tl;dr Edward Dowden (1875).
Equating Prospero with Shakespeare, and vice-versa
The specific equation of Prospero's breaking his staff with Shakespeare's renouncing his art is part of a much larger pattern of identification of Prospero with Shakespeare. In an Oxford University podcast, Emma Smith shows that this pattern goes as far back as the late 17th century. In 1667, when John Dryden and William D'Avenant rewrote The Tempest for the Restoration stage, their preface praised “Shakespeare's magic”, thereby linking him to the magician Prospero.
This identification was well-established by 1740, when the statue of Shakespeare was installed in the Poet's Corner at Westminster Abbey. Here's the statue:
The scroll that Shakespeare is pointing to quotes part of Prospero's famous speech:
Our revels now are ended. These our actors,
As I foretold you, were all spirits, and
Are melted into air, into thin air:
And, like the baseless fabric of this vision,
The cloud-capp'd towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Yea, all which it inherit, shall dissolve,
And, like this insubstantial pageant faded,
Leave not a wrack behind. We are such stuff
As dreams are made on; and our little life
Is rounded with a sleep. (4.1.148–158)
These lines highlight the metatheatricality of The Tempest. Smith points out that throughout the play, Prospero's actions are those of a playwright: he conjures up situations and manipulates characters within them. For example, the eponymous tempest with which the play opens turns out to be only an illusion of his making. Through Prospero's magic, the other characters experience it as real; through the magic of theater, the audience shares that experience. When the next scene reveals that the storm is just a trick of Prospero's, it's easy to extend the parallel. Prospero creates the tempest; Shakespeare creates The Tempest; Prospero is Shakespeare.
During the Romantic era, Samuel Taylor Coleridge explicitly makes this parallel. He delivered a series of lectures on Shakespeare between 1811 and 1818. His lecture on The Tempest has a parenthetical remark where he calls Prospero "the very Shakspeare [sic] himself, as it were, of the tempest". Coleridge leaves it nicely ambiguous as to whether by "the tempest" he means the storm or the play. Prospero is the Shakespeare of the storm, in that he is its maker. He is the Shakespeare of the play, in that he's the playwright's representative in The Tempest.
The Tempest, the Chronology of Shakespeare's Plays, and the Biographical Approach
These early identifications of Prospero with Shakespeare did not, however, make the specific connection between Prospero's breaking his staff and Shakespeare's renouncing his art. Since The Tempest is the first play printed in the First Folio, it was often assumed to be an early play. Scholars such as Edmond Malone and Edward Capell began tackling the chronology of Shakespeare's plays in the late 1700s. Malone's essay "An Attempt to Ascertain the Order in which the Plays Attributed to Shakspeare were Written" was first published in 1788. He says that The Tempest could not have been written before 1609, as the play draws upon descriptions of the Bermudas that began to circulate after Sir George Somers's shipwreck on one of those islands in that year. He also says that the play "exhibits ... strong internal marks of having been a late production", without elaborating on what those marks are. Malone does not, however, say that this was Shakespeare's last play, seeing Twelfth Night as later.
Malone claims that The Tempest was probably written in Summer 1612 and first performed in 1613. He argues that Shakespeare must have chosen this title in order to capitalize on a particularly destructive tempest that had occurred in England late in 1612. Were it not for this historical tempest, the argument goes, Shakespeare would not have named his play The Tempest, and the play must have been performed in 1613, when the storm would still be fresh in the minds of the audience.
Coleridge himself indulged in some rather idiosyncratic speculation about the plays' order from time to time. He came up with three different orderings at three different times, without providing reasoned arguments for any of those arrangements. Initially, he attributed The Tempest to Shakespeare's late period; around the time of his lecture on The Tempest, he considered that it belonged to the middle of Shakespeare's career; eventually, he placed it among Shakespeare's later, but not final, plays. At no time, however, does he claim that The Tempest is Shakespeare's last play.
Since Malone and Coleridge do not position The Tempest at the very end of Shakespeare's career, neither of them makes the equation of Prospero's staff-breaking with the playwright's retirement. This view arose only in the Victorian period. Over the course of the 19th century, stylistic approaches to Shakespeare's language (the proportion of poetry to prose, the prevalence of feminine endings, etc.) began to be used as a way to order the plays chronologically. This approach was what positioned The Tempest as one of Shakespeare's last plays. A footnote to Edward Dowden's first book, Shakspere: His Mind and Art (1875), describes the development of this approach:
Mr. Spedding, in his article "Who Wrote Henry VIII?" (Gentleman's Magazine, August, 1850), first applied quantitative criticism of verse peculiarities to the study of Shakspere's writings. Mr. Charles Bathurst, in "Remarks on the Differences of Shakspere's Versification in Different Periods of his Life" (London, 1867), called attention to the change "from broken to interrupted verse" which took place as Shakspere advanced in his dramatic career; and observed, also, the increase in the use of double-endings in his later plays. Professor Craik,in his "English of Shakspere," and Professor J. K. Ingram, in a lecture upon Shakspere published in "Afternoon Lectures" (Bell and Daldy, 1863), again called attention to these peculiarities of versification as affording evidence for the ascertainment of the chronology of the plays. Finally, about the same time in England and in Germany, two investigators—Rev. F. G. Fleay and Professor Hertzberg—began to apply "quantitative criticism" of the characteristics of verse to the determination of the dates of plays. The test on which Hertzberg chiefly relies is the feminine (double) ending ; he gives the percentage of such endings in seventeen plays, and believes that the percentage indicates their chronological order. See the preface to Cymbeline in the German Shakespeare Society's edition of Tieck and Schlegel's translation. Mr. Fleay's results, independently ascertained, were published subsequently to Hertzberg's. See Trans. New Sh. Soc, and Macmillan's Magazine Sept., 1874. In 1873 Mr. Furnivall, in founding the New Shakspere Society—before he was aware that Mr. Fleay's work was in progress—insisted on the importance of metrical tests for determining the chronology, and gave the proportion of stopped to unstopped lines in three early and three late plays. The latest contribution to the subject is Professor Ingram's valuable paper read before the New Sh. Soc, on the "Weak-ending" Test.
By the Victorian period, these methods had led to a widely accepted chronology of Shakespeare's plays that placed The Tempest firmly as one of the last. It was recognized that the chronology was not exact, but approximate. Nevertheless, the approximate order was well-suited to biographical approaches that sought to show how Shakespeare's art and personality matured hand-in-hand. Dowden writes:
In these chapters we have been chiefly concerned with observing the growth of Shakspere's mind and art. The essential prerequisite of such a study was a scheme of the chronological succession of Shakspere's plays which could be accepted as trustworthy in the main. But for such a study it is fortunately not necessary that we should in every case determine how play followed play. It would for many reasons be important and interesting to ascertain the date at which each work of Shakspere came into existence; but as a fact this has not been accomplished, and we may safely say that it never will be accomplished. To understand in all essentials the history of Shakspere's character and Shakspere's art, we have obtained what is absolutely necessary when we have made out the succession, not of Shakspere's plays, but of Shakspere's chief visions of truth, his most intense moments of inspiration, his greater discoveries about human life.
Retiring from the Stage, having Conquered All
Given that The Tempest was (as it happens, accurately) placed as one of Shakespeare's last plays, given the long-standing identification of Shakespeare and Prospero, and given Dowden's frankly biographical approach to criticism, the temptation to see this as the culmination of the bard's career was perhaps too hard for Dowden to resist. He uses various rhetorical and argumentative strategies to position The Tempest as the last of Shakespeare's plays. In another footnote, he returns to the previously mentioned "latest contribution to the subject" of Shakespeare's chronology, "Professor Ingram's valuable paper":
Professor Ingram, in his paper "On the 'Weak-endings' of Shakspere," arranges the plays of the weak-ending period in the following order: Antony and Cleopatra, Coriolanus, Pericles, Tempest, Cymbeline, Winter's Tale, Two Noble Kinsmen, Henry VIII. From an aesthetic point of view, Antony and Cleopatra and Coriolanus seem to me connected with the plays that immediately precede, not with those that follow them. Professor Ingram is disposed to place Macbeth immediately before Antony and Cleopatra. I had independently arrived at the same opinion. ... Observe that Pericles, Two Noble Kinsmen, and Henry VIII are Shaksperian fragments. Thus The Tempest, Winter's Tale, and Cymbeline remain as the three complete plays which represent the final period of Shakspere's authorship.
In the first step toward positioning The Tempest as Shakespeare's last play, Dowden accepts Ingram's ordering, but brackets Antony and Cleopatra and Coriolanus with immediately preceding plays such as Macbeth. He then dismisses Pericles, Two Noble Kinsmen, and Henry VIII, collaborative plays, as mere "fragments" not worth much consideration. That leaves The Tempest, Winter's Tale, and Cymbeline as Shakespeare's final plays.
Next, in the main body of his text, he claims that of these three plays, it is The Tempest which epitomizes "the pathetic yet august serenity of Shakspere's final period". In an argument of breathtaking circularity, he says that The Tempest must be considered Shakespeare's last play because it best provides the sense of a resolution:
For the purposes of such a study as this we may look upon The Tempest as Shakspere's latest play. Perhaps it actually was such; perhaps A Winter's Tale or Cymbeline or both, may have followed it in point of time. It does not matter greatly, for the purposes of the present study, which preceded and which succeeded. These three plays, as we shall see, form a little group by themselves, but it is The Tempest which gives its most perfect expression to the spirit that breathes through these three plays which bring to an end the dramatic career of Shakspere; and therefore for us it is Shakspere's latest play.
Dowden's use of the royal we is rhetorically effective. He's making a bold and quite unsupported argument; there's no objective reason he can give that would establish The Tempest as later than the other two plays. But the reader is made complicit in his strategy via that "we". In the space of a few sentences, the reader progresses hand-in-hand with Dowden from merely "look[ing] upon The Tempest as Shakspere's latest play" to the flat assertion that "for us it is Shakspere's latest play." Now "we" can milk the identification of Prospero and Shakespeare for all it's worth. And voilà, Prospero's breaking his staff becomes Shakespeare's preparation for retirement:
It is not chiefly because Prospero is a great enchanter, now about to break his magic staff, to drown his book deeper than ever plummet sounded, to dismiss his airy spirits, and to return to the practical service of his Dukedom, that we identify Prospero in some measure with Shakspere himself. It is rather because the temper of Prospero, the grave harmony of his character, his self-mastery, his calm validity of will, his sensitiveness to wrong, his unfaltering justice, and, with these, a certain abandonment, a remoteness from the common joys and sorrows of the world, are characteristic of Shakspere as discovered to us in all his latest plays. (p. 371)
Through these rhetorical and argumentative strategies, Dowden makes The Tempest Shakespeare's farewell to the stage. They are so effective that few pages on, Dowden can state without qualification that while writing The Tempest, Shakespeare "was passing from his service as artist to his service as English country gentleman" (p. 376). The possibility that he might have written four plays, independently or collaboratively, subsequent to this one no longer matters.
In justice to Dowden, it's worth mentioning that despite the dubiety of his argumentative strategy, modern approaches to Shakespeare's chronology have proven him correct. The Tempest is in fact the last independent play that Shakespeare is known to have written. (The long history of seeing this as a late play, however, goes back to Malone, and now I'm wondering whether this is just anchoring bias. Okay, I'm kidding.)
So Dowden did get the fact of the relative chronology of The Tempest right. His interpretation of the play is another matter. Some 140 years after Dowden, critics are no longer so sure about Prospero's "sensitiveness to wrong, his unfaltering justice". Postcolonial approaches to The Tempest cast Prospero as colonizer, exercising imperial control over the original inhabitants of the island: Caliban and Ariel. In the Oxford podcast, Smith also draws attention to his pompousness and his obsession with preserving Miranda's virginity until she is married. But Prospero's rule over the natives and his hangups about sexual purity would all be transparent to Dowden, writing as he was during the Victorian era, renowned for both its empire and its prudishness.
Dowden was a professor in the newly established academic field of English Literature. Gauri Viswanathan has pointed out that the academic study of English literature first arose in the context of British rule in India as part of the imperial mission of “educating and civilizing colonial subjects in the literature and thought of England”. This civilizing mission, this white man’s burden, is
serviceable to the dynamic of power relations between the educator and those who are to be educated. A vital if subtle connection exists between a discourse in which those who are to be educated are represented as morally and intellectually deficient and the attribution of moral and intellectual values to the literary works they are assigned to read.
And in this context the greatest, the most moral, of English literary works are, of course, Shakespeare’s plays. Against this imperial background, it is no surprise that Dowden should see Prospero/Shakespeare as someone who "has reached not only the higher levels of moral attainment; he has also reached an altitude of thought from which he can survey the whole of human life, and see how small and yet how great it is" (p. 372). Well, then! When one has reached such rarefied heights, what's left to do but break one's staff and drown one's books? We've already encompassed "the whole of human life", and the rest is silence.