Philosophy chapter
QMRQuadrants and octants[edit]
Main articles: Octant (solid geometry) and Quadrant (plane geometry)
The four quadrants of a Cartesian coordinate system.
The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are +,+), II (−,+), III (−,−), and IV (+,−). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("north-east") quadrant.
Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or octants, according to the signs of the coordinates of the points. The convention used for naming a specific octant is to list its signs, e.g. (+ + +) or (− + −). The generalization of the quadrant and octant to an arbitrary number of dimensions is the orthant, and a similar naming system applies.
QMROrthogonal coordinates
From Wikipedia, the free encyclopedia
In mathematics, orthogonal coordinates are defined as a set of d coordinates q = (q1, q2, ..., qd) in which the coordinate surfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant. For example, the three-dimensional Cartesian coordinates (x, y, z) is an orthogonal coordinate system, since its coordinate surfaces x = constant, y = constant, and z = constant are planes that meet at right angles to one another, i.e., are perpendicular. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates.
orthogonal coordinates are quadrants
Motivation[edit]
A conformal map acting on a rectangular grid. Note that the orthogonality of the curved grid is retained.
While vector operations and physical laws are normally easiest to derive in Cartesian coordinates, non-Cartesian orthogonal coordinates are often used instead for the solution of various problems, especially boundary value problems, such as those arising in field theories of quantum mechanics, fluid flow, electrodynamics and the diffusion of chemical species or heat.
The chief advantage of non-Cartesian coordinates is that they can be chosen to match the symmetry of the problem. For example, the pressure wave due to an explosion far from the ground (or other barriers) depends on 3D space in Cartesian coordinates, however the pressure predominantly moves away from the center, so that in spherical coordinates the problem becomes very nearly one-dimensional (since the pressure wave dominantly depends only on time and the distance from the center). Another example is (slow) fluid in a straight circular pipe: in Cartesian coordinates, one has to solve a (difficult) two dimensional boundary value problem involving a partial differential equation, but in cylindrical coordinates the problem becomes one-dimensional with an ordinary differential equation instead of a partial differential equation.
The reason to prefer orthogonal coordinates instead of general curvilinear coordinates is simplicity: many complications arise when coordinates are not orthogonal. For example, in orthogonal coordinates many problems may be solved by separation of variables. Separation of variables is a mathematical technique that converts a complex d-dimensional problem into d one-dimensional problems that can be solved in terms of known functions. Many equations can be reduced to Laplace's equation or the Helmholtz equation. Laplace's equation is separable in 13 orthogonal coordinate systems, and the Helmholtz equation is separable in 11 orthogonal coordinate systems.[1][2]
Orthogonal coordinates never have off-diagonal terms in their metric tensor. In other words, the infinitesimal squared distance ds2 can always be written as a scaled sum of the squared infinitesimal coordinate displacements
QMRDifferential operators in three dimensions[edit]
Main article: del
Since these operations are common in application, all vector components in this section are presented with respect to the normalised basis: F_i = \mathbf{F} \cdot \hat{\mathbf{e}}_i.
Operator Expression
Gradient of a scalar field
\nabla \phi =
\frac{\hat{ \mathbf e}_1}{h_1} \frac{\partial \phi}{\partial q^1} +
\frac{\hat{ \mathbf e}_2}{h_2} \frac{\partial \phi}{\partial q^2} +
\frac{\hat{ \mathbf e}_3}{h_3} \frac{\partial \phi}{\partial q^3}
Divergence of a vector field
\nabla \cdot \mathbf F =
\frac{1}{h_1 h_2 h_3}
\left[
\frac{\partial}{\partial q^1} \left( F_1 h_2 h_3 \right) +
\frac{\partial}{\partial q^2} \left( F_2 h_3 h_1 \right) +
\frac{\partial}{\partial q^3} \left( F_3 h_1 h_2 \right)
\right]
Curl of a vector field
\begin{align}
\nabla \times \mathbf F & =
\frac{\hat{ \mathbf e}_1}{h_2 h_3}
\left[
\frac{\partial}{\partial q^2} \left( h_3 F_3 \right) -
\frac{\partial}{\partial q^3} \left( h_2 F_2 \right)
\right] +
\frac{\hat{ \mathbf e}_2}{h_3 h_1}
\left[
\frac{\partial}{\partial q^3} \left( h_1 F_1 \right) -
\frac{\partial}{\partial q^1} \left( h_3 F_3 \right)
\right] \\[10pt]
& + \frac{\hat{ \mathbf e}_3}{h_1 h_2}
\left[
\frac{\partial}{\partial q^1} \left( h_2 F_2 \right) -
\frac{\partial}{\partial q^2} \left( h_1 F_1 \right)
\right]
=\frac{1}{h_1 h_2 h_3}
\begin{vmatrix}
h_1\hat{\mathbf{e}}_1 & h_2\hat{\mathbf{e}}_2 & h_3\hat{\mathbf{e}}_3 \\
\dfrac{\partial}{\partial q^1} & \dfrac{\partial}{\partial q^2} & \dfrac{\partial}{\partial q^3} \\
h_1 F_1 & h_2 F_2 & h_3 F_3
\end{vmatrix}
\end{align}
Laplacian of a scalar field
\nabla^2 \phi = \frac{1}{h_1 h_2 h_3}
\left[
\frac{\partial}{\partial q^1} \left( \frac{h_2 h_3}{h_1} \frac{\partial \phi}{\partial q^1} \right) +
\frac{\partial}{\partial q^2} \left( \frac{h_3 h_1}{h_2} \frac{\partial \phi}{\partial q^2} \right) +
\frac{\partial}{\partial q^3} \left( \frac{h_1 h_2}{h_3} \frac{\partial \phi}{\partial q^3} \right)
\right]
QMRIn classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. The numerical output, however, might also be a vector or a shape. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.
Analytic geometry takes place in a quadrant grid
QMRThe multiplication sign ×
(HTML entity is ×)
Multiplication creates quadrants. For instance 5 x 6 creates a 30 quadrant grid. The grid is a rectangle with side 5 and 6
4 × 5 = 20, the rectangle is composed of 20 squares, having dimensions of 4 by 5.
X is the quadrant
QMRA division table is a matrix reference for simple integer division. The references show some examples. It is a quadrant grid
QMROctant (solid geometry)
From Wikipedia, the free encyclopedia
Three axial planes (x=0, y=0, z=0) divide space into eight equal octant domains, each with a coordinate signs from (-,-,-) to (+,+,+).
An octant in solid geometry is one of the eight divisions of a Euclidean three-dimensional coordinate system defined by the signs of the coordinates. It is similar to the two-dimensional quadrant and the one-dimensional ray.[1]
The generalization of an octant is called orthant.
An octant is made up of quadrants
Naming and Numbering[edit]
For z > 0, the octants have the same numbers as the corresponding quadrants in the plane.
A convention for naming an octant is to give its list of signs, e.g. ( + - - ) or ( - + - ). Octant ( + + + ) is sometimes referred to as the first octant, although similar ordinal name descriptors are not defined for the other seven octants. The advantages of using the ( + - - ) notation are its unambiguousness, and extensibility for higher dimensions.
Number Name x y z Octal (+=0,zyx) Octal (+=1,zyx)
I top-front-right + + + 0 7
II top-back-right − + + 1 6
III top-back-left − − + 3 4
IV top-front-left + − + 2 5
V bottom-front-right + + − 4 3
VI bottom-back-right − + − 5 2
VII bottom-back-left − − − 7 0
VIII bottom-front-left + − − 6 1
In plane geometry, octant is a circular sector obtained by division of the full circle into 8 equal parts, hence the name. It is a sector with the central angle of 45°.
Usage[edit]
Most commonly, octants are seen on the compass rose.
Traditionally wind direction is given as one of the 8 octants (N, NE, E, SE, S, SW, W, NW) because that is more precise than merely giving one of the 4 quadrants, and the wind vane typically does not have enough accuracy to bother with more precise indication.
The name of the instrument "octant" comes from the fact that it is based on 1/8th of the cirle.
It is made up of two quadrants overlapping
QMRThe octant, also called reflecting quadrant, is a measuring instrument used primarily in navigation. It is a type of reflecting instrument.
QMRFor continuous functions f on the whole plane which are smooth in Ω and the complementary region Ωc, the first derivative can have a jump across the boundary of Ω. The value of the normal derivative at a boundary point can be computed from inside or outside Ω. The interior normal derivative will be denoted by ∂n− and the exterior normal derivative by ∂n+. With this terminology the four basic problems of classical potential theory are as follows:[3]
Interior Dirichlet problem: ∆u = 0 in Ω, u = f on ∂Ω
Interior Neumann problem: ∆u = 0 in Ω, ∂n− u = f on ∂Ω
Exterior Dirichlet problem: ∆u = 0 in Ωc, u = f on ∂Ω, u continuous at ∞
Exterior Neumann problem: ∆u = 0 in Ωc, ∂n+ u = f on ∂Ω, u continuous at ∞
QMRA geographic coordinate system is a coordinate system that enables every location on the Earth to be specified by a set of numbers or letters, or symbols.[n 1] The coordinates are often chosen such that one of the numbers represents vertical position, and two or three of the numbers represent horizontal position. A common choice of coordinates is latitude, longitude and elevation.[1]
To specify a location on a two-dimensional map requires a map projection.[2]
It is based on attitude and longitude lines that create quadrants
QMRFor two thousand years, many attempts were made to prove the parallel postulate using Euclid's first four postulates. The main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate isn't self-evident. If the order the postulates were listed in the Elements is significant, it indicates that Euclid included this postulate only when he realised he could not prove it or proceed without it.[7] Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods of time until the mistake was found. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate (Playfair's axiom). Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by his own axiom.
Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However he did give a postulate which is equivalent to the fifth postulate.
Ibn al-Haytham (Alhazen) (965-1039), an Arab mathematician, made an attempt at proving the parallel postulate using a proof by contradiction,[8] in the course of which he introduced the concept of motion and transformation into geometry.[9] He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral",[10] and his attempted proof contains elements similar to those found in Lambert quadrilaterals and Playfair's axiom.[11]
Omar Khayyám (1050–1123), a Persian, attempted to prove the fifth postulate from another explicitly given postulate (based on the fourth of the five principles due to the Philosopher (Aristotle), namely, "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge."[12] He derived some of the earlier results belonging to elliptical geometry and hyperbolic geometry, though his postulate excluded the latter possibility.[13] The Saccheri quadrilateral was also first considered by Omar Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.[10] Unlike many commentators on Euclid before and after him (including Giovanni Girolamo Saccheri), Khayyám was not trying to prove the parallel postulate as such but to derive it from his equivalent postulate. He recognized that three possibilities arose from omitting Euclid's fifth postulate; if two perpendiculars to one line cross another line, judicious choice of the last can make the internal angles where it meets the two perpendiculars equal (it is then parallel to the first line). If those equal internal angles are right angles, we get Euclid's fifth postulate, otherwise, they must be either acute or obtuse. He showed that the acute and obtuse cases led to contradictions using his postulate, but his postulate is now known to be equivalent to the fifth postulate.
Nasir al-Din al-Tusi (1201–1274), in his Al-risala al-shafiya'an al-shakk fi'l-khutut al-mutawaziya (Discussion Which Removes Doubt about Parallel Lines) (1250), wrote detailed critiques of the parallel postulate and on Khayyám's attempted proof a century earlier. Nasir al-Din attempted to derive a proof by contradiction of the parallel postulate.[14] He also considered the cases of what are now known as elliptical and hyperbolic geometry, though he ruled out both of them. The fifth is always questionable
The fourth square is always different and transcendent, yet encompasses the previous three. The fourth square is about right angles in Euclid's postulates. Right angles are quadrants.
QMRSaccheri quadrilateral
From Wikipedia, the free encyclopedia
Saccheri quadrilaterals
A Saccheri quadrilateral (also known as a Khayyam–Saccheri quadrilateral) is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book Euclides ab omni naevo vindicatus (literally Euclid Freed of Every Flaw) first published in 1733, an attempt to prove the parallel postulate using the method Reductio ad absurdum.
The first known consideration of the Saccheri quadrilateral was by Omar Khayyam in the late 11th century, and it may occasionally be referred to as the Khayyam-Saccheri quadrilateral.[1]
For a Saccheri quadrilateral ABCD, the sides AD and BC (also called the legs) are equal in length and perpendicular to the base AB. The top CD is the summit or upper base and the angles at C and D are called the summit angles.
The advantage of using Saccheri quadrilaterals when considering the parallel postulate is that they place the mutually exclusive options in very clear terms:
Are the summit angles right angles, obtuse angles, or acute angles?
As it turns out:
when the summit angles are right angles, the existence of this quadrilateral is equivalent to the statement expounded by Euclid's fifth postulate.
When the summit angles are acute, this quadrilateral leads to hyperbolic geometry, and
when the summit angles are obtuse, the quadrilateral leads to elliptical or spherical geometry(provided that also some other modifications are made to the postulates[2]).
Saccheri himself, however, thought that both the obtuse and acute cases could be shown to be contradictory. He did show that the obtuse case was contradictory, but failed to properly handle the acute case.[3]
History[edit]
Saccheri quadrilaterals were first considered by Omar Khayyam (1048-1131) in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.[1] Unlike many commentators on Euclid before and after him (including of course Saccheri), Khayyam was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle):
Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge.[4]
Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid.
It was not until 600 years later that Giordano Vitale made an advance on Khayyam in his book Euclide restituo (1680, 1686), when he used the quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant.
Saccheri himself based the whole of his long, heroic, and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way.
Saccheri quadrilaterals in Hyperbolic geometry[edit]
Let ABCD be a Saccheri quadrilateral having AB as base, CA and DB the equal sides that are perpendicular to the base and CD the summit. The following properties are valid in any Saccheri quadrilateral in hyperbolic geometry.[5]
The summit angles (at C and D) are equal and acute.
The summit is longer than the base.
The line segment joining the midpoint of the base and the midpoint of the summit is mutually perpendicular to the base and summit.
The line segment joining the midpoints of the sides is not perpendicular to either side.
The above two line segments are perpendicular to each other.
The line segment joining the midpoint of the base and the midpoint of the summit divides the Saccheri quadrilateral into two Lambert quadrilaterals.
Two Saccheri quadrilaterals with congruent bases and congruent summit angles are congruent (i.e., the remaining pairs of corresponding parts are congruent).
Two Saccheri quadrilaterals with congruent summits and congruent summit angles are congruent.
Equations[edit]
In the hyperbolic plane of constant curvature -1, the summit s of a Saccheri quadrilateral can be calculated from the leg l and the base b using the formula
\cosh s = \cosh b \cdot \cosh^2 l - \sinh^2 l.[6][dubious – discuss]
\sinh \left( \frac{s}{2} \right) = \cosh\left( l \right) \sinh\left( \frac{b}{2} \right) [7]
Tilings in the Poincaré disk model[edit]
Tilings of the Poincaré disk model of the Hyperbolic plane exist having Saccheri quadrilaterals as fundamental domains. Besides the 2 right angles, these quadrilaterals have acute summit angles. The tilings exhibit a *nn22 symmetry (orbifold notation), and include:
QMRLambert quadrilateral
From Wikipedia, the free encyclopedia
A Lambert quadrilateral
In geometry, a Lambert quadrilateral,[1] named after Johann Heinrich Lambert, is a quadrilateral three of whose angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel postulate could be proved as a theorem. It is now known that the type of the fourth angle depends upon the geometry in which the quadrilateral exists. In hyperbolic geometry the fourth angle is acute, in Euclidean geometry it is a right angle and in elliptic geometry it is an obtuse angle.
A Lambert quadrilateral can be constructed from a Saccheri quadrilateral by joining the midpoints of the base and summit of the Saccheri quadrilateral. This line segment is perpendicular to both the base and summit and so either half of the Saccheri quadrilateral is a Lambert quadrilateral.Lambert quadrilateral
From Wikipedia, the free encyclopedia
A Lambert quadrilateral
In geometry, a Lambert quadrilateral,[1] named after Johann Heinrich Lambert, is a quadrilateral three of whose angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel postulate could be proved as a theorem. It is now known that the type of the fourth angle depends upon the geometry in which the quadrilateral exists. In hyperbolic geometry the fourth angle is acute, in Euclidean geometry it is a right angle and in elliptic geometry it is an obtuse angle.
A Lambert quadrilateral can be constructed from a Saccheri quadrilateral by joining the midpoints of the base and summit of the Saccheri quadrilateral. This line segment is perpendicular to both the base and summit and so either half of the Saccheri quadrilateral is a Lambert quadrilateral.
Lambert quadrilateral in hyperbolic geometry[edit]
A Lambert quadrilateral, angle C is an acute angle
In hyperbolic geometry a Lambert quadrilateral ABCD where the angles \angle CAB , \angle ABD , \angle BDC are right, and B is opposite C and the curvature = -1 (as in the figure) the following relations hold:[2]
\sinh DC = \sinh AB \cosh AC
\tanh DC = \cosh BD \tanh AB
\sin \angle CBD = \frac {\cosh AB}{ \cosh DC} = \frac {\cosh BD}{ \cosh AC }
\cos \angle CBD = \sinh BD \sinh AB = \tanh DC \tanh AC
\cot \angle CBD = \tanh BD \sinh DC = \tanh AB \sinh AC
Where \tanh , \cosh , \sinh are hyperbolic functions
QMRWhile Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century.
The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited number of assumptions (23 definitions, five common notions, and five postulates) and sought to prove all the other results (propositions) in the work. The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate," or simply the "parallel postulate", which in Euclid's original formulation is:
If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclid's other postulates:
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.
For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other four. Many attempted to find a proof by contradiction, including Ibn al-Haytham (Alhazen, 11th century),[1] Omar Khayyám (12th century), Nasīr al-Dīn al-Tūsī (13th century), and Giovanni Girolamo Saccheri (18th century).
The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were "the first few theorems of the hyperbolic and the elliptic geometries." These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the later development of non-Euclidean geometry. These early attempts at challenging the fifth postulate had a considerable influence on its development among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis and Saccheri.[2] All of these early attempts made at trying to formulate non-Euclidean geometry however provided flawed proofs of the parallel postulate, containing assumptions that were essentially equivalent to the parallel postulate. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries.
Khayyam, for example, tried to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle): "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge."[3] Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he correctly refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid which he didn't realize was equivalent to his own postulate. Another example is al-Tusi's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), who wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented another hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements."[4][5] His work was published in Rome in 1594 and was studied by European geometers, including Saccheri[4] who criticised this work as well as that of Wallis.[6]
Giordano Vitale, in his book Euclide restituo (1680, 1686), used the Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant.
In a work titled Euclides ab Omni Naevo Vindicatus (Euclid Freed from All Flaws), published in 1733, Saccheri quickly discarded elliptic geometry as a possibility (some others of Euclid's axioms must be modified for elliptic geometry to work) and set to work proving a great number of results in hyperbolic geometry.
He finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have been based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, but did not realize it.
In 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral, a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius. He did not carry this idea any further.[7]
At this time it was widely believed that the universe worked according to the principles of Euclidean geometry.[8]
Discovery of non-Euclidean geometry[edit]
The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Circa 1813, Carl Friedrich Gauss and independently around 1818, the German professor of law Ferdinand Karl Schweikart[9] had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Then, around 1830, the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky separately published treatises on hyperbolic geometry. Consequently, hyperbolic geometry is called Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before,[10] though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter k. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences.
Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature. He constructed an infinite family of geometries which are not Euclidean by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. The simplest of these is called elliptic geometry and it is considered to be a non-Euclidean geometry due to its lack of parallel lines.[11]
By formulating the geometry in terms of a curvature tensor, Riemann allowed non-Euclidean geometry to be applied to higher dimensions.
QMRIn geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate):
In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.
It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry[1] and was named after the Scottish mathematician John Playfair. 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. The statement is often written with the phrase, "there is one and only one parallel". In Euclid's Elements, two lines are said to be parallel if they never meet and other characterizations of parallel lines are not used.[2][3]
This axiom is used not only in Euclidean geometry but also in the broader study of affine geometry where the concept of parallelism is central. In the affine geometry setting, the stronger form of Playfair's axiom (where "at most" is replaced by "one and only one") is needed since the axioms of neutral geometry are not present to provide a proof of existence. Playfair's version of the axiom has become so popular that it is often referred to as Euclid's parallel axiom,[4] even though it was not Euclid's version of the axiom.
QMREuclid's fifth postulate implies Playfair's postulate[edit]
The easiest way to show this is using the Euclidean theorem (equivalent to the fifth postulate) that states that the angles of a triangle sum to two right angles. Given a line \ell and a point P not on that line, construct a line, t, perpendicular to the given one through the point P, and then a perpendicular to this perpendicular at the point P. This line is parallel because it cannot meet \ell and form a triangle.[12] Now it can be seen that no other parallels exist. If n was a second line through P, then n makes an acute angle with t (since it is not the perpendicular) and the hypothesis of the fifth postulate holds, and so, n meets \ell.[13]
Playfair's implies Euclid's fifth postulate[edit]
Given that Playfair's postulate implies that only the perpendicular to the perpendicular is a parallel, the lines of the Euclid construction will have to cut each other in a point. It is also necessary to prove that they will do it in the side where the angles sum to less than two right angles, but this is more difficult.[14]
QMRAngles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").
An angle equal to
1
/
4
turn (90° or
π
/
2
radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular.
Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").
An angle equal to
1
/
2
turn (180° or π radians) is called a straight angle.
Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles.
An angle equal to 1 turn (360° or 2π radians) is called a full angle, complete angle, or a perigon.
Angles that are not right angles or a multiple of a right angle are called oblique angles.
Right angles are quadrant angles
QMRVertical and adjacent angle pairs[edit]
Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles.
When two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other.
A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles. They are abbreviated as vert. opp. ∠s.[5]
The equality of vertically opposite angles is called the vertical angle theorem. Eudemus of Rhodes attributed the proof to Thales of Miletus.[6][7] The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical Note,[7] when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: all straight angles are equal, equals added to equals are equal, and equals subtracted from equals are equal.
In the figure, assume the measure of Angle A = x. When two adjacent angles form a straight line, they are supplementary. Therefore, the measure of Angle C = 180 − x. Similarly, the measure of Angle D = 180 − x. Both Angle C and Angle D have measures equal to 180 − x and are congruent. Since Angle B is supplementary to both Angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either Angle C or Angle D we find the measure of Angle B = 180 − (180 − x) = 180 − 180 + x = x. Therefore, both Angle A and Angle B have measures equal to x and are equal in measure.
Angles A and B are adjacent.
Adjacent angles, often abbreviated as adj. ∠s, are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles that are side by side, or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle or full angle are special and are respectively called complementary, supplementary and explementary angles (see "Combine angle pairs" below).
A transversal is a line that intersects a pair of (often parallel) lines and is associated with alternate interior angles, corresponding angles, interior angles, and exterior angles.[8]
Xs are like quadrants, with four parts
Complementary angles are angle pairs whose measures sum to one right angle (
1
/
4
turn, 90°, or
π
/
2
radians). If the two complementary angles are adjacent their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary, because the sum of internal angles of a triangle is 180 degrees, and the right angle itself accounts for ninety degrees.
The adjective complementary is from Latin complementum, associated with the verb complere, "to fill up". An acute angle is "filled up" by its complement to form a right angle.
The difference between an angle and a right angle is termed the complement of the angle.
If angles A and B are complementary, the following relationships hold:
\sin^2A + \sin^2B = 1. \quad \cos^2A + \cos^2B = 1.
\tan A = \cot B. \quad \quad \quad \sec A = \csc B.
(The tangent of an angle equals the cotangent of its complement and its secant equals the cosecant of its complement.)
The prefix "co-" in the names of some trigonometric ratios refers to the word "complementary".
Two angles that sum to a straight angle (
1
/
2
turn, 180°, or π radians) are called supplementary angles.
If the two supplementary angles are adjacent (i.e. have a common vertex and share just one side), their non-shared sides form a straight line. Such angles are called a linear pair of angles.[9] However, supplementary angles do not have to be on the same line, and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary.
If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.
The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.
In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third, because the sum of internal angles of a triangle is a straight angle.
Two angles that sum to a complete angle (1 turn, 360°, or 2π radians) are called explementary angles or conjugate angles.
The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle.
90 degrees is the angle of a quadrant
QMRUnits[edit]
See also: Angular unit
Units used to represent angles are listed below in descending magnitude order. Of these units, the degree and the radian are by far the most commonly used. Angles expressed in radians are dimensionless for the purposes of dimensional analysis.
Most units of angular measurement are defined such that one turn (i.e. one full circle) is equal to n units, for some whole number n. The two exceptions are the radian and the diameter part.
Quadrant (n = 4)
The quadrant is
1
/
4
of a turn, i.e. a right angle. It is the unit used in Euclid's Elements. 1 quad. = 90° =
π
/
2
rad =
1
/
4
turn = 100 grad. In German the symbol ∟ has been used to denote a quadrant
QMRIn mathematics, real coordinate space of n dimensions, written Rn (/ɑrˈɛn/ ar-en) (R with superscript n, also written ℝn with blackboard bold R) or R^n with keyboard is a coordinate space that allows several (n) real variables to be treated as a single variable. With various numbers of dimensions (sometimes unspecified), Rn is used in many areas of pure and applied mathematics, as well as in physics. It is the prototypical real vector space and a frequently used representation of Euclidean n-space. Due to the latter fact, geometric metaphors are widely used for Rn, namely a plane for R2 and three-dimensional space for R3.
It is comprised of quadrants
QMRIn geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space.[1][2] The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.[3]
It is made up of quadrants
QMRCartesian coordinate system[edit]
Main article: Cartesian coordinate system
The Cartesian coordinate system in the plane.
The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines.
Rectangular coordinates.svg
In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes.[5] This can be generalized to create n coordinates for any point in n-dimensional Euclidean space.
Depending on the direction and order of the coordinate axis the system may be a right-hand or a left-hand system. This is one of many coordinate systems.
It is made up of quadrants
QMRQuaternions[edit]
Unit quaternions, also known as Euler–Rodrigues parameters, provide another mechanism for representing 3D rotations. This is equivalent to the special unitary group description.
Expressing rotations in 3D as unit quaternions instead of matrices has some advantages:
Concatenating rotations is computationally faster and numerically more stable.
Extracting the angle and axis of rotation is simpler.
Interpolation is more straightforward. See for example slerp.
Quaternions do not suffer from gimbal lock as Euler angles do.
QMRAdding a fourth rotational axis can solve the problem of gimbal lock, but it requires the outermost ring to be actively driven so that it stays 90 degrees out of alignment with the innermost axis (the flywheel shaft). Without active driving of the outermost ring, all four axes can become aligned in a plane as shown above, again leading to gimbal lock and inability to roll.
QMRThe MU puzzle is a puzzle stated by Douglas Hofstadter and found in Gödel, Escher, Bach. As stated, it is an example of a Post canonical system and can be reformulated as a string rewriting system.
Contents [hide]
1 The puzzle
2 Solution
3 Relationship to logic
4 See also
5 Notes
6 References
The puzzle[edit]
Has the dog Buddha-nature? MU!
— Zen Koan[1]
Suppose there are the symbols M, I, and U which can be combined to produce strings of symbols. The MU puzzle asks one to start with the "axiomatic" string MI and transform it into the string MU using in each step one of the following transformation rules:[2][3]
Nr. Formal rule[note 1] Informal explanation Example
1. xI → xIU Add a U to the end of any string ending in I MI to MIU
2. Mx → Mxx Double the string after the M MIU to MIUIU
3. xIIIy → xUy Replace any III with a U MUIIIU to MUUU
4. xUUy → xy Remove any UU MUUU to MU
Solution[edit]
The puzzle's solution is no. It is impossible to change the string MI into MU by repeatedly applying the given rules.
In order to prove assertions like this, it is often beneficial to look for an invariant, that is some quantity or property that doesn't change while applying the rules.
In this case, one can look at the total number of I in a string. Only the second and third rules change this number. In particular, rule two will double it while rule three will reduce it by 3. Now, the invariant property is that the number of I is not divisible by 3:
In the beginning, the number of Is is 1 which is not divisible by 3.
Doubling a number that is not divisible by 3 does not make it divisible by 3.
Subtracting 3 from a number that is not divisible by 3 does not make it divisible by 3 either.
Thus, the goal of MU with zero I cannot be achieved because 0 is divisible by 3.
In the language of modular arithmetic, the number n of I obeys the congruence
n \equiv 2^a \not\equiv 0 \pmod 3.\,
where a counts how often the second rule is applied.
Relationship to logic[edit]
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The MIU system illustrates several important concepts in logic by means of analogy.
It can be interpreted as an analogy for a formal system — an encapsulation of mathematical and logical concepts using symbols. The MI string is akin to a single axiom, and the four transformation rules are akin to rules of inference (see this online interface for producing derivations in the MIU-system]).
The MU string and the impossibility of its derivation is then analogous to a statement of mathematical logic which cannot be proven or disproven by the formal system.
It also demonstrates the contrast between interpretation on the "syntactic" level of symbols and on the "semantic" level of meanings. On the syntactic level, there is no knowledge of the MU puzzle's insolubility. The system does not refer to anything: it is simply a game involving meaningless strings. Working within the system, an algorithm could successively generate every valid string of symbols in an attempt to generate MU, and though it would never succeed, it would search forever, never deducing that the quest was futile. For a human player however, after a number of attempts, one soon begins to suspect that the puzzle may be impossible. One then "jumps out of the system" and starts to reason about the system, rather than working within it. Eventually, one realises that the system is in some way about divisibility by three. This is the "semantic" level of the system — a level of meaning that the system naturally attains. On this level, the MU puzzle can be seen to be impossible.
The inability of the MIU system to express or deduce facts about itself, such as the inability to derive MU, is a consequence of its simplicity. However, more complex formal systems, such as systems of mathematical logic, may possess this ability. This is the key idea behind Godel's Incompleteness Theorem.
QMRThe MU puzzle is a good example of a logical problem where determining an invariant is useful. The puzzle asks one to start with the word MI and transform it into the word MU using in each step one of the following transformation rules:
If a string ends with an I, a U may be appended (xI → xIU)
The string after the M may be completely duplicated (Mx → Mxx)
Any three consecutive I's (III) may be replaced with a single U (xIIIy → xUy)
Any two consecutive U's may be removed (xUUy → xy)
An example derivation (superscripts indicating the applied rules) is
MI →2 MII →2 MIIII →3 MUI →2 MUIUI →1 MUIUIU →2 MUIUIUUIUIU →4 MUIUIIUIU → ...
Is it possible to convert MI into MU using these four transformation rules only?
One could spend many hours applying these transformation rules to strings. However, it might be quicker to find a property that is invariant to all rules (i.e. that isn't changed by any of them), and demonstrates that getting to MU impossible. Logically looking at the puzzle, the only way to get rid of any I's is to have three consecutive I's in the string. This makes the following invariant interesting to consider:
The number of I's in the string is not a multiple of 3.
This is an invariant to the problem if for each of the transformation rules the following holds: if the invariant held before applying the rule, it will also hold after applying it. If we look at the net effect of applying the rules on the number of I's and U's we can see this actually is the case for all rules:
1 +0 +1 Number of I's is unchanged. If the invariant held, it still does.
2 ×2 ×2 If n is not a multiple of 3, then 2×n isn't either. The invariant still holds.
3 −3 +1 If n is not a multiple of 3, n−3 isn't either. The invariant still holds.
4 +0 −2 Number of I's is unchanged. If the invariant held, it still does.
The table above shows clearly that the invariant holds for each of the possible transformation rules, which basically means that whichever rule we pick, at whatever state, if the number of I's was not a multiple of three before applying the rule, it won't be afterwards either.
Given that there is a single I in the starting string MI, and one is not a multiple of three, it's impossible to go from MI to MU as zero is a multiple of three.
QMRIn classical modal logic, a proposition is said to be
possible if and only if it is not necessarily false (regardless of whether it is actually true or actually false);
necessary if and only if it is not possibly false; and
contingent if and only if it is not necessarily false and not necessarily true (i.e. possible but not necessarily true);
impossible if and only if it is not possibly true (i.e. false and necessarily false).
QMRJustification via truth table[edit]
The validity of modus ponens in classical two-valued logic can be clearly demonstrated by use of a truth table.
p q p → q
T T T
T F F
F T T
F F T
In instances of modus ponens we assume as premises that p → q is true and p is true. Only one line of the truth table—the first—satisfies these two conditions (p and p → q). On this line, q is also true. Therefore, whenever p → q is true and p is true, q must also be true.
QMRProper equilibrium is a refinement of Nash Equilibrium due to Roger B. Myerson. Proper equilibrium further refines Reinhard Selten's notion of a trembling hand perfect equilibrium by assuming that more costly trembles are made with significantly smaller probability than less costly ones.
QMRThe first pupils of Aristotle commentated on his writings, but often with a view to expand his work. Thus Theophrastus invented five moods of syllogism in the first figure, in addition to the four invented by Aristotle, and stated with additional accuracy the rules of hypothetical syllogisms. He also often differed with his master,[1] including in collecting much information concerning animals and natural events, which Aristotle had omitted.
QMREfficiency[edit]
Situations are considered to have distributive efficiency when goods are distributed to the people who can gain the most utility from them.
Many economists use Pareto efficiency as their efficiency goal. According to this measure of social welfare, a situation is optimal only if no individuals can be made better off without making someone else worse off.
This ideal state of affairs can only come about if four criteria are met:
The marginal rates of substitution in consumption are identical for all consumers. This occurs when no consumer can be made better off without making others worse off.
The marginal rate of transformation in production is identical for all products. This occurs when it is impossible to increase the production of any good without reducing the production of other goods.
The marginal resource cost is equal to the marginal revenue product for all production processes. This takes place when marginal physical product of a factor must be the same for all firms producing a good.
The marginal rates of substitution in consumption are equal to the marginal rates of transformation in production, such as where production processes must match consumer wants.
QMRvon Neumann–Morgenstern expected utility[edit]
Main article: Von Neumann–Morgenstern utility theorem
Von Neumann and Morgenstern addressed situations in which the outcomes of choices are not known with certainty, but have probabilities attached to them.
A notation for a lottery is as follows: if options A and B have probability p and 1 − p in the lottery, we write it as a linear combination:
L = p A + (1-p) B
More generally, for a lottery with many possible options:
L = \sum_i p_i A_i,
where \sum_i p_i =1.
By making some reasonable assumptions about the way choices behave, von Neumann and Morgenstern showed that if an agent can choose between the lotteries, then this agent has a utility function such that the desirability of an arbitrary lottery can be calculated as a linear combination of the utilities of its parts, with the weights being their probabilities of occurring.
This is called the expected utility theorem. The required assumptions are four axioms about the properties of the agent's preference relation over 'simple lotteries', which are lotteries with just two options. Writing B\preceq A to mean 'A is weakly preferred to B' ('A is preferred at least as much as B'), the axioms are:
completeness: For any two simple lotteries L and M, either L\preceq M or M\preceq L (or both, in which case they are viewed as equally desirable).
transitivity: for any three lotteries L,M,N, if L\preceq M and M\preceq N, then L\preceq N.
convexity/continuity (Archimedean property): If L \preceq M\preceq N, then there is a p between 0 and 1 such that the lottery pL + (1-p)N is equally desirable as M.
independence: for any three lotteries L,M,N and any probability p, L \preceq M if and only if pL+(1-p)N \preceq pM+(1-p)N. Intuitively, if the lottery formed by the probabilistic combination of L and N is no more preferable than the lottery formed by the same probabilistic combination of M and N, then and only then L \preceq M.
Axioms 3 and 4 enable us to decide about the relative utilities of two assets or lotteries.
In more formal language: A von Neumann–Morgenstern utility function is a function from choices to the real numbers:
u\colon X\to \R
which assigns a real number to every outcome in a way that captures the agent's preferences over simple lotteries. Under the four assumptions mentioned above, the agent will prefer a lottery L_2 to a lottery L_1 if and only if, for the utility function characterizing that agent, the expected utility of L_2 is greater than the expected utility of L_1:
L_1\preceq L_2 \text{ iff } u(L_1)\leq u(L_2).
Repeating in category language: u is a morphism between the category of preferences with uncertainty and the category of reals as an additive group.
Of all the axioms, independence is the most often discarded. A variety of generalized expected utility theories have arisen, most of which drop or relax the independence axiom.
QMRAn aggregate demand curve is the sum of individual demand curves for different sectors of the economy. The aggregate demand is usually described as a linear sum of four separable demand sources:[4]
AD = C + I + G + (X - M)
where
C is consumption (may also be known as consumer spending), which is given by a_c + b_c(Y - T) where Y is consumers' income and T the taxes paid by consumers,
I is investment,
G is government spending,
NX = X - M is net export, where
X is total exports, and
M total imports, given bya_m + b_m(Y - T).
These four major parts, which can be stated in either 'nominal' or 'real' terms, are:
personal consumption expenditures (C) or "consumption," demand by households and unattached individuals; its determination is described by the consumption function. The consumption function is C = a + MPC \times (Y - T), where
a is autonomous consumption, MPC the marginal propensity to consume, and (Y - T) the disposable income.
gross private domestic investment (I), such as spending by business firms on factory construction. This includes all private sector spending aimed at the production of some future consumable.
In Keynesian economics, not all of gross private domestic investment counts as part of aggregate demand. Much or most of the investment in inventories can be due to a short-fall in demand (unplanned inventory accumulation or "general over-production"). The Keynesian model forecasts a decrease in national output and income when there is unplanned investment. (Inventory accumulation would correspond to an excess supply of products; in the National Income and Product Accounts, it is treated as a purchase by its producer.) Thus, only the planned or intended or desired part of investment (I_p) is counted as part of aggregate demand. (So, I does not include the 'investment' in running up or depleting inventory levels.)
Investment is affected by the output and the interest rate (i). Consequently, we can write it as I(Y, i). Investment has positive relationship with the output and negative relationship with the interest rate. For example, an increase in the interest rate will cause aggregate demand to decline. Interest costs are part of the cost of borrowing and as they rise, both firms and households will cut back on spending. This shifts the aggregate demand curve to the left. This lowers equilibrium GDP below potential GDP. As production falls for many firms, they begin to lay off workers, and unemployment rises. The declining demand also lowers the price level. The economy is in recession.
gross government investment and consumption expenditures (G).
net exports (NX and sometimes (X - M)), i.e., net demand by the rest of the world for the country's output.
In sum, for a single country at a given time, aggregate demand (D or AD) is given by C + I_p + G + (X - M).
QMRAnthropology[edit]
There is an ongoing dispute as to whether anthropology is intrinsically holistic. Supporters of this concept consider anthropology holistic in two senses. First, it is concerned with all human beings across times and places, and with all dimensions of humanity (evolutionary, biophysical, sociopolitical, economic, cultural, psychological, etc.) Further, many academic programs following this approach take a "four-field" approach to anthropology that encompasses physical anthropology, archeology, linguistics, and cultural anthropology or social anthropology.[19]
Some leading anthropologists disagree, and consider anthropological holism to be an artifact from 19th century social evolutionary thought that inappropriately imposes scientific positivism upon cultural anthropology.[20]
The term "holism" is additionally used within social and cultural anthropology to refer to an analysis of a society as a whole which refuses to break society into component parts. One definition says: "as a methodological ideal, holism implies ... that one does not permit oneself to believe that our own established institutional boundaries (e.g. between politics, sexuality, religion, economics) necessarily may be found also in foreign societies."[21
QMRLudwig von Bertalanffy outlines systems inquiry into three major domains: Philosophy, Science, and Technology. In his work with the Primer Group, Béla H. Bánáthy generalized the domains into four integratable domains of systemic inquiry:
Domain Description
Philosophy the ontology, epistemology, and axiology of systems;
Theory a set of interrelated concepts and principles applying to all systems
Methodology the set of models, strategies, methods, and tools that instrumentalize systems theory and philosophy
Application the application and interaction of the domains
QMR four phases of a pure substance were in equilibrium (P = 4), the phase rule would give F = −1, which is meaningless, since there cannot be −1 independent variables. This explains the fact that four phases of a pure substance (such as ice I, ice III, liquid water and water vapour) are not found in equilibrium at any temperature and pressure. In terms of chemical potentials there are now three equations, which cannot in general be satisfied by any values of the two variables T and p, although in principle they might be solved in a special case where one equation is mathematically dependent on the other two. In practice, however, the coexistence of more phases than allowed by the phase rule normally means that the phases are not all in true equilibrium.
QMRTwo-component systems[edit]
For binary mixtures of two chemically independent components, C = 2 so that F = 4 – P. In addition to temperature and pressure, the other degree of freedom is the composition of each phase, often expressed as mole fraction or mass fraction of one component.
Boiling Point Diagram
As an example, consider the system of two completely miscible liquids such as toluene and benzene, in equilibrium with their vapours. This system may be described by a boiling-point diagram which shows the composition (mole fraction) of the two phases in equilibrium as functions of temperature (at a fixed pressure).
Four thermodynamic variables which may describe the system include temperature (T), pressure (p), mole fraction of component 1 (toluene) in the liquid phase (x1L), and mole fraction of component 1 in the vapour phase (x1V). However since two phases are in equilibrium, only two of these variables can be independent (F = 2). This is because the four variables are constrained by two relations: the equality of the chemical potentials of liquid toluene and toluene vapour, and the corresponding equality for benzene.
QMRThe name cynic derives from Ancient Greek κυνικός (kynikos), meaning "dog-like", and κύων (kyôn), meaning "dog" (genitive: kynos).[3] One explanation offered in ancient times for why the cynics were called "dogs" was because the first cynic, Antisthenes, taught in the Cynosarges gymnasium at Athens.[4] The word cynosarges means the "place of the white dog". It seems certain, however, that the word dog was also thrown at the first cynics as an insult for their shameless rejection of conventional manners, and their decision to live on the streets. Diogenes, in particular, was referred to as the "Dog",[5] a distinction he seems to have revelled in, stating that "other dogs bite their enemies, I bite my friends to save them."[6] Later cynics also sought to turn the word to their advantage, as a later commentator explained:
There are four reasons why the Cynics are so named. First because of the indifference of their way of life, for they make a cult of indifference and, like dogs, eat and make love in public, go barefoot, and sleep in tubs and at crossroads. The second reason is that the dog is a shameless animal, and they make a cult of shamelessness, not as being beneath modesty, but as superior to it. The third reason is that the dog is a good guard, and they guard the tenets of their philosophy. The fourth reason is that the dog is a discriminating animal which can distinguish between its friends and enemies. So do they recognize as friends those who are suited to philosophy, and receive them kindly, while those unfitted they drive away, like dogs, by barking at them.[7]
QMR The Roman cavalry saddle had four horns [2] and was believed to have been copied from Celtic peoples.
QMRHistorical method basics
The following questions are used by historians in modern work.
When was the source, written or unwritten, produced (date)?
Where was it produced (localization)?
By whom was it produced (authorship)?
From what pre-existing material was it produced (analysis)?
In what original form was it produced (integrity)?
What is the evidential value of its contents (credibility)?
The first four are known as higher criticism; the fifth, lower criticism; and, together, external criticism. The sixth and final inquiry about a source is called internal criticism.
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