One image immediately sprang to mind when I heard that the actor Heath Ledger had been found dead on his bed in a New York apartment, surrounded by prescription drugs. The image was not from Brokeback Mountain or his other films, but of a much earlier picture: Henry Wallis's 1856 painting, The Death of Chatterton.

In that extraordinary painting, the poet Thomas Chatterton, just 17, lies sprawled across the bed in his garret. Through the open window, dawn is breaking over St Paul's Cathedral. On the table stands the bottle of arsenic, with which he has just killed himself.

Chatterton, penniless and starving, probably committed suicide in despair, although it is possible he was self-medicating for syphilis, and overdid the dose. Ledger, 28, already wealthy and celebrated, may also have killed himself by accident. There is a direct link between them: two gifted individuals, dead long before their time, destined, like butterflies, to live gorgeously for too brief a season.

The notion of the artist doomed to early death, bequeathed by the Romantics and most memorably depicted by Wallis, remains deeply embedded in modern culture. Ledger now joins the roster of the talented young, untimely dead: Percy Bysshe Shelley and John Keats, Wilfred Owen and Rupert Brooke, James Dean and Marilyn Monroe, River Phoenix and Kurt Cobain.

Our relationship with movie stars is as intense and intimate as it once was with poets. Actors live other lives for us on screen. We live through them in other worlds, and we expect to grow old with them. When they die young, we are immediately reminded of our own impending deaths, and the need to seize the day.

The Greeks knew that “Those whom the gods love die young”. The Who sang: “Hope I die before I get old” and Billy Joel lamented that “Only the good die young”. For the Romantics, early death was practically part of the job description. Byron (dead at 36) groused at the prospect of growing old: “If thou regret'st thy youth, why live?”

Artists themselves looked with something close to envy on those who flamed early and brightly, before being snuffed out. A.E.Housman, in his poem To an Athlete Dying Young, wrote: “Smart lad, to slip betimes away/ From fields where glory does not stay”. Wordsworth, living on to the age of 80, could never be adored in the same way as Shelley, Keats and Byron, who assembled just 90 years of life between them.

Statistically speaking, artistic creativity is bad for your health, and often fatal. Film actors actually tend to live slightly longer (an average of 67 years) than composers and musicians (65), and far longer than poets, who have a life expectancy comparable with stuntmen and deep sea divers, pegging out at the average age of 62. Bad habits tend to combine, in artists, with sad habits: “Why is it that all men who are outstanding in philosophy, poetry or the arts are melancholic?” wondered Aristotle (who lived on to the then-ripe age of 62).

The outpouring of grief over the death of Heath Ledger fits squarely into a ritual of public mourning so familiar that it edges close to cliché. Yet there is also something disturbing in the sudden and extreme glamour conferred by an early death. Ledger was a brilliant actor, approaching the height of his talents, but that could not be said of, say, River Phoenix, an average and untested performer, whose early death alone conferred a sort of ersatz greatness. Some great artists die young, but no artist should be rendered great by dying young.

For a younger generation, early death has acquired a grim and growing allure, founded in our cultural veneration for the young dead. The notion that self-destruction is cool has taken firm root in teenage culture. The websites set up to memorialise the seven young people who have killed themselves in Bridgend, South Wales, in the past year echo a tragic desire for recognition in death and a sense that the act of suicide is somehow brave. The postings of admiration and respect are entirely heartbreaking, and so wrong.

At the same time, millions are avidly watching while Amy Winehouse gradually destroys herself with drugs, as if this were another artistic performance. Pete Doherty has become an icon of self-harm: advertising last night's fascinating documentary about a child genius with an incurable disease, Channel 4's trailer joked: “He's had more drugs than Pete Doherty”.

The Romantics have much to answer for here. The Death of Chatterton is a beautiful, arresting painting, but it is also horrible, a morbid celebration of pointless death and wasted genius. The way Wallis luridly depicts the dead poet, and the squalid instruments of his self-destruction, is reminiscent of the drug-haggard, once-beautiful Winehouse, staggering towards oblivion, sucking on a crack pipe.

Housman was wrong. There is nothing “smart” in slipping away too early. Dylan Thomas (dead at 39) was closer when he urged us to “rage against the dying of the light”. Death should have no dominion, no glamour, and no cachet.

Heath Ledger should be mourned for the great roles he will never play, but not celebrated for the nature, and prematurity, of his death. He left a handful of truly inspired film performances. Today, thousands mourn a young man apparently killed by drugs, alone in his bedroom, but the images that deserve to be remembered are those from his films. These, like Ledger himself, will not grow old. Like the figures adorning Keats's Grecian urn, Ledger's films will live on:

For ever warm and still to be enjoy'd

For ever panting, and for ever young.

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