Proposals that Stonehenge was con- structed as an astronomical observatory [1, 2, 3] with the purpose of predicting eclipses [4, 5] imply that the builders of Stonehenge, even Stonehenge I, were possessed of a degree of intellectual sophistication that seems inconsistent with the usual picture of the population of S. England in the and millennium B.C. Either the people were not just primitive farmers or these proposals must be substantially in error. It is not the purpose of this paper to attempt a decision between these alternatives but to explain the nature of the astronomical arguments in more detail than has been done heretofore. My hope is that with a moderate effort it will be possible for the reader to rework the calculations on which the astronomical suggestions have been based; for only when archaeologists and astronomers have understood each other's arguments can we expect to resolve this intriguing dilemma. At several places in the discussion I shall quote the solution of mathematical problems, thereby permitting the whole discussion to be confined to simple trigonometry. These few places will be marked by an asterisk. The reader wishing to cover these gaps will need to consult a text on spherical astronomy, e.g. [6].