But they were induced to take the gamble by the implied promise that the strong economies would help them out if they got into trouble. They survived for as long as times were good; and they tried to keep pace by taking on extra debt. But when the crunch came, the strong economies reneged on their promise.
The result? A debt crisis that has engulfed Greece and Portugal, to a lesser extent Ireland and Spain, and now - potentially - Italy.
But instead of recognising their mistakes, Europe's leaders have continued to ignore economic realities. They have treated debtor countries - not as victims of a failed political doctrine - but as moral lepers who must don hair shirts and pay for their sins.
Even more sadly, they have insisted that their political goals should take priority and be reinforced. Far from acknowledging that the single currency did not wash away economic differences across Europe, but exacerbated them by trying to suppress them, the remedy they now contemplate is an even more determined bid to create a single European state by moving further towards economic union.
They obstinately refuse to recognise that economic weakness cannot be wished away. Even if it is buried in a political framework so that it is out of sight, it will simply manifest itself in another way.
Even if Greece or Portugal or even Italy became simply provinces in a wider Europe governed from Berlin, their lack of competitiveness could not be hidden; it would just mean they became depressed regions, with no prospect of seeking their own salvation.
They would have no chance of pursuing a monetary or exchange-rate policy more suited to their needs. They would have to rely on decisions made in Berlin or Paris; decisions unlikely to be helpful. German and French taxpayers, whose patience has already worn thin, would have little tolerance of regions whose weakness was seen as threatening living standards across the continent.
But the people of Greece and other debtor countries not only have to put up with the loss of self-government and lectures about their profligacy. Remarkably, the measures that Europe's leaders now insist upon will make matters worse, not better - not just for the debtor countries, but for all of Europe and the rest of us as well.
As the Irishman said when asked for directions, "I wouldn't start from here." But what Europe's leaders should do is accept that they have to start from here, and to prescribe policies that offer a chance of turning things around.
What they should do is cut Greece loose from the euro so the Greeks can devalue sharply and then - on the basis of improved competitiveness - trade their way to generating the wealth needed to pay back their debts.
Yet, Europe's leaders insist that Greece - already going backwards at a rapid rate, with national output dropping like a stone - must cut a further 100,000 public-sector jobs, slash salaries and pensions, cut health spending, raise taxes and sell off assets. It is impossible to see how - by slashing and burning - an economy that is already overburdened by debt, and with no capacity to service that debt, let alone repay it, can hope to work and trade its way out of its problems.
And that problem will be compounded as other debtor countries are given the same advice. The whole European economy faces a bleak future if that advice is taken.
If the European economy tips back into recession, and if European bank failures were to provoke a renewed financial crisis, we would all suffer. Global liquidity would dry up, interest rates would rise sharply, markets would contract.
But there would be longer-term implications as well. We might, for example, pause to consider, in the transtasman context, whether a currency union as advocated by some would really be the panacea it is said to be.
And, in case we feel a sense of superior wisdom contemplating European difficulties, let us not delude ourselves. If we were to find ourselves in Europe, our Government would support the same failed nostrums as are insisted upon by Europe's leaders and for the same ideological reasons.
