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32 partly also eastward, away from the longer axis. Already they have had to unmake their eastward progress and come back to the longer axis. They have still, however, been able to congratulate themselves on a considerable amount of progress southward; and the optimists among them no doubt set up a theory that, after all, true progress is southward, and the eastward motion was only an accidental concomitant of the southward. But when the extremity of the long axis is reached, a terrible conflict sets in; the upward path is beginning to turn towards the north-west. The southward progress, therefore, is being lost! A most dramatic novel could be woven of the conflict of opinion and feeling that would arise, as soon as the nature of the situation came to be realized; for all the tragedy of History is summed up in the prosaic fact that "the actual progress is not statable; the recorded progress will all have to be unmade."

Whether the spiral wire was actually used in ancient times to teach the true principle of development, will perhaps never be known. In this connection it may be well to notice that the Spiral of Ascent, the winding pathway, would naturally be represented to the early Geometrician (who lived where snakes were common and corkscrew wires not yet invented) by a snake coiled round a branch and hanging freely. The Geometry of the snake-coil is a more advanced stage of the Logic of the Divining-rod. In the fable of the rods that were shown to Pharaoh, the rods of both parties turned into snakes before that of the good prophet could swallow those of the bad magicians.

The Pentagram and the Magic Squares and other