Infinite and Geometry

“The knowledge of which geometry aims is the knowledge of the eternal." (Plato)

"You say that just as space consists of an infinity of contiguous points, so time is but an infinite collection of contiguous instants? Good! Consider, then, an arrow in its flight. At any instant its extremity occupies a definite point in its path. Now, while occupying this position it must be at rest there. But how can a point be motionless and yet in motion at the same time?” (Zeno)

"Time and space are divided into the same and equal divisions. Wherefore also, Zeno’s argument, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time, is fallacious. For there are two senses in which the term ‘infinte’ is applied both to length and to time and in fact to all continuous things: either in regard to divisibility or in regard to number. Now it is not possible to touch things infinite as to number in a finite time, but it is possible to touch things infinite in regard to divisibility; for time itself is also infinite in this sense." (Aristotle)

“Our account does not rob mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraceable. In point of fact they do not need the infinite and do not use it. They postulate any that the finite straight line may be produced as far as they wish.” (Aristotle, Physics)

"A finite straight line can be extended indefinitely to make an infinitely long straight line." (Euclid’s postulate)

"Given a straight line and any point off to the side of it, there is, through that point, one and only one line that is parallel to the given line." (Euclid’s postulate)

"We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads." (Voltaire)

“Of late the speculations about Infinities have run so high, and grown to such strange notions, as have occasioned no small scruples and disputes among the geometers of the present age. Some there are of great note who, not contented with holding that finite lines may be divided into an infinite number of parts, do yet further maintain that each of these infinitesimals is itself subdivisible into an infinity of other parts or infinitesimals of a second order, and so on ad infinitum. These I say assert there are infinitesimals of infinitesimals, etc., without ever coming to an end; so that according to them an inch does not barely contain an infinite number of parts, but an infinity of an infinity of an infinity ad infinitum of parts.” (George Berkeley, “The Principles of Human Knowledge”, 1710)

“The introduction into geometrical work of conceptions such as the infinite, the imaginary, and the relations of hyperspace, none of which can be directly imagined, has a psychological significance well worthy of examination. It gives a deep insight into the resources and working of the human mind. We arrive at the borderland of mathematics and psychology.” (John Theodore Merz, “History of European Thought in the Nineteenth Century”, 1903)

The Infinite from Poet’s Pen

“Nature that framed us of four elements,
Warring within our breasts for regiment,
Doth teach us all to have aspiring minds:
Our souls, whose faculties can comprehend
The wondrous architecture of the world:
And measure every wand’ring planet’s course,
Still climbing after knowledge infinite,
And always moving as the restless spheres,
Wills us to wear ourselves and never rest,
Until we reach the ripest fruit of all,
That perfect bliss and sole felicity,
The sweet fruition of an earthly crown.”
(Christopher Marlowe, “Tamburlaine the Great”, 1590)

 “I could be bounded in a nutshell, and count myself a king of infinite space.” (William Shakespeare, “Hamlet”, cca. 1600)

“There’s nothing of so infinite vexation
As man’s own thoughts.”
(John Webster, “The White Devil”, 1612)

“[Infinity is] […] a dark
llimitable ocean, without bound,
Without dimension; where length, breadth, and height,
And time, and place, are lost…”
(John Milton, “Paradise Lost”, 1667)

“As lines (so loves) oblique may well
Themselves in every angle greet:
But ours so truly parallel,
Though infinite, can never meet.”
(Andrew Marvell, “The Definition of Love”, 1681)

“Even as the finite encloses an infinite series
And in the unlimited limits appear,
So the soul of immensity dwells in minutia
And in narrowest limits no limit in here.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!”
(Jacques Bernoulli, “Ars Conjectandi”, 1713)

“As lines, so loves oblique, may well
Themselves in every angle greet
But ours, so truly parallel,
Though infinite, can never meet.”
(Andrew Marvell, “The Definition of Love”)

“So, Nat’ralists observe, a Flea
Hath smaller Fleas that on him prey,
And these have smaller Fleas to bite ’em
And so proceed, ad infinitum.”
(Jonathan Swift", “On Poetry: a Rhapsody”, 1733)

“If in the infinite you want to stride,
Just walk in the finite to every side.” (Johann Wolfgang von Goethe)
“To see the world in a grain of sand,
And a heaven in a wild flower;
Hold infinity in the palm of your hand,
And eternity in an hour.”
(William Blake, “Auguries of Innocence”, 1803)

“Action is transitory, - a step, a blow,
The motion of a muscle, this way or that -
’Tis done, and in the after-vacancy
We wonder at ourselves like men betrayed:
Suffering is permanent, obscure and dark,
And shares the nature of infinity.”
(William Wordsworth, “The Borderers”, 1842)

“God puts his finger in the other scale,
And up we bounce, a bubble. Nought is great
Nor small, with God; for none but he can make
The atom imperceptible, and none
But he can make a world; he counts the orbs,
He counts the atoms of the universe,
And makes both equal; both are infinite.”
(Philip James Bailey, “Festus", 1845)

“What miracle of weird transforming
Is this wild work of frost and light,
This glimpse of glory infinite!”
(John Greenleaf Whittier, “The Pageant", 1869)

“Great fleas have little fleas upon their backs to bite 'em,
And little fleas have lesser fleas, and so ad infinitum.
And the great fleas themselves, in turn have greater fleas to go on;
While these again have greater still, and greater still, and so on.”
(Augustus de Morgan, “A Budget of Paradoxes”, 1915)

“But the star-glistered salver of infinity,
The circle, blind crucible of endless space,
Is sluiced by motion,-subjugated never.”
(Hart Crane. “The Bridge”, 1930)

“Big whorls have little whorls
Which feed on their velocity,
And little whorls have lesser whorls,
And so on to viscosity.”
(Lewis Richardson)

“All finite things reveal infinitude:
The mountain with its singular bright shade
Like the blue shine on freshly frozen snow,
The after-light upon ice-burdened pines;
Odor of basswood on a mountain-slope,
A scent beloved of bees;
Silence of water above a sunken tree :
The pure serene of memory in one man, --
A ripple widening from a single stone
Winding around the waters of the world.”
(Theodore Roethke, “The Far Field” IV, 1964)