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Euclidean geometry

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.

The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.

For more than two thousand years, the adjective "Euclidean" was unnecessary because Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field).

Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects. This is in contrast to analytic geometry, introduced almost 2,000 years later by René Descartes, which uses coordinates to express geometric properties by means of algebraic formulas.

The Elements
The Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost.

There are 13 books in the Elements:

Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, for example, "In any triangle, two angles taken together in any manner are less than two right angles." (Book I proposition 17) and the Pythagorean theorem "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." (Book I, proposition 47)

Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of surface regions. Notions such as prime numbers and rational and irrational numbers are introduced. It is proved that there are infinitely many prime numbers.

Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The platonic solids are constructed.

Axioms

Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a small number of simple axioms. Until the advent of non-Euclidean geometry, these axioms were considered to be obviously true in the physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality.

Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):

 * Let the following be postulated:

 * 1) To draw a straight line from any point to any point.
 * 2) To produce (extend) a finite straight line continuously in a straight line.
 * 3) To describe a circle with any centre and distance (radius).
 * 4) That all right angles are equal to one another.
 * 5) [The parallel postulate]: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

Although Euclid explicitly only asserts the existence of the constructed objects, in his reasoning he also implicitly assumes them to be unique.

The Elements also include the following five "common notions":

 * 1) Things that are equal to the same thing are also equal to one another (the transitive property of a Euclidean relation).
 * 2) If equals are added to equals, then the wholes are equal (Addition property of equality).
 * 3) If equals are subtracted from equals, then the differences are equal (subtraction property of equality).
 * 4) Things that coincide with one another are equal to one another (reflexive property).
 * 5) The whole is greater than the part.

Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more extensive and complete sets of axioms.

Parallel postulate
To the ancients, the parallel postulate seemed less obvious than the others. They aspired to create a system of absolutely certain propositions, and to them, it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible since one can construct consistent systems of geometry (obeying the other axioms) in which the parallel postulate is true, and others in which it is false. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: his first 28 propositions are those that can be proved without it.

Many alternative axioms can be formulated which are logically equivalent to the parallel postulate (in the context of the other axioms). For example, Playfair's axiom states:

 * In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the given line.

The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists.

Methods of proof
Euclidean Geometry is constructive. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge. In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory. Strictly speaking, the lines on paper are models of the objects defined within the formal system, rather than instances of those objects. For example, a Euclidean straight line has no width, but any real drawn line will have. Though nearly all modern mathematicians consider nonconstructive methods just as sound as constructive ones, Euclid's constructive proofs often supplanted fallacious nonconstructive ones—e.g., some of the Pythagoreans' proofs that involved irrational numbers, which usually required a statement such as "Find the greatest common measure of ..."

Euclid often used proof by contradiction. Euclidean geometry also allows the method of superposition, in which a figure is transferred to another point in space. For example, proposition I.4, side–angle–side congruence of triangles, is proved by moving one of the two triangles so that one of its sides coincides with the other triangle's equal side, and then proving that the other sides coincide as well. Some modern treatments add a sixth postulate, the rigidity of the triangle, which can be used as an alternative to superposition.

Naming of points and figures
Points are customarily named using capital letters of the alphabet. Other figures, such as lines, triangles, or circles, are named by listing a sufficient number of points to pick them out unambiguously from the relevant figure, e.g., triangle ABC would typically be a triangle with vertices at points A, B, and C.

Complementary and supplementary angles
Angles whose sum is a right angle are called complementary. Complementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the right angle. The number of rays in between the two original rays is infinite.

Angles whose sum is a straight angle are supplementary. Supplementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the straight angle (180 degree angle). The number of rays in between the two original rays is infinite.

Modern versions of Euclid's notation
In modern terminology, angles would normally be measured in degrees or radians.

Modern school textbooks often define separate figures called lines (infinite), rays (semi-infinite), and line segments (of finite length). Euclid, rather than discussing a ray as an object that extends to infinity in one direction, would normally use locutions such as "if the line is extended to a sufficient length", although he occasionally referred to "infinite lines". A "line" in Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary.

Pons asinorum
The pons asinorum (bridge of asses) states that in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. Its name may be attributed to its frequent role as the first real test in the Elements of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross.

Congruence of triangles

Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent.

Triangle angle sum
The sum of the angles of a triangle is equal to a straight angle (180 degrees). This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one obtuse or right angle.

Pythagorean theorem
The celebrated Pythagorean theorem (book I, proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

Thales' theorem
Thales' theorem, named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I, Prop. 32 after the manner of Euclid Book III, Prop. 31.

Scaling of area and volume
In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions, $$A \propto L^2$$, and the volume of a solid to the cube, $$V \propto L^3$$. Euclid proved these results in various special cases such as the area of a circle and the volume of a parallelepipedal solid. Euclid determined some, but not all, of the relevant constants of proportionality. E.g., it was his successor Archimedes who proved that a sphere has 2/3 the volume of the circumscribing cylinder.

System of measurement and arithmetic
Euclidean geometry has two fundamental types of measurements: angle and distance. The angle scale is absolute, and Euclid uses the right angle as his basic unit, so that, for example, a 45-degree angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction.

Measurements of area and volume are derived from distances. For example, a rectangle with a width of 3 and a length of 4 has an area that represents the product, 12. Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, for example in the proof of book IX, proposition 20.

Euclid refers to a pair of lines, or a pair of planar or solid figures, as "equal" (ἴσος) if their lengths, areas, or volumes are equal respectively, and similarly for angles. The stronger term "congruent" refers to the idea that an entire figure is the same size and shape as another figure. Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly. (Flipping it over is allowed.) Thus, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their differing sizes are referred to as similar. Corresponding angles in a pair of similar shapes are equal and corresponding sides are in proportion to each other.

Design and Analysis

 * Stress Analysis: Stress Analysis - Euclidean geometry is pivotal in determining stress distribution in mechanical components, which is essential for ensuring structural integrity and durability.

 * Gear Design: Gear - The design of gears, a crucial element in many mechanical systems, relies heavily on Euclidean geometry to ensure proper tooth shape and engagement for efficient power transmission.

 * Heat Exchanger Design: Heat exchanger - In thermal engineering, Euclidean geometry is used to design heat exchangers, where the geometric configuration greatly influences thermal efficiency. See shell-and-tube heat exchangers and plate heat exchangers for more details.

 * Lens Design: Lens - In optical engineering, Euclidean geometry is critical in the design of lenses, where precise geometric shapes determine the focusing properties. Geometric optics analyzes the focusing of light by lenses and mirrors.

Dynamics

 * Vibration Analysis: Vibration - Euclidean geometry is essential in analyzing and understanding the vibrations in mechanical systems, aiding in the design of systems that can withstand or utilize these vibrations effectively.

 * Wing Design: Aircraft Wing Design - The application of Euclidean geometry in aerodynamics is evident in aircraft wing design, airfoils, and hydrofoils where geometric shape directly impacts lift and drag characteristics.

 * Satellite Orbits: Satellite Orbits - Euclidean geometry helps in calculating and predicting the orbits of satellites, essential for successful space missions and satellite operations. Also see astrodynamics, celestial mechanics, and elliptic orbit.

CAD Systems

 * 3D Modeling: In CAD (computer-aided design) systems, Euclidean geometry is fundamental for creating accurate 3D models of mechanical parts. These models are crucial for visualizing and testing designs before manufacturing.

 * Design and Manufacturing: Much of CAM (computer-aided manufacturing) relies on Euclidean geometry. The design geometry in CAD/CAM typically consists of shapes bounded by planes, cylinders, cones, tori, and other similar Euclidean forms. Today, CAD/CAM is essential in the design of a wide range of products, from cars and airplanes to ships and smartphones.

 * Evolution of Drafting Practices: Historically, advanced Euclidean geometry, including theorems like Pascal's theorem and Brianchon's theorem, was integral to drafting practices. However, with the advent of modern CAD systems, such in-depth knowledge of these theorems is less necessary in contemporary design and manufacturing processes.

Circuit Design

 * PCB Layouts: Printed Circuit Board (PCB) Design utilizes Euclidean geometry for the efficient placement and routing of components, ensuring functionality while optimizing space. Efficient layout of electronic components on PCBs is critical for minimizing signal interference and optimizing circuit performance.

Electromagnetic and Fluid Flow Fields

 * Antenna Design: Antenna Design - Euclidean geometry of antennas helps in designing antennas, where the spatial arrangement and dimensions directly affect antenna and array performance in transmitting and receiving electromagnetic waves.

 * Field Theory: Complex Potential Flow - In the study of inviscid flow fields and electromagnetic fields, Euclidean geometry aids in visualizing and solving potential flow problems. This is essential for understanding fluid velocity field and electromagnetic field interactions in three-dimensional space. The relationship of which is characterized by an irrotational solenoidal field or a conservative vector field.

Controls

 * Control System Analysis: Control Systems - The application of Euclidean geometry in control theory helps in the analysis and design of control systems, particularly in understanding and optimizing system stability and response.

 * Calculation Tools: Jacobian - Euclidean geometry is integral in using Jacobian matrices for transformations and control systems in both mechanical and electrical engineering fields, providing insights into system behavior and properties. The Jacobian serves as a linearized design matrix in statistical regression and curve fitting; see non-linear least squares. The Jacobian is also used in random matrices, moment, statistics, and diagnostics.

Other general applications
Because of Euclidean geometry's fundamental status in mathematics, it is impractical to give more than a representative sampling of applications here.

Geometry is used extensively in architecture.

Geometry can be used to design origami. Some classical construction problems of geometry are impossible using compass and straightedge, but can be solved using origami.

Archimedes and Apollonius
Archimedes (c. 287 BCE), a colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians. Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original. He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the Archimedean property of finite numbers.

Apollonius of Perga (c. 240 BCE) is mainly known for his investigation of conic sections.

17th century: Descartes
René Descartes (1596–1650) developed analytic geometry, an alternative method for formalizing geometry which focused on turning geometry into algebra.

In this approach, a point on a plane is represented by its Cartesian (x, y) coordinates, a line is represented by its equation, and so on.

In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.

The equation
 * $$|PQ|=\sqrt{(p_x-q_x)^2+(p_y-q_y)^2} \, $$

defining the distance between two points P = (px, py) and Q = (qx, qy) is then known as the Euclidean metric, and other metrics define non-Euclidean geometries.

In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., y = 2x + 1 (a line), or x2 + y2 = 7 (a circle).

Also in the 17th century, Girard Desargues, motivated by the theory of perspective, introduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry, projective geometry, but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.

18th century
Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763, at least 28 different proofs had been published, but all were found incorrect.

Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include doubling the cube and squaring the circle. In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two, while doubling a cube requires the solution of a third-order equation.

Euler discussed a generalization of Euclidean geometry called affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint).

19th century
In the early 19th century, Carnot and Möbius systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.

Higher dimensions
In the 1840s William Rowan Hamilton developed the quaternions, and John T. Graves and Arthur Cayley the octonions. These are normed algebras which extend the complex numbers. Later it was understood that the quaternions are also a Euclidean geometric system with four real Cartesian coordinates. Cayley used quaternions to study rotations in 4-dimensional Euclidean space.

At mid-century Ludwig Schläfli developed the general concept of Euclidean space, extending Euclidean geometry to higher dimensions. He defined polyschemes, later called polytopes, which are the higher-dimensional analogues of polygons and polyhedra. He developed their theory and discovered all the regular polytopes, i.e. the $$n$$-dimensional analogues of regular polygons and Platonic solids. He found there are six regular convex polytopes in dimension four, and three in all higher dimensions.

Schläfli performed this work in relative obscurity and it was published in full only posthumously in 1901. It had little influence until it was rediscovered and fully documented in 1948 by H.S.M. Coxeter.

In 1878 William Kingdon Clifford introduced what is now termed geometric algebra, unifying Hamilton's quaternions with Hermann Grassmann's algebra and revealing the geometric nature of these systems, especially in four dimensions. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modeled to new positions. The Clifford torus on the surface of the 3-sphere is the simplest and most symmetric flat embedding of the Cartesian product of two circles (in the same sense that the surface of a cylinder is "flat").

Non-Euclidean geometry
The century's most influential development in geometry occurred when, around 1830, János Bolyai and Nikolai Ivanovich Lobachevsky separately published work on non-Euclidean geometry, in which the parallel postulate is not valid. Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates.

In the 19th century, it was also realized that Euclid's ten axioms and common notions do not suffice to prove all of the theorems stated in the Elements. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself. The very first geometric proof in the Elements, shown in the figure above, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third vertex. His axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the completeness property of the real numbers. Starting with Moritz Pasch in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of Hilbert, George Birkhoff, and Tarski.

20th century and relativity

Einstein's theory of special relativity involves a four-dimensional space-time, the Minkowski space, which is non-Euclidean. This shows that non-Euclidean geometries, which had been introduced a few years earlier for showing that the parallel postulate cannot be proved, are also useful for describing the physical world.

However, the three-dimensional "space part" of the Minkowski space remains the space of Euclidean geometry. This is not the case with general relativity, for which the geometry of the space part of space-time is not Euclidean geometry. For example, if a triangle is constructed out of three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity. A relatively weak gravitational field, such as the Earth's or the Sun's, is represented by a metric that is approximately, but not exactly, Euclidean. Until the 20th century, there was no technology capable of detecting these deviations in rays of light from Euclidean geometry, but Einstein predicted that such deviations would exist. They were later verified by observations such as the slight bending of starlight by the Sun during a solar eclipse in 1919, and such considerations are now an integral part of the software that runs the GPS system.

As a description of the structure of space
Euclid believed that his axioms were self-evident statements about physical reality. Euclid's proofs depend upon assumptions perhaps not obvious in Euclid's fundamental axioms, in particular that certain movements of figures do not change their geometrical properties such as the lengths of sides and interior angles, the so-called Euclidean motions, which include translations, reflections and rotations of figures. Taken as a physical description of space, postulate 2 (extending a line) asserts that space does not have holes or boundaries; postulate 4 (equality of right angles) says that space is isotropic and figures may be moved to any location while maintaining congruence; and postulate 5 (the parallel postulate) that space is flat (has no intrinsic curvature).

As discussed above, Albert Einstein's theory of relativity significantly modifies this view.

The ambiguous character of the axioms as originally formulated by Euclid makes it possible for different commentators to disagree about some of their other implications for the structure of space, such as whether or not it is infinite (see below) and what its topology is. Modern, more rigorous reformulations of the system typically aim for a cleaner separation of these issues. Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1–4 are consistent with either infinite or finite space (as in elliptic geometry), and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or a torus for two-dimensional Euclidean geometry).

Infinite objects
Euclid sometimes distinguished explicitly between "finite lines" (e.g., Postulate 2) and "infinite lines" (book I, proposition 12). However, he typically did not make such distinctions unless they were necessary. The postulates do not explicitly refer to infinite lines, although for example some commentators interpret postulate 3, existence of a circle with any radius, as implying that space is infinite.

The notion of infinitesimal quantities had previously been discussed extensively by the Eleatic School, but nobody had been able to put them on a firm logical basis, with paradoxes such as Zeno's paradox occurring that had not been resolved to universal satisfaction. Euclid used the method of exhaustion rather than infinitesimals.

Later ancient commentators, such as Proclus (410–485 CE), treated many questions about infinity as issues demanding proof and, e.g., Proclus claimed to prove the infinite divisibility of a line, based on a proof by contradiction in which he considered the cases of even and odd numbers of points constituting it.

At the turn of the 20th century, Otto Stolz, Paul du Bois-Reymond, Giuseppe Veronese, and others produced controversial work on non-Archimedean models of Euclidean geometry, in which the distance between two points may be infinite or infinitesimal, in the Newton–Leibniz sense. Fifty years later, Abraham Robinson provided a rigorous logical foundation for Veronese's work.

Infinite processes
Ancient geometers may have considered the parallel postulate – that two parallel lines do not ever intersect – less certain than the others because it makes a statement about infinitely remote regions of space, and so cannot be physically verified.

The modern formulation of proof by induction was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid's proofs, e.g., the proof of the infinitude of primes.

Supposed paradoxes involving infinite series, such as Zeno's paradox, predated Euclid. Euclid avoided such discussions, giving, for example, the expression for the partial sums of the geometric series in IX.35 without commenting on the possibility of letting the number of terms become infinite.

Classical logic
Euclid frequently used the method of proof by contradiction, and therefore the traditional presentation of Euclidean geometry assumes classical logic, in which every proposition is either true or false, i.e., for any proposition P, the proposition "P or not P" is automatically true.

Modern standards of rigor
Placing Euclidean geometry on a solid axiomatic basis was a preoccupation of mathematicians for centuries. The role of primitive notions, or undefined concepts, was clearly put forward by Alessandro Padoa of the Peano delegation at the 1900 Paris conference: "...when we begin to formulate the theory, we can imagine that the undefined symbols are completely devoid of meaning and that the unproved propositions are simply conditions imposed upon the undefined symbols.

Then, the system of ideas that we have initially chosen is simply one interpretation of the undefined symbols; but..this interpretation can be ignored by the reader, who is free to replace it in his mind by another interpretation.. that satisfies the conditions...

Logical questions thus become completely independent of empirical or psychological questions...

The system of undefined symbols can then be regarded as the abstraction obtained from the specialized theories that result when...the system of undefined symbols is successively replaced by each of the interpretations..."

That is, mathematics is context-independent knowledge within a hierarchical framework. As said by Bertrand Russell: "If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus, mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

Such foundational approaches range between foundationalism and formalism.

Axiomatic formulations
"Geometry is the science of correct reasoning on incorrect figures."
 * Euclid's axioms: In his dissertation to Trinity College, Cambridge, Bertrand Russell summarized the changing role of Euclid's geometry in the minds of philosophers up to that time. It was a conflict between certain knowledge, independent of experiment, and empiricism, requiring experimental input. This issue became clear as it was discovered that the parallel postulate was not necessarily valid and its applicability was an empirical matter, deciding whether the applicable geometry was Euclidean or non-Euclidean.
 * Hilbert's axioms: Hilbert's axioms had the goal of identifying a simple and complete set of independent axioms from which the most important geometric theorems could be deduced. The outstanding objectives were to make Euclidean geometry rigorous (avoiding hidden assumptions) and to make clear the ramifications of the parallel postulate.
 * Birkhoff's axioms: Birkhoff proposed four postulates for Euclidean geometry that can be confirmed experimentally with scale and protractor. This system relies heavily on the properties of the real numbers.  The notions of angle and distance become primitive concepts.
 * Tarski's axioms: Alfred Tarski (1902–1983) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets. Tarski proved that his axiomatic formulation of elementary Euclidean geometry is consistent and complete in a certain sense: there is an algorithm that, for every proposition, can be shown either true or false. (This does not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply. ) This is equivalent to the decidability of real closed fields, of which elementary Euclidean geometry is a model.

Classical theorems

 * Angle bisector theorem
 * Butterfly theorem
 * Ceva's theorem
 * Heron's formula
 * Menelaus' theorem
 * Nine-point circle
 * Pythagorean theorem

title: Euclidean geometry


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Epic poetry

An epic poem, or simply an epic, is a lengthy narrative poem typically about the extraordinary deeds of extraordinary characters who, in dealings with gods or other superhuman forces, gave shape to the mortal universe for their descendants.

Etymology
The English word epic comes from Latin epicus, which itself comes from the Ancient Greek adjective ἐπικός (epikos), from ἔπος (epos), "word, story, poem."

In ancient Greek, 'epic' could refer to all poetry in dactylic hexameter (epea), which included not only Homer but also the wisdom poetry of Hesiod, the utterances of the Delphic oracle, and the strange theological verses attributed to Orpheus. Later tradition, however, has restricted the term 'epic' to heroic epic, as described in this article.

Overview
Originating before the invention of writing, primary epics, such as those of Homer, were composed by bards who used complex rhetorical and metrical schemes by which they could memorize the epic as received in tradition and add to the epic in their performances. Later writers like Virgil, Apollonius of Rhodes, Dante, Camões, and Milton adopted and adapted Homer's style and subject matter, but used devices available only to those who write.

The oldest epic recognized is the Epic of Gilgamesh (c. 2500–1300 BCE), which was recorded in ancient Sumer during the Neo-Sumerian Empire. The poem details the exploits of Gilgamesh, the king of Uruk. Although recognized as a historical figure, Gilgamesh, as represented in the epic, is a largely legendary or mythical figure.

The longest written epic from antiquity is the ancient Indian Mahabharata (c. 3rd century BC–3rd century AD), which consists of 100,000 ślokas or over 200,000 verse lines (each shloka is a couplet), as well as long prose passages, so that at ~1.8 million words it is roughly twice the length of Shahnameh, four times the length of the Rāmāyaṇa, and roughly ten times the length of the Iliad and the Odyssey combined.

Famous examples of epic poetry include the Sumerian Epic of Gilgamesh, the ancient Indian Mahabharata and Rāmāyaṇa in Sanskrit and Silappatikaram and Manimekalai in Tamil, the Persian Shahnameh, the Ancient Greek Odyssey and Iliad, Virgil's Aeneid, the Old English Beowulf, Dante's Divine Comedy, the Finnish Kalevala, the German Nibelungenlied, the French Song of Roland, the Spanish Cantar de mio Cid, the Portuguese Os Lusíadas, the Armenian Daredevils of Sassoun, the Old Russian The Tale of Igor's Campaign, John Milton's Paradise Lost, The Secret History of the Mongols, the Kyrgyz Manas, and the Malian Sundiata. Epic poems of the modern era include Derek Walcott's Omeros, Mircea Cărtărescu's The Levant and Adam Mickiewicz's Pan Tadeusz. Paterson by William Carlos Williams, published in five volumes from 1946 to 1958, was inspired in part by another modern epic, The Cantos by Ezra Pound.

Oral epics
The first epics were products of preliterate societies and oral history poetic traditions. Oral tradition was used alongside written scriptures to communicate and facilitate the spread of culture. In these traditions, poetry is transmitted to the audience and from performer to performer by purely oral means. Early 20th-century study of living oral epic traditions in the Balkans by Milman Parry and Albert Lord demonstrated the paratactic model used for composing these poems. What they demonstrated was that oral epics tend to be constructed in short episodes, each of equal status, interest and importance. This facilitates memorization, as the poet is recalling each episode in turn and using the completed episodes to recreate the entire epic as he performs it. Parry and Lord also contend that the most likely source for written texts of the epics of Homer was dictation from an oral performance.

Milman Parry and Albert Lord have argued that the Homeric epics, the earliest works of Western literature, were fundamentally an oral poetic form. These works form the basis of the epic genre in Western literature. Nearly all of Western epic (including Virgil's Aeneid and Dante's Divine Comedy) self-consciously presents itself as a continuation of the tradition begun by these poems.

Composition and conventions
In his work Poetics, Aristotle defines an epic as one of the forms of poetry, contrasted with lyric poetry and drama (in the form of tragedy and comedy).

 * Epic poetry agrees with Tragedy in so far as it is an imitation in verse of characters of a higher type. They differ in that Epic poetry admits but one kind of meter and is narrative in form. They differ, again, in their length: for Tragedy endeavors, as far as possible, to confine itself to a single revolution of the sun, or but slightly to exceed this limit, whereas the Epic action has no limits of time. This, then, is a second point of difference; though at first the same freedom was admitted in Tragedy as in Epic poetry.

 * Of their constituent parts some are common to both, some peculiar to Tragedy: whoever, therefore knows what is good or bad Tragedy, knows also about Epic poetry. All the elements of an Epic poem are found in Tragedy, but the elements of a Tragedy are not all found in the Epic poem. – Aristotle, Poetics Part V

Harmon & Holman (1999) define an epic:
 * Epic: A long narrative poem in elevated style presenting characters of high position in adventures forming an organic whole through their relation to a central heroic figure and through their development of episodes important to the history of a nation or race.
 * — Harmon & Holman (1999)

Harmon and Holman delineate ten main characteristics of an epic:
 * 1) Begins in medias res ("in the thick of things").
 * 2) The setting is vast, covering many nations, the world or the universe.
 * 3) Begins with an invocation to a muse (epic invocation).
 * 4) Begins with a statement of the theme.
 * 5) Includes the use of epithets.
 * 6) Contains long lists, called an epic catalogue.
 * 7) Features long and formal speeches.
 * 8) Shows divine intervention in human affairs.
 * 9) Features heroes that embody the values of the civilization.
 * 10) Often features the tragic hero's descent into the underworld or hell.

The hero generally participates in a cyclical journey or quest, faces adversaries that try to defeat them in their journey, and returns home significantly transformed by their journey. The epic hero illustrates traits, performs deeds, and exemplifies certain morals that are valued by the society the epic originates from. Many epic heroes are recurring characters in the legends of their native cultures.

Conventions of the Indian Epic
In the Indian mahākāvya epic genre, more emphasis was laid on description than on narration. Indeed, the traditional characteristics of a mahākāvya are listed as:
 * It must take its subject matter from the epics (Ramayana or Mahabharata), or from history,
 * It must help further the four goals of man (purusharthas),
 * It must contain descriptions of cities, seas, mountains, moonrise and sunrise, and accounts of merrymaking in gardens, of bathing parties, drinking bouts, and love-making.
 * It should tell the sorrow of separated lovers and should describe a wedding and the birth of a son.
 * It should describe a king's council, an embassy, the marching forth of an army, a battle, and the victory of a hero.

Themes
Classical epic poetry recounts a journey, either physical (as typified by Odysseus in the Odyssey) or mental (as typified by Achilles in the Iliad) or both. Epics also tend to highlight cultural norms and to define or call into question cultural values, particularly as they pertain to heroism.

Proem
In the proem or preface, the poet may begin by invoking a Muse or similar divinity. The poet prays to the Muses to provide them with divine inspiration to tell the story of a great hero.

Example opening lines with invocations:
 * Sing goddess the baneful wrath of Achilles son of Peleus – Iliad 1.1

 * Muse, tell me in verse of the man of many wiles – Odyssey 1.1

 * From the Heliconian Muses let us begin to sing – Hesiod, Theogony 1.1

 * Beginning with thee, Oh Phoebus, I will recount the famous deeds of men of old – Argonautica 1.1

 * Muse, remember to me the causes – Aeneid 1.8

 * Sing Heav'nly Muse, that on the secret top
 * of Oreb, or of Sinai, didst inspire – Paradise Lost 1.6–7

An alternative or complementary form of proem, found in Virgil and his imitators, opens with the performative verb "I sing". Examples:

 * I sing arms and the man – Aeneid 1.1

 * I sing pious arms and their captain – Gerusalemme liberata 1.1

 * I sing ladies, knights, arms, loves, courtesies, audacious deeds – Orlando Furioso 1.1–2

This Virgilian epic convention is referenced in Walt Whitman's poem title / opening line "I sing the body electric".

Compare the first six lines of the Kalevala:
 * Mastered by desire impulsive,
 * By a mighty inward urging,
 * I am ready now for singing,
 * Ready to begin the chanting
 * Of our nation's ancient folk-song
 * Handed down from by-gone ages.

These conventions are largely restricted to European classical culture and its imitators. The Epic of Gilgamesh, for example, or the Bhagavata Purana do not contain such elements, nor do early medieval Western epics that are not strongly shaped by the classical traditions, such as the Chanson de Roland or the Poem of the Cid.

In medias res
Narrative opens "in the middle of things", with the hero at his lowest point. Usually flashbacks show earlier portions of the story. For example, the Iliad does not tell the entire story of the Trojan War, starting with the judgment of Paris, but instead opens abruptly on the rage of Achilles and its immediate causes. So too, Orlando Furioso is not a complete biography of Roland, but picks up from the plot of Orlando Innamorato, which in turn presupposes a knowledge of the romance and oral traditions.

Enumeratio
Epic catalogues and genealogies are given, called enumeratio. These long lists of objects, places, and people place the finite action of the epic within a broader, universal context, such as the catalog of ships. Often, the poet is also paying homage to the ancestors of audience members. Examples:
 * In The Faerie Queene, the list of trees I.i.8–9.
 * In Paradise Lost, the list of demons in Book I.
 * In the Aeneid, the list of enemies the Trojans find in Etruria (Central Italy) in Book VII. Also, the list of ships in Book X.
 * In the Iliad, the Catalogue of Ships, the most famous epic catalogue, and the Trojan Battle Order

Stylistic features
In the Homeric and post-Homeric tradition, epic style is typically achieved through the use of the following stylistic features:
 * Heavy use of repetition or stock phrases: e.g., Homer's "rosy-fingered dawn" and "wine-dark sea".
 * Epic similes

Form
Many verse forms have been used in epic poems through the ages, but each language's literature typically gravitates to one form, or at least to a very limited set.

Ancient Sumerian epic poems did not use any kind of poetic meter and lines did not have consistent lengths; instead, Sumerian poems derived their rhythm solely through constant repetition and parallelism, with subtle variations between lines. Indo-European epic poetry, by contrast, usually places strong emphasis on the importance of line consistency and poetic meter. Ancient Greek epics were composed in dactylic hexameter. Very early Latin epicists, such Livius Andronicus and Gnaeus Naevius, used Saturnian meter. By the time of Ennius, however, Latin poets had adopted dactylic hexameter.

Dactylic hexameter has been adapted by a few anglophone poets such as Longfellow in "Evangeline", whose first line is as follows:

 * This is the | forest pri | meval. The | murmuring | pines and the | hemlocks

Old English, German and Norse poems were written in alliterative verse, usually without rhyme. The alliterative form can be seen in the Old English "Finnsburg Fragment" (alliterated sounds are in bold):

While the above classical and Germanic forms would be considered stichic, Italian, Spanish and Portuguese long poems favored stanzaic forms, usually written in terza rima or especially ottava rima. Terza rima is a rhyming verse stanza form that consists of an interlocking three-line rhyme scheme. An example is found in the first lines of the Divine Comedy by Dante, who originated the form:

Nel mezzo del cammin di nostra vita (A) mi ritrovai per una selva oscura (B) ché la diritta via era smarrita. (A)

Ahi quanto a dir qual era è cosa dura (B) esta selva selvaggia e aspra e forte (C) che nel pensier rinnova la paura! (B)

In ottava rima, each stanza consists of three alternate rhymes and one double rhyme, following the ABABABCC rhyme scheme. Example:

From the 14th century English epic poems were written in heroic couplets, and rhyme royal, though in the 16th century the Spenserian stanza and blank verse were also introduced. The French alexandrine is currently the heroic line in French literature, though in earlier literature – such as the chanson de geste – the decasyllable grouped in laisses took precedence. In Polish literature, couplets of Polish alexandrines (syllabic lines of 7+6 syllables) prevail. In Russian, iambic tetrameter verse is the most popular. In Serbian poetry, the decasyllable is the only form employed.

Balto-Finnic (e.g. Estonian, Finnish, Karelian) folk poetry uses a form of trochaic tetrameter that has been called the Kalevala meter. The Finnish and Estonian national epics, Kalevala and Kalevipoeg, are both written in this meter. The meter is thought to have originated during the Proto-Finnic period.

In Indic epics such as the Ramayana and Mahabharata, the shloka form is used.

Genres and related forms
The primary form of epic, especially as discussed in this article, is the heroic epic, including such works as the Iliad and Mahabharata. Ancient sources also recognized didactic epic as a category, represented by such works as Hesiod's Works and Days and Lucretius's De rerum natura.

A related type of poetry is the epyllion (plural: epyllia), a brief narrative poem with a romantic or mythological theme. The term, which means "little epic", came into use in the nineteenth century. It refers primarily to the erudite, shorter hexameter poems of the Hellenistic period and the similar works composed at Rome from the age of the neoterics; to a lesser degree, the term includes some poems of the English Renaissance, particularly those influenced by Ovid. The most famous example of classical epyllion is perhaps Catullus 64.

Epyllion is to be understood as distinct from mock epic, another light form.

Romantic epic is a term used to designate works such as Morgante, Orlando Innamorato, Orlando Furioso and Gerusalemme Liberata, which freely lift characters, themes, plots and narrative devices from the world of prose chivalric romance.

Non-European forms

Long poetic narratives that do not fit the traditional European definition of the heroic epic are sometimes known as folk epics. Indian folk epics have been investigated by Lauri Honko (1998), Brenda Beck (1982) and John Smith, amongst others. Folk epics are an important part of community identities. For example, in Egypt, the folk genre known as al-sira relates the saga of the Hilālī tribe and their migrations across the Middle East and north Africa, see Bridget Connelly (1986). In India, folk epics reflect the caste system of Indian society and the life of the lower levels of society, such as cobblers and shepherds, see C.N. Ramachandran, "Ambivalence and Angst: A Note on Indian folk epics," in Lauri Honko (2002. p. 295). Some Indian oral epics feature strong women who actively pursue personal freedom in their choice of a romantic partner (Stuart, Claus, Flueckiger and Wadley, eds, 1989, p. 5). Japanese traditional performed narratives were sung by blind singers. One of the most famous, The Tale of the Heike, deals with historical wars and had a ritual function to placate the souls of the dead (Tokita 2015, p. 7). A variety of epic forms are found in Africa. Some have a linear, unified style while others have a more cyclical, episodic style (Barber 2007, p. 50). People in the rice cultivation zones of south China sang long narrative songs about the origin of rice growing, rebel heroes, and transgressive love affairs (McLaren 2022). The borderland ethnic populations of China sang heroic epics, such as the Epic of King Gesar of the Mongols, and the creation-myth epics of the Yao people of south China.

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title: Epic poetry


text:
Eocene

The Eocene is a geological epoch that lasted from about 56 to 33.9 million years ago (Ma). It is the second epoch of the Paleogene Period in the modern Cenozoic Era. The name Eocene comes from the Ancient Greek (Ēṓs, "Dawn") and  (kainós, "new") and refers to the "dawn" of modern ('new') fauna that appeared during the epoch.

The Eocene spans the time from the end of the Paleocene Epoch to the beginning of the Oligocene Epoch. The start of the Eocene is marked by a brief period in which the concentration of the carbon isotope 13C in the atmosphere was exceptionally low in comparison with the more common isotope 12C. The average temperature of Earth at the beginning of the Eocene was about 27 degrees Celsius. The end is set at a major extinction event called the Grande Coupure (the "Great Break" in continuity) or the Eocene–Oligocene extinction event, which may be related to the impact of one or more large bolides in Siberia and in what is now Chesapeake Bay. As with other geologic periods, the strata that define the start and end of the epoch are well identified, though their exact dates are slightly uncertain.

Etymology
The term "Eocene" is derived from Ancient Greek Ἠώς (Ēṓs) meaning "Dawn", and καινός kainos meaning "new" or "recent", as the epoch saw the dawn of recent, or modern, life.

Scottish geologist Charles Lyell (ignoring the Quaternary) divided the Tertiary Epoch into the Eocene, Miocene, Pliocene, and New Pliocene (Holocene) Periods in 1833. British geologist John Phillips proposed the Cenozoic in 1840 in place of the Tertiary, and Austrian paleontologist Moritz Hörnes introduced the Paleogene for the Eocene and Neogene for the Miocene and Pliocene in 1853. After decades of inconsistent usage, the newly formed International Commission on Stratigraphy (ICS), in 1969, standardized stratigraphy based on the prevailing opinions in Europe: the Cenozoic Era subdivided into the Tertiary and Quaternary sub-eras, and the Tertiary subdivided into the Paleogene and Neogene periods. In 1978, the Paleogene was officially defined as the Paleocene, Eocene, and Oligocene epochs; and the Neogene as the Miocene and Pliocene epochs. In 1989, Tertiary and Quaternary were removed from the time scale due to the arbitrary nature of their boundary, but Quaternary was reinstated in 2009.

Boundaries
The Eocene is a dynamic epoch that represents global climatic transitions between two climatic extremes, transitioning from the hot house to the cold house. The beginning of the Eocene is marked by the Paleocene–Eocene Thermal Maximum, a short period of intense warming and ocean acidification brought about by the release of carbon en masse into the atmosphere and ocean systems, which led to a mass extinction of 30–50% of benthic foraminifera (single-celled species which are used as bioindicators of the health of a marine ecosystem)—one of the largest in the Cenozoic. This event happened around 55.8 Ma, and was one of the most significant periods of global change during the Cenozoic.

The middle Eocene was characterized by the shift towards a cooler climate at the end of the EECO, around 47.8 Ma, which was briefly interrupted by another warming event called the middle Eocene climatic optimum (MECO). Lasting for about 400,000 years, the MECO was responsible for a globally uniform 4° to 6°C warming of both the surface and deep oceans, as inferred from foraminiferal stable oxygen isotope records. The resumption of a long-term gradual cooling trend resulted in a glacial maximum at the late Eocene/early Oligocene boundary.

The end of the Eocene was also marked by the Eocene–Oligocene extinction event, also known as the Grande Coupure.

Stratigraphy
The Eocene is conventionally divided into early (56–47.8 Ma), middle (47.8–38 Ma), and late (38–33.9 Ma) subdivisions. The corresponding rocks are referred to as lower, middle, and upper Eocene. The Ypresian Stage constitutes the lower, the Priabonian Stage the upper; and the Lutetian and Bartonian stages are united as the middle Eocene.

The Western North American floras of the Eocene were divided into four floral "stages" by Jack Wolfe (1968) based on work with the Puget Group fossils of King County, Washington. The four stages, Franklinian, Fultonian, Ravenian, and Kummerian covered the Early Eocene through early Oligocene, and three of the four were given informal early/late substages. Wolfe tentatively deemed the Franklinian as Early Eocene, the Fultonian as Middle Eocene, the Ravenian as Late, and the Kummerian as Early Oligocene. The beginning of the Kummerian was refined by Gregory Retallack et al (2004) as 40 Mya, with a refined end at the Eocene-Oligocene boundary where the younger Angoonian floral stage starts.

Palaeogeography and tectonics
During the Eocene, the continents continued to drift toward their present positions.

At the beginning of the period, Australia and Antarctica remained connected, and warm equatorial currents may have mixed with colder Antarctic waters, distributing the heat around the planet and keeping global temperatures high. When Australia split from the southern continent around 45 Ma, the warm equatorial currents were routed away from Antarctica. An isolated cold water channel developed between the two continents. However, modeling results call into question the thermal isolation model for late Eocene cooling, and decreasing carbon dioxide levels in the atmosphere may have been more important. Once the Antarctic region began to cool down, the ocean surrounding Antarctica began to freeze, sending cold water and icefloes north and reinforcing the cooling.

The northern supercontinent of Laurasia began to fragment, as Europe, Greenland and North America drifted apart.

In western North America, the Laramide Orogeny came to an end in the Eocene, and compression was replaced with crustal extension that ultimately gave rise to the Basin and Range Province. The Kishenehn Basin, around 1.5 km in elevation during the Lutetian, was uplifted to an altitude of 2.5 km by the Priabonian. Huge lakes formed in the high flat basins among uplifts, resulting in the deposition of the Green River Formation lagerstätte.

At about 35 Ma, an asteroid impact on the eastern coast of North America formed the Chesapeake Bay impact crater.

The Tethys Ocean finally closed with the collision of Africa and Eurasia, while the uplift of the Alps isolated its final remnant, the Mediterranean, and created another shallow sea with island archipelagos to the north. Planktonic foraminifera in the northwestern Peri-Tethys are very similar to those of the Tethys in the middle Lutetian but become completely disparate in the Bartonian, indicating biogeographic separation. Though the North Atlantic was opening, a land connection appears to have remained between North America and Europe since the faunas of the two regions are very similar.

Eurasia was separated in three different landmasses 50 Ma; Western Europe, Balkanatolia and Asia. About 40 Ma, Balkanatolia and Asia were connected, while Europe was connected 34 Ma. The Fushun Basin contained large, suboxic lakes known as the paleo-Jijuntun Lakes.

India collided with Asia, folding to initiate formation of the Himalayas. The incipient subcontinent collided with the Kohistan–Ladakh Arc around 50.2 Ma and with Karakoram around 40.4 Ma, with the final collision between Asia and India occurring ~40 Ma.

Climate
The Eocene Epoch contained a wide variety of different climate conditions that includes the warmest climate in the Cenozoic Era, and arguably the warmest time interval since the Permian-Triassic mass extinction and Early Triassic, and ends in an icehouse climate. The evolution of the Eocene climate began with warming after the end of the Paleocene–Eocene Thermal Maximum (PETM) at 56 Ma to a maximum during the Eocene Optimum at around 49 Ma. During this period of time, little to no ice was present on Earth with a smaller difference in temperature from the equator to the poles. Because of this the maximum sea level was 150 meters higher than current levels. Following the maximum was a descent into an icehouse climate from the Eocene Optimum to the Eocene–Oligocene transition at 34 Ma. During this decrease, ice began to reappear at the poles, and the Eocene–Oligocene transition is the period of time when the Antarctic ice sheet began to rapidly expand.

Early Eocene
Greenhouse gases, in particular carbon dioxide and methane, played a significant role during the Eocene in controlling the surface temperature. The end of the PETM was met with very large sequestration of carbon dioxide into the forms of methane clathrate, coal, and crude oil at the bottom of the Arctic Ocean, that reduced the atmospheric carbon dioxide. This event was similar in magnitude to the massive release of greenhouse gasses at the beginning of the PETM, and it is hypothesized that the sequestration was mainly due to organic carbon burial and weathering of silicates. For the early Eocene there is much discussion on how much carbon dioxide was in the atmosphere. This is due to numerous proxies representing different atmospheric carbon dioxide content. For example, diverse geochemical and paleontological proxies indicate that at the maximum of global warmth the atmospheric carbon dioxide values were at 700–900 ppm, while model simulations suggest a concentration of 1,680 ppm fits best with deep sea, sea surface, and near-surface air temperatures of the time. Other proxies such as pedogenic (soil building) carbonate and marine boron isotopes indicate large changes of carbon dioxide of over 2,000 ppm over periods of time of less than 1 million years. This large influx of carbon dioxide could be attributed to volcanic out-gassing due to North Atlantic rifting or oxidation of methane stored in large reservoirs deposited from the PETM event in the sea floor or wetland environments. For contrast, today the carbon dioxide levels are at 400 ppm or 0.04%.

During the early Eocene, methane was another greenhouse gas that had a drastic effect on the climate. Methane has 30 times more of a warming effect than carbon dioxide on a 100-year scale (i.e., methane has a global warming potential of 29.8±11). Most of the methane released to the atmosphere during this period of time would have been from wetlands, swamps, and forests. The atmospheric methane concentration today is 0.000179% or 1.79 ppmv. As a result of the warmer climate and the sea level rise associated with the early Eocene, more wetlands, more forests, and more coal deposits would have been available for methane release. If we compare the early Eocene production of methane to current levels of atmospheric methane, the early Eocene would have produced triple the amount of methane. The warm temperatures during the early Eocene could have increased methane production rates, and methane that is released into the atmosphere would in turn warm the troposphere, cool the stratosphere, and produce water vapor and carbon dioxide through oxidation. Biogenic production of methane produces carbon dioxide and water vapor along with the methane, as well as yielding infrared radiation. The breakdown of methane in an atmosphere containing oxygen produces carbon monoxide, water vapor and infrared radiation. The carbon monoxide is not stable, so it eventually becomes carbon dioxide and in doing so releases yet more infrared radiation. Water vapor traps more infrared than does carbon dioxide. At about the beginning of the Eocene Epoch (55.8–33.9 Ma) the amount of oxygen in the Earth's atmosphere more or less doubled.

During the warming in the early Eocene between 55 and 52 Ma, there were a series of short-term changes of carbon isotope composition in the ocean. These isotope changes occurred due to the release of carbon from the ocean into the atmosphere that led to a temperature increase of 4 – at the surface of the ocean. Recent analysis of and research into these hyperthermals in the early Eocene has led to hypotheses that the hyperthermals are based on orbital parameters, in particular eccentricity and obliquity. The hyperthermals in the early Eocene, notably the Palaeocene–Eocene Thermal Maximum (PETM), the Eocene Thermal Maximum 2 (ETM2), and the Eocene Thermal Maximum 3 (ETM3), were analyzed and found that orbital control may have had a role in triggering the ETM2 and ETM3. An enhancement of the biological pump proved effective at sequestering excess carbon during the recovery phases of these hyperthermals. These hyperthermals led to increased perturbations in planktonic and benthic foraminifera, with a higher rate of fluvial sedimentation as a consequence of the warmer temperatures. Unlike the PETM, the lesser hyperthermals of the Early Eocene had negligible consequences for terrestrial mammals. These Early Eocene hyperthermals produced a sustained period of extremely hot climate known as the Early Eocene Climatic Optimum (EECO). During the early and middle EECO, the superabundance of the euryhaline dinocyst Homotryblium in New Zealand indicates elevated ocean salinity in the region.

Equable climate problem
One of the unique features of the Eocene's climate as mentioned before was the equable and homogeneous climate that existed in the early parts of the Eocene. A multitude of proxies support the presence of a warmer equable climate being present during this period of time. A few of these proxies include the presence of fossils native to warm climates, such as crocodiles, located in the higher latitudes, the presence in the high latitudes of frost-intolerant flora such as palm trees which cannot survive during sustained freezes,  and fossils of snakes found in the tropics that would require much higher average temperatures to sustain them. TEX86 BAYSPAR measurements indicate extremely high sea surface temperatures of 40 C to 45 C at low latitudes, although clumped isotope analyses point to a maximum low latitude sea surface temperature of 36.3 C ± 1.9 C during the EECO. Relative to present-day values, bottom water temperatures are 10 C-change higher according to isotope proxies. With these bottom water temperatures, temperatures in areas where deep water forms near the poles are unable to be much cooler than the bottom water temperatures.

An issue arises, however, when trying to model the Eocene and reproduce the results that are found with the proxy data. Using all different ranges of greenhouse gasses that occurred during the early Eocene, models were unable to produce the warming that was found at the poles and the reduced seasonality that occurs with winters at the poles being substantially warmer. The models, while accurately predicting the tropics, tend to produce significantly cooler temperatures of up to 20 C-change colder than the actual determined temperature at the poles. This error has been classified as the "equable climate problem". To solve this problem, the solution would involve finding a process to warm the poles without warming the tropics. Some hypotheses and tests which attempt to find the process are listed below.

Large lakes
Due to the nature of water as opposed to land, less temperature variability would be present if a large body of water is also present. In an attempt to try to mitigate the cooling polar temperatures, large lakes were proposed to mitigate seasonal climate changes. To replicate this case, a lake was inserted into North America and a climate model was run using varying carbon dioxide levels. The model runs concluded that while the lake did reduce the seasonality of the region greater than just an increase in carbon dioxide, the addition of a large lake was unable to reduce the seasonality to the levels shown by the floral and faunal data.

Ocean heat transport
The transport of heat from the tropics to the poles, much like how ocean heat transport functions in modern times, was considered a possibility for the increased temperature and reduced seasonality for the poles. With the increased sea surface temperatures and the increased temperature of the deep ocean water during the early Eocene, one common hypothesis was that due to these increases there would be a greater transport of heat from the tropics to the poles. Simulating these differences, the models produced lower heat transport due to the lower temperature gradients and were unsuccessful in producing an equable climate from only ocean heat transport.

Orbital parameters
While typically seen as a control on ice growth and seasonality, the orbital parameters were theorized as a possible control on continental temperatures and seasonality. Simulating the Eocene by using an ice free planet, eccentricity, obliquity, and precession were modified in different model runs to determine all the possible different scenarios that could occur and their effects on temperature. One particular case led to warmer winters and cooler summer by up to 30% in the North American continent, and it reduced the seasonal variation of temperature by up to 75%. While orbital parameters did not produce the warming at the poles, the parameters did show a great effect on seasonality and needed to be considered.

Polar stratospheric clouds
Another method considered for producing the warm polar temperatures were polar stratospheric clouds. Polar stratospheric clouds are clouds that occur in the lower stratosphere at very low temperatures. Polar stratospheric clouds have a great impact on radiative forcing. Due to their minimal albedo properties and their optical thickness, polar stratospheric clouds act similar to a greenhouse gas and traps outgoing longwave radiation. Different types of polar stratospheric clouds occur in the atmosphere: polar stratospheric clouds that are created due to interactions with nitric or sulfuric acid and water (Type I) or polar stratospheric clouds that are created with only water ice (Type II).

Methane is an important factor in the creation of the primary Type II polar stratospheric clouds that were created in the early Eocene. Since water vapor is the only supporting substance used in Type II polar stratospheric clouds, the presence of water vapor in the lower stratosphere is necessary where in most situations the presence of water vapor in the lower stratosphere is rare. When methane is oxidized, a significant amount of water vapor is released. Another requirement for polar stratospheric clouds is cold temperatures to ensure condensation and cloud production. Polar stratospheric cloud production, since it requires the cold temperatures, is usually limited to nighttime and winter conditions. With this combination of wetter and colder conditions in the lower stratosphere, polar stratospheric clouds could have formed over wide areas in Polar Regions.

To test the polar stratospheric clouds effects on the Eocene climate, models were run comparing the effects of polar stratospheric clouds at the poles to an increase in atmospheric carbon dioxide. The polar stratospheric clouds had a warming effect on the poles, increasing temperatures by up to 20 °C in the winter months. A multitude of feedbacks also occurred in the models due to the polar stratospheric clouds' presence. Any ice growth was slowed immensely and would lead to any present ice melting. Only the poles were affected with the change in temperature and the tropics were unaffected, which with an increase in atmospheric carbon dioxide would also cause the tropics to increase in temperature. Due to the warming of the troposphere from the increased greenhouse effect of the polar stratospheric clouds, the stratosphere would cool and would potentially increase the amount of polar stratospheric clouds.

While the polar stratospheric clouds could explain the reduction of the equator to pole temperature gradient and the increased temperatures at the poles during the early Eocene, there are a few drawbacks to maintaining polar stratospheric clouds for an extended period of time. Separate model runs were used to determine the sustainability of the polar stratospheric clouds. It was determined that in order to maintain the lower stratospheric water vapor, methane would need to be continually released and sustained. In addition, the amounts of ice and condensation nuclei would need to be high in order for the polar stratospheric cloud to sustain itself and eventually expand.

Middle Eocene
The Eocene is not only known for containing the warmest period during the Cenozoic; it also marked the decline into an icehouse climate and the rapid expansion of the Antarctic ice sheet. The transition from a warming climate into a cooling climate began at around 49 Ma. Isotopes of carbon and oxygen indicate a shift to a global cooling climate. The cause of the cooling has been attributed to a significant decrease of >2,000 ppm in atmospheric carbon dioxide concentrations. One proposed cause of the reduction in carbon dioxide during the warming to cooling transition was the azolla event. With the equable climate during the early Eocene, warm temperatures in the arctic allowed for the growth of azolla, which is a floating aquatic fern, on the Arctic Ocean. The significantly high amounts of carbon dioxide also acted to facilitate azolla blooms across the Arctic Ocean. Compared to current carbon dioxide levels, these azolla grew rapidly in the enhanced carbon dioxide levels found in the early Eocene. The isolation of the Arctic Ocean, evidenced by euxinia that occurred at this time, led to stagnant waters and as the azolla sank to the sea floor, they became part of the sediments on the seabed and effectively sequestered the carbon by locking it out of the atmosphere for good. The ability for the azolla to sequester carbon is exceptional, and the enhanced burial of azolla could have had a significant effect on the world atmospheric carbon content and may have been the event to begin the transition into an ice house climate. The azolla event could have led to a draw down of atmospheric carbon dioxide of up to 470 ppm. Assuming the carbon dioxide concentrations were at 900 ppmv prior to the Azolla Event they would have dropped to 430 ppmv, or 30 ppmv more than they are today, after the Azolla Event. This cooling trend at the end of the EECO has also been proposed to have been caused by increased siliceous plankton productivity and marine carbon burial, which also helped draw carbon dioxide out of the atmosphere. Cooling after this event, part of a trend known as the Middle-Late Eocene Cooling (MLEC), continued due to continual decrease in atmospheric carbon dioxide from organic productivity and weathering from mountain building. Many regions of the world became more arid and cold over the course of the stage, such as the Fushun Basin. In East Asia, lake level changes were in sync with global sea level changes over the course of the MLEC.

Global cooling continued until there was a major reversal from cooling to warming in the Bartonian. This warming event, signifying a sudden and temporary reversal of the cooling conditions, is known as the Middle Eocene Climatic Optimum (MECO). At around 41.5 Ma, stable isotopic analysis of samples from Southern Ocean drilling sites indicated a warming event for 600,000 years. A similar shift in carbon isotopes is known from the Northern Hemisphere in the Scaglia Limestones of Italy. Oxygen isotope analysis showed a large negative change in the proportion of heavier oxygen isotopes to lighter oxygen isotopes, which indicates an increase in global temperatures. The warming is considered to be primarily due to carbon dioxide increases, because carbon isotope signatures rule out major methane release during this short-term warming. A sharp increase in atmospheric carbon dioxide was observed with a maximum of 4,000 ppm: the highest amount of atmospheric carbon dioxide detected during the Eocene. Other studies suggest a more modest rise in carbon dioxide levels. The increase in atmospheric carbon dioxide has also been hypothesised to have been driven by increased seafloor spreading rates and metamorphic decarbonation reactions between Australia and Antarctica and increased amounts of volcanism in the region. One possible cause of atmospheric carbon dioxide increase could have been a sudden increase due to metamorphic release due to continental drift and collision of India with Asia and the resulting formation of the Himalayas; however, data on the exact timing of metamorphic release of atmospheric carbon dioxide is not well resolved in the data. Recent studies have mentioned, however, that the removal of the ocean between Asia and India could have released significant amounts of carbon dioxide. Another hypothesis still implicates a diminished negative feedback of silicate weathering as a result of continental rocks having become less weatherable during the warm Early and Middle Eocene, allowing volcanically released carbon dioxide to persist in the atmosphere for longer. Yet another explanation hypothesises that MECO warming was caused by the simultaneous occurrence of minima in both the 400 kyr and 2.4 Myr eccentricity cycles. During the MECO, sea surface temperatures in the Tethys Ocean jumped to 32–36 °C, and Tethyan seawater became more dysoxic. A decline in carbonate accumulation at ocean depths of greater than three kilometres took place synchronously with the peak of the MECO, signifying ocean acidification took place in the deep ocean. On top of that, MECO warming caused an increase in the respiration rates of pelagic heterotrophs, leading to a decreased proportion of primary productivity making its way down to the seafloor and causing a corresponding decline in populations of benthic foraminifera. An abrupt decrease in lakewater salinity in western North America occurred during this warming interval. This warming is short lived, as benthic oxygen isotope records indicate a return to cooling at ~40 Ma.

Late Eocene
At the end of the MECO, the MLEC resumed. Cooling and the carbon dioxide drawdown continued through the late Eocene and into the Eocene–Oligocene transition around 34 Ma. The post-MECO cooling brought with it a major aridification trend in Asia, enhanced by retreating seas. The cooling during the initial stages of the opening of the Drake Passage ~38.5 Ma was not global, as evidenced by an absence of cooling in the North Atlantic. During the cooling period, benthic oxygen isotopes show the possibility of ice creation and ice increase during this later cooling. The end of the Eocene and beginning of the Oligocene is marked with the massive expansion of area of the Antarctic ice sheet that was a major step into the icehouse climate. Multiple proxies, such as oxygen isotopes and alkenones, indicate that at the Eocene–Oligocene transition, the atmospheric carbon dioxide concentration had decreased to around 750–800 ppm, approximately twice that of present levels. Along with the decrease of atmospheric carbon dioxide reducing the global temperature, orbital factors in ice creation can be seen with 100,000-year and 400,000-year fluctuations in benthic oxygen isotope records. Another major contribution to the expansion of the ice sheet was the creation of the Antarctic Circumpolar Current. The creation of the Antarctic circumpolar current would isolate the cold water around the Antarctic, which would reduce heat transport to the Antarctic along with creating ocean gyres that result in the upwelling of colder bottom waters. The issue with this hypothesis of the consideration of this being a factor for the Eocene-Oligocene transition is the timing of the creation of the circulation is uncertain. For Drake Passage, sediments indicate the opening occurred ~41 Ma while tectonics indicate that this occurred ~32 Ma. Solar activity did not change significantly during the greenhouse-icehouse transition across the Eocene-Oligocene boundary.

Flora
During the early-middle Eocene, forests covered most of the Earth including the poles. Tropical forests extended across much of modern Africa, South America, Central America, India, South-east Asia and China. Paratropical forests grew over North America, Europe and Russia, with broad-leafed evergreen and broad-leafed deciduous forests at higher latitudes.

Polar forests were quite extensive. Fossils and even preserved remains of trees such as swamp cypress and dawn redwood from the Eocene have been found on Ellesmere Island in the Arctic. Even at that time, Ellesmere Island was only a few degrees in latitude further south than it is today. Fossils of subtropical and even tropical trees and plants from the Eocene also have been found in Greenland and Alaska. Tropical rainforests grew as far north as northern North America and Europe.

Palm trees were growing as far north as Alaska and northern Europe during the early Eocene, although they became less abundant as the climate cooled. Dawn redwoods were far more extensive as well.

The earliest definitive Eucalyptus fossils were dated from 51.9 Ma, and were found in the Laguna del Hunco deposit in Chubut province in Argentina.

Cooling began mid-period, and by the end of the Eocene continental interiors had begun to dry, with forests thinning considerably in some areas. The newly evolved grasses were still confined to river banks and lake shores, and had not yet expanded into plains and savannas.

The cooling also brought seasonal changes. Deciduous trees, better able to cope with large temperature changes, began to overtake evergreen tropical species. By the end of the period, deciduous forests covered large parts of the northern continents, including North America, Eurasia and the Arctic, and rainforests held on only in equatorial South America, Africa, India and Australia.

Antarctica began the Eocene fringed with a warm temperate to sub-tropical rainforest. Pollen found in Prydz Bay from the Eocene suggest taiga forest existed there. It became much colder as the period progressed; the heat-loving tropical flora was wiped out, and by the beginning of the Oligocene, the continent hosted deciduous forests and vast stretches of tundra.

Fauna
During the Eocene, plants and marine faunas became quite modern. Many modern bird orders first appeared in the Eocene. The Eocene oceans were warm and teeming with fish and other sea life.

Mammals

The oldest known fossils of most of the modern mammal orders appear within a brief period during the early Eocene. At the beginning of the Eocene, several new mammal groups arrived in North America. These modern mammals, like artiodactyls, perissodactyls, and primates, had features like long, thin legs, feet, and hands capable of grasping, as well as differentiated teeth adapted for chewing. Dwarf forms reigned. All the members of the new mammal orders were small, under 10 kg; based on comparisons of tooth size, Eocene mammals were only 60% of the size of the primitive Palaeocene mammals that preceded them. They were also smaller than the mammals that followed them. It is assumed that the hot Eocene temperatures favored smaller animals that were better able to manage the heat.

Both groups of modern ungulates (hoofed animals) became prevalent because of a major radiation between Europe and North America, along with carnivorous ungulates like Mesonyx. Early forms of many other modern mammalian orders appeared, including horses (most notably the Eohippus), bats, proboscidians (elephants), primates, and rodents. Older primitive forms of mammals declined in variety and importance. Important Eocene land fauna fossil remains have been found in western North America, Europe, Patagonia, Egypt, and southeast Asia. Marine fauna are best known from South Asia and the southeast United States.

Established megafauna of the Eocene include the Uintatherium, Arsinoitherium, and brontotheres, in which the former two, unlike the latter, did not belong to ungulates but groups that became extinct shortly after their establishments.

Large terrestrial mammalian predators had already existed since the Paleocene, but new forms now arose like Hyaenodon and Daphoenus (the earliest lineage of a once-successful predatory family known as bear dogs). Entelodonts meanwhile established themselves as some of the largest omnivores. The first nimravids, including Dinictis, established themselves as amongst the first feliforms to appear. Their groups became highly successful and continued to live past the Eocene.

Basilosaurus is a very well-known Eocene whale, but whales as a group had become very diverse during the Eocene, which is when the major transitions from being terrestrial to fully aquatic in cetaceans occurred. The first sirenians were evolving at this time, and would eventually evolve into the extant manatees and dugongs.

It is thought that millions of years after the Cretaceous-Paleogene extinction event, brain sizes of mammals now started to increase, "likely driven by a need for greater cognition in increasingly complex environments".

Birds
Eocene birds include some enigmatic groups with resemblances to modern forms, some of which continued from the Paleocene. Bird taxa of the Eocene include carnivorous psittaciforms, such as Messelasturidae, Halcyornithidae, large flightless forms such as Gastornis and Eleutherornis, long legged falcon Masillaraptor, ancient galliforms such as Gallinuloides, putative rail relatives of the family Songziidae, various pseudotooth birds such as Gigantornis, the ibis relative Rhynchaeites, primitive swifts of the genus Aegialornis, and primitive penguins such as Archaeospheniscus and Inkayacu.

Reptiles
Reptile fossils from this time, such as fossils of pythons and turtles, are abundant.

Insects and arachnids
Several rich fossil insect faunas are known from the Eocene, notably the Baltic amber found mainly along the south coast of the Baltic Sea, amber from the Paris Basin, France, the Fur Formation, Denmark, and the Bembridge Marls from the Isle of Wight, England. Insects found in Eocene deposits mostly belong to genera that exist today, though their range has often shifted since the Eocene. For instance the bibionid genus Plecia is common in fossil faunas from presently temperate areas, but only lives in the tropics and subtropics today.

title: Eocene