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insects for receiving emanations, no doubt, of their own sixth sense. This wire—and here my invention is unique—I have wound into a flat coil mounted on a frame that can be rotated, so that it intercepts the maximum amount of aeroelectricity when facing toward it and none at all when it is turned on edge. In this way the bearing of a meteorological disturbance from the operator can be precisely determined. Turning this contrivance back and forth, I establish that the direction of the loudest cracklings is east-northeast. Out with the chart again. There is undoubtedly a region of rarefied air in the direction of the cracklings. The wind turns dumbly about it in accordance with the well-known laws of rotating bodies, and the Prinzess, like a bemused lover, just as dumbly follows the winds. I dismantle the apparatus, put it away in the case, and strap the lid shut.
    Neither of my companions says anything, but clearly they are waiting for my prophecy from the invisible world.
    â€œThere’s a mass of fluxuous air—here.”
    I draw in a rough oblong on the chart, to the east and somewhat to the north of us.
    â€œThis is bending the wind in a slightly obliquedirection. Consequently our course is taking us a little to the east of north.”
    â€œAhah. H’mm.”
    He looks at the chart, a little troubled.
    â€œNever get to the Pole that way.”
    Here Waldemer is right, but in a far more complicated way than he thinks. In a pedagogical spirit only slightly tinged with malice, I decide to offer him a little lecture in mathematics.
    â€œOn the contrary, an airship moving north-northeast will in the end—
must
inevitably—reach the Pole.”
    I look about for a scrap of paper and, not finding any, sketch rapidly on the edge of the chart. A rough polar projection: circles for the parallels of latitude, radials for the meridians.

    â€œIf the airship continues to move north-northeast it will, by definition, cross all meridians at the same angle—the angle θ that I’ve indicated here—amounting to twenty-two and a half degrees.” With a winning but treacherous little smile I lead him into this logic. “Thus, viewed from directly above as I’ve drawn it, the course will describe a spiral around the Pole, coming closer and closer to it according to the formula for asymptotes as presented in the theorem of Geminus. After a time it will be only a mile from it. Then a foot. Then an inch. Then a millionth of an inch.”
    Waldemer knows there is something wrong here but he is not sure what. He is not attuned to these abstractions—asymptotes, theorems of Geminus, spirals that approach infinity.
    â€œCome off it, Major. You and your paradoxes of Zeno. I’ll settle for a millionth of an inch.”
    Theodor at thispoint gravely intervenes to demonstrate his own competence in mathematics, not so much at Waldemer’s expense as at my own.
    â€œExcuse me, Gustav. There is a slight error in your assumptions. The problem is not really geometric but trigonometric.”
    He takes the pencil himself and begins sketching on another part of the chart.

    â€œLet
r
indicate our distance from the Pole, V our forward velocity, and θ our course, or angle to the meridian. Now we may break down our velocity into two components.

    â€œThe linear velocity at which we approach the Pole is constant and we must inevitably reach it in time. But what you have neglected is that the
angular
motion—the rate at which we are whirling about the spiral—becomes infinite as the airship approaches the Pole, so that the centrifugal forces involved will either destroy the airship or the laws of physics, one or the other.”
    Congratulations, Theodor. You are correct and have beaten your teacher in an open demonstration, even though the audience may not appreciate it.
    â€œH’mm.”
    Waldemer regards all these hieroglyphics and insect tracks with increasing