A brief history of work on the 4 $\times $ 4 magic square is presented, with particular reference to Frenicle's achievement over 300 years ago of establishing 880 as the number of essentially different squares by using the method of exhaustion (not convincingly repeated except by computer in 1976). He also established several central theorems. Our paper confirms the number 880 by a wholly new method of `Frenicle quads' and `part sums', which leads to the classification of all solutions into, initially, six `genera' one of which has no members and thence to the enumeration of all possible solutions by analytical methods only. The working leads also to the first analytical proof independent of solutions that 12 and only 12 patterns formed by linking `complementary' numbers within a square are necessary and sufficient to describe all solutions -- a fact which has been known since 1908, but not hitherto proved. A second method of construction and partial proof, greatly shortened by what has gone before, is also described. This yields a highly symmetrical list of the 880 magic squares. Together the two methods combine to explain many of the special characteristics and otherwise mysterious properties of these fascinating squares. The complete symmetrical list of squares ends the paper.