The great Argentine writer Jorge Luis Borges, who translated William Faulkner, André Gide, Franz Kafka and Virginia Woolf into Spanish, drew the line at Shakespeare. Speaking of the moment when Hamlet asks the ghost why it returns to haunt “the glimpses of the moon”, Borges commented: “I don’t think it can be translated. Perhaps the words can be translated. Certainly Shakespeare cannot be translated. ‘The glimpses of the moon’ means exactly ‘the glimpses of the moon’.”

All, however, is not lost. “It has been said that Shakespeare cannot be translated into any other language,” Borges added. “But Shakespeare cannot be translated into English, either, since he wrote what [Robert Louis] Stevenson called ‘that amazing dialect, the Shakespeare-ese’.” This might not be entirely true, as the translator Daniel Hahn points out in this superbly diverting book. Recalling a hip-hop production of Romeo and Juliet he once saw, he persuades us instantly that “the phrase ‘Do you kiss your teeth at me, fam?’ proved to be a perfect translation of ‘Do you bite your thumb at us, sir?’”

And if into English, then why not into Portuguese, or French, or Māori? Hahn’s project is to argue that “Shakespeare with every word changed can still be great, and can remain Shakespeare”, and to that end he reproduces chunks of Dutch, Russian, Welsh, Thai, Arabic, Japanese, and a dozen other languages, betting that by simply counting syllables or observing alliteration in a language one doesn’t understand (as he cheerfully admits, he doesn’t understand Danish), one can learn something about the quality of a translation. I wasn’t convinced that wager worked much of the time, but the typesetters, as you can imagine, were certainly getting a decent workout, and the gambit does finally pay off when a long passage from Twelfth Night is annotated by boxes mentioning dozens of different translators’ choices.

What really illuminates the book are Hahn’s conversations with his fellow translators, who can explain their choices directly. In Māori, we learn, Lady Macbeth’s question to her husband, “Are you a man?”, makes no sense at all, so the translator Te Haumihiata Mason renders it as something roughly meaning “Have you got balls?” – “which is,” Hahn notes contentedly, “exactly what Lady M is asking.” Meanwhile, Prince Hal’s name means “fish” in Hungarian, which would be unhelpfully distracting, so it gets changed to Riki, short for Henrik.

Hahn also offers many asides about the annoyances and pleasures of translation in general. “The word ‘literal’ is annoyingly overused to suggest a sort of ‘neutral’ translation, which cannot exist,” he complains; and he shows that, in many cases, a non-literal choice would be better. When Mark Antony imagines Caesar’s spirit to “cry ‘Havoc’”, for example, the closest Portuguese word is the rather weak-sounding “devastação”; a better choice, Hahn shows, is “matança” (killing), because it’s shorter and more easily shoutable.

Each chapter addresses a different question translators face, for example whether to translate into verse (careful: as one French translator observes, you risk making “a genius into a talented versifier”), or how to translate jokes: it’s usually best, everyone agrees, to create an entirely new joke – “being faithful to the laugh”, as Hahn calls it. In a German Midsummer Night’s Dream, to preserve the doggerel rhymes, we are promised not that Thisbe will be in “mulberry shade” but that she will be “hiding like a newt”. Translators might even embrace the possibility of a joke where none previously existed – which Hahn illustrates brightly by mentioning that the “sorting hat” in Harry Potter has become, in French, le choixpeau (the chapeau that chooses).

Can you even preserve alliteration? Sometimes, if you’re lucky: Love’s Labour’s Lost received the surely unimprovable Greek title of “Agapēs Agōnas Agonos” (“the struggles of love are barren”). But when no such fortunate tricks are available, you can simply replace one idiom with another: so, in Spanish, Much Ado About Nothing is often called “A lot of noise, not many nuts”.

There are quibbles to be made here and there. Hahn calls a line from Richard III “irregular” after counting syllables, but it’s a perfectly regular line that begins with an anapest (da-da-dum). And when Juliet says to Romeo “You kiss by th’book”, Hahn glosses this as her approvingly noting his “formal courtship”, but she is surely issuing a flirtatious challenge. And – this being the publisher’s rather than the author’s fault – the book has been produced, inexplicably, without an index.

All may be forgiven, though, for the delight and endless curiosity displayed in these pages. “In Shakespeare, people get sad with precision,” Hahn enthuses. And he is cherishably bitchy about certain literary “translators” who somehow produce new English versions of Chekhov or Ibsen without speaking the source language – the process being, as he surmises, “a sort of high-status prettying up of a so-called ‘literal’ translation”. By the end of the book, Hahn has amply demonstrated not only the treasures of other languages, but also the rich and strange inexhaustibility of Shakespeare himself.