While I was sitting in the Ivor Novello Theatre this week watching an excellent new production of The Tempest, a 63-year-old man with a serious heart condition and amputated fingers on his left hand, was preparing to scale the North Face of the Eiger.

We had reached the scene where Ariel talks of collecting magic dew from the “still-vexed Bermoothes”, Shakespeare’s hypothetical island that may be an early reference to Bermuda, and I found myself thinking of the explorer Sir Ranulph Fiennes, about to embark on his latest, and possibly maddest, adventure.

The Tempest was written at the dawn of the age of exploration. The story — of shipwreck, survival and so much more — may have been partly inspired by the wreck of a ship off the Bermuda coast, the Sea Adventure, part of the fleet sent out to Virginia in 1609. A tract entitled A Discovery of the Bermudas, otherwise called the Isle of Devils, was published by one Silvester Jourdain, a survivor from the Sea Adventure, at around the time that Shakespeare wrote The Tempest.

“Bermoothes”, like the island on which the play takes place, is an island of the imagination. It comes from an age when so much of the brave new world remained unknown and magical. A new exhibition opening next week at the British Museum captures that sense of wonderment in discovery. John White, gentleman and artist, joined Raleigh’s expeditions of the 1580s, and his exquisite drawings, on display for the first time in 40 years, represent virtually the first recorded glimpse of a new world through English eyes.

By pleasing coincidence, the National Portrait Gallery is mounting the same exhibition in reverse, to tell the stories of the earliest visitors to these shores from Africa, the Americas, the South Pacific and India. The amazement was mutual.

Today, of course, there is barely a patch of the world that remains unknown: we can swoop down into the planet’s remotest corners through Google Earth, and pinpoint our precise whereabouts though GPS and computer. The world of 1611 was impossibly vast and magical; today it often seems shrunken and knowable.

But it is people like Sir Ranulph Twisleton-Wykeham-Fiennes who keep the sense of wonder and achievement alive by heading off, on our behalf, to perform ever more arcane quests in ever more inhospitable places.

Explorers may voyage for profit, or publicity, or “because it is there”; but at a time when “it” is increasingly elusive and overvisited, the few who still battle on through snow, sand or surf perform a vital human function — the need to believe that there are still aspects of the physical world that we cannot know for sure, and which may defeat us.

Sir Ranulph’s search for the next quest has taken him across the Jostedalsbreen glacier in Norway, to both poles, up Everest and to the lost city of Ubar in Oman, but the places he sets out to discover and vanquish these days are as much personal as geographical. He is scared of heights, so the summit to be scaled on the North Face of the Eiger, raising money for Marie Curie Cancer Care, is partly his own fear. He has difficulty holding a normal ice pick in his left hand, having removed the upper part of his fingers himself in self-surgery after getting frostbite during his solo walk to the North Pole in 2000.

Modern exploration requires lateral thinking. As Sir Ranulph observes of the poles, there are only two and they have been reached in so many ways in the past 30 years that “you have to go by camel or motorbike or something to be first”. He is the heir to a long, strange British tradition of exploration, of brave, vain, articulate and often quite bonkers people staggering into the wilder parts in search of danger. The discoverers went in the name of empire, or science, or for God’s sake, or self-aggrandisement, but there was a spiritual and cultural dimension to the questing, a belief that being human required finding something new, or climbing to the top of something big, in order, as Eliot put it, to “arrive where we started, and know the place for the first time”.

Perhaps there is postimperial vanity in our enduring collective fascination with exploration: since we no longer rule the waves, we can at least row single-handedly across them in search of new records. But there is also an affirmation of a particular sort of Britishness. All nations celebrate their explorers, but here we celebrate failure as much as success (Scott), and eccentricity over efficiency (think of Livingston, riding his ox through the African wilderness wearing his bus conductor’s cap).

My favourite explorer is one who got it utterly wrong, and failed entirely: John Evans, a Welshman, who voyaged alone up the Mississippi in 1792 into the wilds of Dakota, convinced that Prince Madoc had brought the Welsh language to the Americas in the 12th century and that he would eventually find a tribe of Welsh-speaking Indians. He didn’t, and died in a bar in St Louis, after an expedition that was deeply heroic and thoroughly foolish in precisely equal measure.

It is hard to imagine any other nation allowing its heart to beat faster because a man of retirement age with a dicky heart and half the regulation number of fingers insists on climbing one of the nastiest mountains in the world. That we do may hark back to an earlier age of exploration and discovery, an endless cultural search for undiscovered islands of the mind.

In the next few days, weather permitting, an ageing explorer will begin to haul himself up the Eiger, in search of adventure and the “still-vexed Bermoothes” that Shakespeare imagined four centuries ago when the world was brave and new.

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