Hence f ( z) = a 0.Ĭonversely, the slightest deviation of an analytic function from a constant value implies that there must be at least one singularity somewhere in the infinite complex plane. If now we choose to let r approach ∞, we may conclude that for all n > 0, | a n| = 0. In fact, if | f ( z)| ≤ M for all z, then Cauchy's inequality Eq. (11.37), is Liouville's theorem: If f ( z) is analytic and bounded in the entire complex plane it is a constant. Īn immediate consequence of the inequality, Eq. This is an instance of the following fairly obvious theorem, whose proof we skip. Glide reflection, since it is the composition of a reflection that switches sense and a translation that does not, must also switch sense. (Draw a picture to illustrate this.)īy contrast, translation and rotation clearly preserve the sense of any sequence of three points. Reflection switches the sense of any sequence of three points. Imagine that we apply a reflection to ABC in order to get A′ B′ C′. Will the two ordered triangles and have the same sense? Not necessarily. Suppose A′ B′ C′ is a triangle that is congruent to ABC, with the correspondence A→ A′, B→ B′, C→ C′. If we take the same set of points but put them into a sequence in another way, say,, then we may get a sequence with the opposite sense to the sense of. Notice that the idea of sense depends on the order in which we list the points, not just on the points themselves. If we travel clockwise, then we say the sequence has clockwise sense.įor example, in Figure 4.23, has clockwise sense while has counterclockwise sense. Given three points A, B, C not lying on the same line, if we travel in a counterclockwise direction as we go in order from A to B and then to C, then we say has counterclockwise sense. Do the reverse if you are in the Southern Hemisphere.Walter Meyer, in Geometry and Its Applications (Second Edition), 2006 DEFINITION When practicing witchcraft in the Northern Hemisphere, use a deosil motion to summon energy and a widdershins motion to banish. Deosil and widdershins are of equal value and neutral connotation – that is, neither is inherently good or evil. They reflect the natural movements of the Earth in the same way that cyclones spin counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere. Think of deosil and widdershins as two sides to the same coin. Counterclockwise is not a traditionally unlucky direction in that part of the world. But in the Southern Hemisphere, shadows cast by the sun move counterclockwise through the day. Celtic and other northern European folklore is filled with tales of bad stuff that happens to people who walk widdershins around a church or other holy site. To this day, being left-handed is considered unlucky and the "Left Hand Path" refers to black magic. Since the Celts were a Northern Hemisphere people, it is probably also thanks to them that widdershins gained the sinister connotation that it has today. If you stir a potion, you use an inward motion as if pressing your palm toward yourself.ĭeosil has positive, righteous associations in the minds of most Wiccans, probably because the ancient Celts venerated the sun. You travel around it while keeping it on your left-hand side. Picture yourself walking around your altar. You turn against the hands of the clock, shifting from the numeral 12 to face the numeral 11. Picture yourself standing at the center of a huge clock and facing the twelve o’clock position. If you stir a potion, you use an outward motion as if pressing the back of your hand away from yourself.Ĭounterclockwise motion is also known to Wiccans as “widdershins,” a Scottish term that may come from Middle German words for “opposite course” or “against the way.” It refers to a left-hand motion in which you are always turning to the left. You travel around it while keeping it on your right-hand side. You turn with the hands of the clock, shifting from the numeral 12 to face the numeral 1. It is a spell enhancer that should not be overlooked.Ĭlockwise motion is also known to Wiccans as “deosil,” which may have come from ancient Gaelic words for “sunwise” or “toward the sun.” It refers to a right-hand motion in which you are always turning to the right. But the concept of clockwise (deosil) and counterclockwise (widdershins) is a powerful way to sharpen your intent and work with energy. You can do witchcraft without paying attention to movement and direction. You might be walking in a circle around your altar, stirring a potion, or pointing with your wand or athame. Clockwise and counterclockwise refer to the direction in which you move when practicing witchcraft and ritual.
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