As part of his somewhat esoteric belief system, Yeats believed that historical change could be conceived in terms of gyres. These are historical epochs, lasting for roughly 2,000 years or so. To help his readers understand this highly complicated concept, Yeats visualizes gyres as interpenetrating cones, with their interpenetration representing the passing of one historical epoch into another.

In “The Second Coming,” one gyre, or epoch of history, is about to come to an end, giving way to another. This is the epoch that has lasted since the birth of Christ. In its place will arise a completely different era, one characterized by bloodshed, chaos, and violence. Yeats senses that this gyre is in the offing by the fact that the falcon cannot hear the falconer. In other words, ordinary people—the falcon—no longer pay heed to the wisdom of their alleged social betters—the falconer—whom Yeats regards as born to rule.

Yeats's theory of the gyre, as set out in “The Second Coming,” is generally thought of as being an anticipation of the rise of fascism, in which all the old certainties of the Judeo-Christian tradition would come under a full-frontal assault. Particular lines, such as “the blood-dimmed tide is loosed,” certainly appear to lend credence to this interpretation.

In any case, it's clear that “turning and turning in the widening gyre” is leading inexorably to a period of human history in which very few of us would want to live.

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A gyre in "The Second Coming" refers to a spiral or a circular motion, but it also stands for the larger cycles of history. Yeats believed that an orderly gyre or cycle of history that began with the birth of Christ was ending, about to be replaced with a new historical cycle of chaos and cruelty. As the speaker puts it in impassioned terms near the beginning of the poem:

Mere anarchy is loosed upon the world,

The blood-dimmed tide is loosed, and everywhere

The ceremony of innocence is drowned

A "gyre," in its simplest interpretation, is simply a spiral: this line, then, refers to the ever-widening spiral formation in which the falcon flies out of the falconer's reach. But Yeats was very interested in the concept of "gyres," which to him had a more philosophical meaning. A gyre, according to Yeats, represented "the precise movement" of the human mind, according to the introduction to his 1921 publication The Second Coming. He also wrote a poem called "The Gyres," which is very instructive when we are trying to interpret the meaning of the word in "The Second Coming."

In "The Gyres," Yeats says, "Things thought too long can be no longer thought, / For beauty dies of beauty, worth of worth..." In this poem, all the inevitable decay and entropy he describes is encompassed by the first line's "The gyres! the gyres!" In this poem, as in "The Second Coming," the "gyre" is straightforwardly a spiral, but symbolically a representation of the inevitable onward motion of life toward a form of necessary chaos: "Things fall apart, the centre cannot hold."

I have moved this question to this group, as the reference to the word "gyre" obviously refers to this famous poem by W. B. Yeats and its first line. The poem concerns the poet's prediction of the grim future that awaits humanity, and the first startling image that is used to introduce this theme uses the word "gyre" to indicate the way that culture is disintegrating. Consider the following lines:

Turnign and turning in the widening gyre

The falcon cannot hear the falconer;

We are presented therefore with the somewhat mysterious image of a falcon going round and round in an ever widening "gyre," or a radiating spiral, moving further and further away beyond the reach of its falconer. Of course, the further that the falcon moves away from its falconer, the harder it is for the falconer to regain control over the bird and the falcon cannot be diverted or controlled in its direction. The word "gyre" therefore refers to the spiral motion of the falcon as it flies.