There are museums, and then there is the Ashmolean Museum in Oxford. This was the first of its kind, in 1683: the first purpose-built place to present a collection of interesting stuff, gathered from all over the world, to the public. Now the Ashmolean is about to reopen after a £61m expansion. So you have the entire 326-year history of a cultural phenomenon in one building. Worth a visit, I think.

It goes back further than that, though. The crafty old antiquarian Elias Ashmole, who presented the collection to the university, was a freemason, royalist, alchemist and opportunist, who married money to fund his researches. His prize acquisition was the “cabinet of curiosities” of the John Tradescants, father and son. These famous botanists had amassed their treasures — from Japan to North America — on naval expeditions. From the late 1620s, they would show these to anyone who turned up at their home in Lambeth, nicknamed “the Ark”, for an entry fee of sixpence. So the Ashmolean, which still possesses the original cabinet of curiosities among its amazing collections of early Egyptian and Minoan artefacts, Raphaels, Titians, Asian, Far Eastern and Islamic art, should really be the Tradescanteum. That makes the concept of the museum more than 50 years older. But Ashmole, who had helped finance the Ark, prised the collection out of the younger Tradescant’s widow in a lawsuit, threw in his own collection of antiquities and manuscripts, and reaped the glory. Oxford (a royalist stronghold during the civil war) trumped London: it was to be another 76 years before the British Museum opened its doors.

And today? What you see from the street is, in fact, the second main Ashmolean building, a fine neoclassical affair by Sir Charles Cockerell, opened in 1845. There is no sign of the colossal new six-storey extension, which has been shoehorned into the site behind by the architect Rick Mather, a veteran of museums and university buildings alike. The Ashmolean is nationally important and now has the building to match, complete with a cunningly concealed rooftop restaurant giving you views across the skyline of Oxford.

Some had wanted the new building to be neoclassical, but that ran the danger of looking a bit feeble next to the richness of Cockerell. So the new building is in the lyrical-modern mode and almost wholly internal. Hemmed in by other buildings, constructed on the site of some small late-Victorian extensions, it presents no public face to the outside world apart from oblique glimpses down a side alley. What Mather has done — along with reopening Cockerell’s vast original front door — is to take his cue from the dimensions of the old galleries, connect new with old on the level at several points and create an interlocking sequence of 39 new galleries, both lofty and intimate. Every- thing is connected by two full-height atriums at each end, one large, one small, which rise more than 80ft from basement floor to glass roof. Links are made via delicate steel-and-glass bridges and a cascading sequence of stairs.

Mather is an architect for detail. You don’t need to know how on earth he and his engineers have squeezed in modern museum air-handling equipment without lowering the ceilings, but the clue is in the very thick walls and in the narrow slots — little more than shadow-gaps — at top and bottom. What’s a lot more obvious is his big idea here: to set glass display cases into the depth of some of the walls so they can be seen from both sides. These, along with strategically placed windows, make you curious about what lies beyond — say, when you find yourself looking at the backs of a lot of oriental ceramics; and they help you find your way around. You are constantly getting glimpses of the rest of the building.

There’s a lot of nostalgia right now for very old-fashioned museums that display things in cases in a strictly typological and chronological way, making no concessions to populism. If you like that, you may not like the rather modish exhibition layout here, designed by the exhibition specialists Metaphor. The words “new display and interpretation strategy” strike a chill to the heart, especially when the whole display is given the title Crossing Cultures Crossing Time. Isn’t it enough just to be the Ashmolean? This is the Tate Modern way of pointing out connections by sometimes startling juxtapositions of new with old, west and east, and so on. But the themed galleries that do this are only part of the whole, and, besides, it’s well done. Nice that there’s one room full of ancient display cases, for old times’ sake.

What makes a museum tick is more than what you see. The Ashmolean, for instance, had no loading dock. When it lent out or borrowed objects — Titians included — they had to be manhandled across the forecourt to a van parked outside on a double yellow line. Not terribly secure. Now they have a theatre-sized get-in, proper conservation labs, the lot. It all means the quality and range of exhibitions here, temporary and permanent, can be that much better, as well as bigger.

Whatever your views on the slippery Ashmole, what’s undisputed is that the Tradescants invented the idea of the world brought together in one place, and he developed that idea. The Ashmolean has always been a national treasure, and at intervals it has reinvented itself. Its latest incarnation mixes intelligence with show- manship. That sounds like old Elias.

The Ashmolean Museum, Beaumont Street, Oxford, reopens to the public on Saturday; admission free; www.ashmolean.org

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