Earth has been habitable through most of its history, but the anthropogenically mediated greenhouse effect, if sufficiently strong, can threaten Earth's long-standing equability. This paper's main aim is to determine the strength of the anthropogenic greenhouse effect (the climate sensitivity) from observational data and basic physics alone, without recourse to the parameterisations of earth-system models and their inevitable uncertainties. A key finding is that the sensitivity can be constrained by harmonising historical records of land and ocean temperatures with observations of potential climate-change drivers in a non-steady state, energy-balance equation via a least-squares optimisation. The global temperature increase, for a CO2 doubling, is found to lie (95 % confidence limits) between 3.0oC and 6.3oC, with a best estimate of +4oC. Under a business-as-usual scenario, which assumes that there will be no significant change in people's attitudes and priorities, Earth's surface temperature is forecast to rise by 7.9oC over the land, and by 3.6oC over the oceans, by the year 2100. Global temperature rise has slowed in the last decade, leading some to question climate predictions of substantial 21st-Century warming. A formal runs test, however, shows that the recent slowdown is part of the normal behaviour of the climate system.