the audience who must keep score between the two. Laertes’ language is held up for ridicule, in a game of rhetorical point-scoring — ‘A hit, a very palpable hit’ (5.2.262).
 Hamlet leaves behind the rhetoric of numbers in the final scene, only to be faced with a more difficult problem than Antonio’s in Marston’s play. How is he to be revenged on Claudius, who is both doubly guilty — ‘a father killed, a mother stained’ (4.4.56) — and infinitely unworthy of Old Hamlet, ‘a slave that is not twentieth part the kith/ Of your precedent lord’ (3.4.95)? A more fundamental question is whether or not Hamlet’s slaying of Claudius can be deemed revenge at all, an argument that has found some favour over the years (Ure 1974: 42; Mercer 1987: 247; Gurnham 2009: 13). It is my contention that the mathematics of revenge traced thus far can help us to interrogate this proposition, by framing the question in terms of why Hamlet kills Claudius twice. Our familiarity with the text of Hamlet should not blind us to the strangeness of Hamlet’s choice to poison Claudius by two different means, within the space of a few short lines. Does this suggest that Claudius is doubly guilty and, if so, of what crimes?
 Let us then turn to the final fencing match to establish exactly what happens, in order to see the logic of Hamlet’s choice in those last crucial moments. When the instruments for the fencing match are brought forward, it must be remembered that Hamlet has no knowledge of Claudius’s and Laertes’s foul play. The fact that he does not think twice about entering a contest organised between his enemy and the son of the man he has killed is of a piece with his trust in providence, while also confirming Claudius’s claim that he is ‘[m]ost generous and free from all contriving’ (4.7.133). Throughout the scene Hamlet plays his part unwittingly, and it is not until Laertes informs him of what has actually been happening (5.2.298-305) that he can act with full knowledge. On learning that ‘the King’s to blame’ (5.2.305), Hamlet immediately points his weapon in the direction of his uncle: ‘The point envenomed too? Then venom to thy work!’ (5.2.306). But this is not sufficient for Hamlet, and he proceeds:
Here, thou incestuous, damned Dane!
Drink of this potion. Is the union here?
Follow my mother.
Even in the earliest printed version of Shakespeare’s play, the quarto of 1603 which is half the length of the ‘enlarged’ 1604 edition, Claudius is subjected to multiple deaths; first Hamlet stabs the king saying ‘The poisoned instrument within my hand?/ Then venom to thy venom — die damned villain!’ (17.95). Hamlet’s speech continues uninterrupted as he forces the king to drink from the poisoned chalice: ‘Come drink – here lies thy union, here!’ (17.97). Thompson and Taylor point out that the ‘union’ pun is somewhat defunct since the pearl is only present in the longer version of the play. However, the ‘double-killing’ of Claudius remains intact, a constant feature of Hamlet’s revenge, if revenge is the right word for it.
 Claudius is fatally poisoned twice by Hamlet, but whose deaths precisely are being requited? Hamlet’s final word to Claudius, along with the punning on union, seems to suggest that the poisoned chalice is commended to his own lips for having poisoned his wife, albeit unintentionally. Another mirror image is created, as the poisoner becomes the poisoned. Does this mean that the stabbing of Claudius with a poisoned rapier equates with Hamlet’s revenge for his father? If we look at Laertes’s speech which prompted Hamlet’s actions, we see that having first absolved Hamlet of murder (‘I am justly killed with mine own treachery’ (5.2.292)), Laertes then tells Hamlet that his own death is imminent:
Hamlet, thou art slain.
No medicine in the world can do thee good:
In thee there is not half an hour’s life;
The treacherous instrument is in thy hand.
Hamlet is dead, but yet still living (Sale 2007); which means that his revenge on his uncle is not for Old Hamlet, but for his father’s namesake, Hamlet himself. Hamlet has been stabbed on Claudius’ instructions, and for that he stabs Claudius: an eye for an eye.
 In the aftermath of the King’s death, it is important to note the reasons Hamlet gives for killing Claudius. Where we might conventionally be reminded of the various crimes and motivations that have led up to this moment, Hamlet’s silence on the matter is deafening. Hamlet delivers twenty-eight lines between poisoning Claudius and his own death, yet he neglects to mention the two regicides that form the core of the play’s action. De Grazia observes that Hamlet’s death speech ‘manages to cram in a great deal’ (2007: 203), which makes the omission all the more significant. One of the ways in which Hamlet’s killing of Claudius is remarkable is how little reference is made to the actions and motivations that started the revenge tragedy. In the Folio text of Hamlet, the word ‘father’ appears over seventy times in total. Except for mentioning his ‘father’s signet in my purse’ when recounting his sea-voyage (5.2.49), Hamlet does not once use the word ‘father’ before, during, or after the duel. This in a play ‘whose common theme/ Is death of fathers’ (1.2.103).
 What is lacking in all three early printed versions of the text is any indication that Hamlet’s killing of Claudius can be directly related to the ghost’s command to be revenged. As a literary critic, it is dangerous to play the numbers game with Shakespeare. Yet Hamlet is a play that is overtly conscious of numbers and their signification, or lack thereof. Earlier in the play, before Hamlet is faced with split-second decisions about the death of Claudius, he observes the army of Fortinbras crossing through Danish territory. This prompts his ‘How all occasions do inform against me’ soliloquy (4.4.31), as he meditates on the value of the soldiers’ lives in relation to the ‘little patch of ground/ That hath no profit but the name’ (4.4.17). He had been told by the Norwegian captain that he personally has no interest in the land:
To pay five ducats — five — I would not farm it
Nor will it yield to Norway or the Pole
A ranker rate should it be sold in fee.
The strange repetition of ‘five’, along with the unmistakeable economic language of ‘yield’, ‘rate’ and ‘fee’, would appear to conform to Woodbridge’s theory that economic instability underwrites much of the drama of the period (2010). But it should also be noted that in Hamlet’s private response to this state of affairs, he wonders at a situation ‘[w]hereon the numbers cannot try the cause’ (4.4.62). Obscure in its phrasing, Hamlet’s line suggests that there is a limitation to the uses to which numbers can be put. When it comes to valuing a human life, it is hard indeed to try the cause using only numbers.
‘We cannot justly be revenged too much’ (The Revenger’s Tragedy, 5.2.9)
 A final example of the intersection of revenge and numeracy comes from Thomas Middleton’s The Revenger’s Tragedy. This is revenge tragedy at its most refined, as attested to by the protagonist’s full and enthusiastic identification with the role laid out for him: ‘’Tis I, ’tis Vindice, ’tis I’ (3.5.167). At once generic self-identification and intertextual echo of Hamlet, the line is indicative of a play that is well aware of the genre in which it is operating. This awareness extends to the multiplication of revenge plots before our very eyes: Vindice has a double motivation provided by father and fiancé (and is seconded by his brother Hippolito), the Duke’s four sons plot each others’ demise, while a noble Antonio apparently recruits ‘five hundred gentlemen in the action’ (5.2.29) to be revenged for the rape of his virtuous wife. In the play’s opening scene, the protagonist even suggests that the seven cardinal sins are too few for this drama:
the uprightest man – if such there be,
That sin but seven times a day – broke custom,
And made up eight with looking after her.
Taken together, such details suggest that Middleton’s play is far more generically aware than anything we have seen hitherto on the early modern stage; and Middleton shows his interest in using numbers to prove it.
 The play culminates in an elaborate double masque in the final act where two sets of revengers dance their way to death. Rather than focusing on the intricate symmetry of the final scene, where four revengers replace four revengers who have killed four victims and are killed in turn — (4+4)-(4+4) = 0 — I want to examine a single death, which contains ‘[m]illions of deaths’ (3.5.188) at the play’s centre. Vindice’s revenge on the Duke is both coup de théâtre and commentary, as Middleton/Vindice seek to present a revenge that cannot be equalled. For Vindice to outdo not only the many competing revengers within his own play, but also his generic predecessors, his revenge on the Duke must be both dramatically unique and excessively violent, while still remaining quantifiable. For if revenge is beyond measure, then how are we to know whose is the most excessive?
 Vindice’s innovation is to use the skull of his beloved fiancé, poisoned by the Duke, as the murder weapon. Smearing the lips with poison, he invites the Duke to meet him in an ‘unsunnéd lodge/ Wherein ’tis night at noon’ (3.5.18), where he promises to introduce the lecher to a country lady with ‘a grave look’ (3.5.139). Vindice has a sharp eye for reciprocity, and he is keen that his love ‘shall be revenged/ In the like strain, and kiss his lips to death’ (3.5.104). Hippolito’s response, ‘Brother, I do applaud thy constant vengeance,/ The quaintness of thy malice above thought’ (3.5.108), moves us towards the ‘witty violence’ and aestheticisation of revenge that is such a staple of the play’s critical reception (Brucher 1981: 270; Dollimore 1989: 149; Hirschfeld 2010: 208). In case the quaintness of his malice is lost on the Duke, Vindice spells it out for him in his dying moments:
|Vindice:||’tis the skull|
|Of Gloriana, whom thou poisonedst last.|
|Duke:||O, ’t’as poisoned me.|
Vindice’s crude moralising continues: ‘Then those that did eat are eaten’ (3.5.162). Direct reciprocity is most dominant here, a ‘witty literalisation of eye-for-an-eye justice’ (Hirschfeld 2010: 205). But balance shortly gives way to excess, as Vindice compounds the Duke’s physical pain with mental agony:
|Vindice:||Puh, ‘tis but early yet. Now I’ll begin.|
|To stick thy soul with ulcers; I will make
Thy spirit grievous sore: it shall not rest,
But like some pestilent man toss in thy breast.
Mark me duke:
Thou’rt a renowned, high, and mighty cuckold.
|Vindice:||Thy bastard, thy bastard rides a-hunting in thy brow.|
|Duke:||Millions of deaths.|
Where other revengers voiced the hope that their revenge would be numerically adequate, here the victim himself freely admits it from his own lips. Leaving no room for speculation, the success of Vindice’s revenge is presented as a statement of fact. The lack of specificity (how many millions?) only underlies how total this revenge has been.
 Middleton’s use of 1,000,000 is both ridiculous and wry — how far we have travelled from the overreacher Marlowe, whose Helen could launch only a thousand ships. Considering the multiplication of revenge plots invented by Middleton in this play, revenge as a term here has become practically meaningless. Can this make sense of the play’s playful self-reflexive title, ‘The Revengers Tragedie’, lacking (as was common) a possessive apostrophe? Just as the movement of a decimal point changes the value of a given number, the early modern printed text’s lack of apostrophe is capable of multiplying ownership and responsibility: one revenger’s tragedy becomes many revengers’ tragedies. The character of Vindice may be deadly serious in inflicting millions of deaths on his opponent, but Middleton appears to be less certain how total one revenge play can really be, positing an inverse relationship between mathematical rhetoric and generic triumph.
‘His epitaph, thus: Ne Plus Ultra’ (Antonio’s Revenge, 2.3.132)
 Before the final act of Hamlet, on hearing the news of Laertes’ return, Claudius turns to his queen and says:
O my dear Gertrude, this,
Like to a murdering piece in many places
Gives me superfluous death.
When Claudius talks of superfluous death, what precisely is he trying to say? In Thompson and Taylor’s Arden 3 edition of Hamlet, they gloss the curious phrase as ‘i.e. kills me many times over’. While this is a perfectly accurate paraphrase, it does not address the paradoxical nature of what Claudius is suggesting; to die more than once is a contradiction in terms. Yet the paradox of suffering multiple deaths is not unique to Hamlet’s antagonist, and indeed recurs with surprising frequency in the genre of revenge tragedy. Variations on the theme of ‘superfluous death’ emerge across a broad spectrum of authors in relation to the staging of revenge, most hyperbolically in the Duke’s million deaths in The Revenger’s Tragedy. In the end, Claudius too succumbs to a ‘double’ death at the hands of the titular hero. The king’s proleptic fear alerts us to the precisely calculated fatalities that characterise the genre of revenge tragedy, while simultaneously suggesting a superfluity to so many of the revengers’ equations of death.
 This necessarily brief survey of revenge by numbers has demonstrated that revenge as a concept may aim towards excess, which in its early modern literary manifestation was rigourously quantified. Alongside the multiplication of deaths necessitated by revenge is the desire to intensify those deaths, whether by a factor of a hundred, a thousand or a million. The reasons for this are themselves double: on the one hand, authors’ conscious intertextuality lead them to numbers as a way of keeping score between those writing in the genre, while on the other, the insufficiency of many of the numbers in the examples given undercuts the very notion of quantifiability that characters are so intent to prove. Early modern revenge tragedies problematise any easy equation of life for life, questioning the very notion of parity between one death and another. This leads to a situation in which the quantification of revenge requires not an economic model of supply and demand, but rather an algebraic one. If X is the death of a loved one, what then is the value of Y, the death of one’s enemy, in the equation X x Y = Revenge?
University of Fribourg
 I would like to thank Laurie Maguire for directing me towards Puttenham for this definition.[back to text]
In Q1, the number chosen to represent Hamlet’s love is twenty (16.153). Q2’s ‘forty thousand’ is two thousand times greater, as even the multiple playtexts seem to compete.[back to text]
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