The snake was designed to narrow the margins of fluctuation between European currencies and to make it easier for businessmen to plan ahead.
But the markets were growing in power in the wake of a decision by the biggest industrial countries in 1971 to abandon fixed exchange rates. Mr Heath bowed to the pressure from currency traders and withdrew. By 1978 only five members were left in the snake, as market forces ruled.
The difficulties of the first attempt at European currency co-ordination led to a search for a better-constructed mechanism, with more effective and durable methods for resisting the pressures of the market. The framework for the European Monetary System was agreed at the end of 1978 and it came into operation the following March.
The system has complex legal agreements and operating rules which set the terms under which central banks help each other support their currencies when they came under attack. And underlying the technical details is an agreement that major decisions are taken jointly.
Britain has been a member of the system since it began. But the Labour government was unwilling to risk a repeat of the snake fiasco at a time when the pound was at the mercy of fluctuating oil prices. The pound remained outside the crucial operational part of the system, the exchange rate mechanism, and continued to float free of any link to the mark and the others.
The EMS proved in practice to be a relatively loose arrangement which used the mark as anchor currency, on the assumption - not explicit but certainly borne out in practice - that it would never be devalued against other members of the system.
Hopes that the ERM would promote economic and monetary policy co-ordination were soon dashed. The economies of Europe, and particularly national inflation rates, were too diverse to avert regular realignments: there were three in 1979, two each year in 1981 and 1982, one each year in 1983 and 1985, two in 1986 and one in 1987.
There was a technical devaluation of the lira in January 1990, when Italy switched from wide bands of fluctuation against other currencies to the tougher discipline of narrow bands. But there was no proper realignment of the system for over five years until the Italian devaluation last weekend.
By the time Britain joined in October 1990, the ERM was undergoing a subtle change, not of operational method but of aspirations.
A year earlier, the Delors report on economic and monetary union proposed a three-stage move towards a single currency, of which the first was closer co-operation inside the ERM. The mechanism was also to be widened as soon as possible to include all the members of the Community.
Further realignments were discouraged as negotiations on monetary union progressed. The architects of union, especially the French, began to see the ERM as a practice run, and the parities agreed in 1987 appeared to be set in stone.
The markets also began to believe the ERM was a halfway stage to union. When the Maastricht agreement set out a detailed timetable, they became much more confident about the prospects of volatile currencies such as the lira and the pound because of their firm links to the mark.
But with Maastricht under threat, investors are once again treating the system as the loose arrangement it was when it began in 1979. The markets which beat Ted Heath have done it again.