hcju/proofwikips huggingface dataset:
Definition:Southern Hemisphere
The '''southern hemisphere''' of Earth is the hemisphere between the equator and the South Pole. Points in the southern hemisphere have latitude between $0 \degrees \, \mathrm S$ (the equator itself) and $90 \degrees \, \mathrm S$ (the South Pole).
Definition:Eastern Hemisphere
The '''eastern hemisphere''' of Earth is the hemisphere immediately to the east of the principal meridian. Points in the eastern hemisphere have longitude between $0 \degrees \, \mathrm E$ (the principal meridian itself) and $180 \degrees \, \mathrm E$.
Definition:Western Hemisphere
The '''western hemisphere''' of Earth is the hemisphere immediately to the west of the principal meridian. Points in the western hemisphere have longitude between $0 \degrees \, \mathrm W$ (the principal meridian itself) and $180 \degrees \, \mathrm W$.
Definition:Latitude
=== Terrestrial Latitude === {{:Definition:Latitude (Terrestrial)}} === Celestial Latitude === {{:Definition:Latitude (Celestial)}}
Definition:Terrestrial Hemisphere
=== Northern Hemisphere === {{:Definition:Northern Hemisphere}} === Southern Hemisphere === {{:Definition:Southern Hemisphere}} === Eastern Hemisphere === {{:Definition:Eastern Hemisphere}} === Western Hemisphere === {{:Definition:Western Hemisphere}}
Definition:Colatitude (Terrestrial)
Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $\phi$ denote the latitude of $J$. The '''colatitude''' of $J$ is the (spherical) angle $90 \degrees - \phi$, that is: :if $J$ is in the northern hemisphere of Earth, the '''colatitude''' is the (spherical) angle $\sphericalangle NOJ$ :if $J$ is in the southern hemisphere of Earth, the '''colatitude''' is the (spherical) angle $\sphericalangle SOJ$. :500px
Definition:Parallel of Latitude
Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $C$ be the small circle through $J$ parallel to the plane containing the equator. Then $C$ is known as the '''parallel of latitude''' of $J$. Thus $C$ consists of all the points on Earth's surface which have the same latitude as $J$. :500px
Definition:Imperial/Length/Mil
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''mil''' }} {{eqn | r = \dfrac 1 {1000}       | c = inch }} {{eqn | r = 25 \cdotp 4       | c = microns }} {{eqn | r = 0 \cdotp 0254       | c = millimetres }} {{eqn | r = 2 \cdotp 54 \times 10^{-5}       | c = metres }} {{end-eqn}}
Definition:Metric System/Length
Various metric units of length which have specific names are as follows: === Micrometre === {{Definition:Metric System/Length/Micrometre}} === Millimetre === {{Definition:Metric System/Length/Millimetre}} === Centimetre === {{Definition:Metric System/Length/Centimetre}} === Decimetre === {{Definition:Metric System/Length/Decimetre}} === Metre === {{Definition:Metric System/Length/Metre}} === Hectometre === {{Definition:Metric System/Length/Hectometre}} === Kilometre === {{Definition:Metric System/Length/Kilometre}}
Definition:Non-Archimedean Norm Axioms
{{begin-axiom}} {{axiom | n = \text N 1         | lc= Positive Definiteness:         | q = \forall x \in R         | ml= \norm x = 0         | mo= \iff         | mr= x = 0_R }} {{axiom | n = \text N 2         | lc= Multiplicativity:         | q = \forall x, y \in R         | ml= \norm {x \circ y}         | mo= =         | mr= \norm x \times \norm y }} {{axiom | n = \text N 4         | lc= Ultrametric Inequality:         | q = \forall x, y \in R         | ml= \norm {x + y}         | mo= \le         | mr= \max \set {\norm x, \norm y} }} {{end-axiom}}
Definition:Non-Archimedean/Norm (Division Ring)/Definition 1
A norm $\norm {\, \cdot \,}$ on $R$ is '''non-Archimedean'''  $\norm {\, \cdot \,}$ satisfies the axiom: {{begin-axiom}} {{axiom | n = \text N 4         | lc= Ultrametric Inequality:         | q = \forall x, y \in R         | ml= \norm {x + y}         | mo= \le         | mr= \max \set {\norm x, \norm y} }} {{end-axiom}}
Definition:Non-Archimedean/Norm (Division Ring)/Definition 2
A '''non-Archimedean norm''' on $R$ is a mapping from $R$ to the non-negative reals: :$\norm {\, \cdot \,}: R \to \R_{\ge 0}$ satisfying the non-Archimedean norm axioms: {{:Definition:Non-Archimedean Norm Axioms}}
Definition:Kilometre/Symbol
The symbol for the '''kilometre''' is $\mathrm {km}$: :$\mathrm k$ for kilo :$\mathrm m$ for metre.
Definition:Nautical Mile
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''nautical mile''' }} {{eqn | r = 1852       | c = metres (by definition) }} {{eqn | r = 10       | c = cables }} {{eqn | o = \approx       | r = 6076       | c = feet }} {{eqn | o = \approx       | r = 1 \cdotp 151       | c = (international) mile }} {{eqn | o = \approx       | r = 0 \cdotp 999 \, 363       | c = admiralty (UK nautical) mile }} {{end-eqn}}
Definition:Admiralty Mile
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''admiralty mile''' }} {{eqn | r = 6080       | c = feet (by definition) }} {{eqn | r = 1853 \cdotp 184       | c = metres }} {{eqn | o = \approx       | r = 1 \cdotp 1515       | c = (international) mile }} {{end-eqn}}
Definition:Inner Angle of Spherical Triangle
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Consider the $4$ consecutive parts of $\triangle ABC$: :$B, a, C, b$ Then $C$ is known as the '''inner angle''' of $B, a, C, b$
Definition:Inner Side of Spherical Triangle
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Consider the $4$ consecutive parts of $\triangle ABC$: :$B, a, C, b$ Then $a$ is known as the '''inner side''' of $B, a, C, b$
Definition:Quadrantal Triangle
Let $\triangle ABC$ be a spherical triangle. Let one of the sides of $\triangle ABC$ be a right angle: $\dfrac \pi 2$. :500px Then $\triangle ABC$ is known as a '''quadrantal (spherical) triangle'''.
Definition:Right Spherical Triangle
Let $\triangle ABC$ be a spherical triangle. Let one of the angles of $\triangle ABC$ be a right angle: $\dfrac \pi 2$. :500px Then $\triangle ABC$ is known as a '''right spherical triangle'''.
Definition:Spherical Excess of Spherical Triangle
Let $\triangle ABC$ be a spherical triangle. Let $E$ be the sum of the angles of $\triangle ABC$. The '''spherical excess''' of $\triangle ABC$ is how much $E$ is greater than $\pi$ radians.
Definition:Polar Triangle
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let $A'$, $B'$ and $C'$ be the poles of the sides $BC$, $AC$ and $AB$ respectively which are in the same hemisphere as the points $A$, $B$ and $C$ respectively. :400px Then the spherical triangle $\triangle A'B'C'$ is the '''polar triangle''' of $\triangle ABC$.
Definition:Length of Arc of Great Circle
Let $E$ be a great circle on a sphere $S$ whose center is $O$. Let $AB$ be an arc of $E$. The '''length''' of $AB$ is defined as the angle subtended at $O$ by $AB$.
Definition:Haversine
The '''haversine''' of an angle is defined as: {{begin-eqn}} {{eqn | l = \hav \theta       | r = \dfrac 1 2 \paren {1 - \cos \theta}       | c =  }} {{eqn | r = \dfrac 1 2 \vers \theta       | c = where $\vers \theta$ denotes the versed sine of $\theta$ }} {{eqn | r = \sin^2 \dfrac {\theta} 2       | c = Double Angle Formula for Cosine: Corollary 2 }} {{end-eqn}}
Definition:Versed Sine
The '''versed sine''' of an angle is defined as: :$\vers \theta = 1 - \cos \theta$
Definition:Knot (Unit of Measurement)
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''knot''' }} {{eqn | r = 1 \, 852       | c = metres per hour (exactly) }} {{eqn | o = \approx       | r = 0 \cdotp 51444       | c = metres per second }} {{eqn | o = \approx       | r = 1 \cdotp 15078       | c = (international) miles per hour }} {{end-eqn}}
Definition:Zenith
Let an observer $O$ on the surface of Earth be assumed to be the center of the celestial sphere $C$. The '''zenith''' $Z$ is the point on $C$ which is vertically overhead.
Definition:Celestial Sphere
The '''celestial sphere''' is an imaginary sphere, an indeterminate distance from Earth, which intersects the straight lines from the celestial bodies to the observer. :480px It is then convenient to identify the directions of the celestial bodies with their position on the '''celestial sphere'''.
Definition:Celestial
In the context of astronomy, the word '''celestial''' is an adjective whose meaning is loosely '''of the sky'''. It is used to define bodies and phenomena vaguely defined as being outside Earth's atmosphere, hence does not apply to meteorological phenomena. It is often used as an antithesis for the word '''terrestrial''', which means '''of Earth'''.
Definition:Terrestrial
The word '''terrestrial''' is an adjective whose meaning is loosely '''of Earth'''. It is usually used in the context of astronomy and geodesy to refer to concepts which need to be distinguished from their celestial counterparts.
Definition:Celestial Sphere/Observer
In the context of astronomy, in particular spherical astronomy, the '''observer''' is considered to be a reference point located at the center of the celestial sphere. In practice, of course, an actual '''observer''' is physically located on the surface of Earth. However, the celestial sphere is considered to be so much larger than the radius of Earth that the '''observer''' is considered to be at the center of Earth.
Definition:Celestial Horizon
The '''celestial horizon''' is the great circle on the celestial sphere whose plane is perpendicular to the line from the zenith to the observer. Thus the '''celestial horizon''' is the edge of the visible part of the celestial sphere. The rest of it is hidden by Earth itself. :480px
Definition:Visible Hemisphere
The '''visible hemisphere''' is the hemisphere of the celestial sphere between the celestial horizon and the zenith. That is, it is the half of the sky that can be seen from Earth at an arbitrary time. The other hemisphere of the celestial sphere is hidden from an observer by Earth itself. :480px
Definition:Vertical Circle
Let $X$ be the position of a star (or other celestial body) on the celestial sphere. The '''vertical circle''' through $X$ is defined as the great circle passing through both $X$ and the zenith $Z$.
Definition:Celestial Altitude
Let $X$ be a point on the celestial sphere. The '''(celestial) altitude''' of $X$ is defined as the angle subtended by the the arc of the vertical circle through $X$ between the celestial horizon and $X$ itself.
Definition:Zenith Distance
Let $X$ be the position of a star (or other celestial body) on the celestial sphere. The '''zenith distance''' of $X$ is defined as the angle subtended by the the arc of the vertical circle through $X$ between $X$ and the zenith.
Definition:Parallel of Altitude
Let $X$ be a point on the celestial sphere. Let $LXM$ be a small circle through $X$ which is parallel to the celestial horizon. Then $LXM$ is a '''parallel of altitude''', and is such that all celestial bodies on it have the same altitude.
Definition:North Celestial Pole
Let $O$ be a celestial observer situated at the surface of Earth in the Northern hemisphere. Let $OP$ be a straight line parallel to Earth's axis. Let $P$ be the point in the visible hemisphere at which $OP$ meets the celestial sphere. Then $P$ is the '''north celestial pole'''. That is, the '''north celestial pole''' is the pole of the celestial equator which is the zenith of the North Pole.
Definition:Polaris
'''Polaris''' is a star whose position in the celestial sphere is very nearly exactly at the north celestial pole. Hence it appears not to move through the sky during the course of a night.
Definition:Principal Vertical Circle
The vertical circle on the celestial sphere which passes through the north celestial pole (and south celestial pole) is called the '''principal vertical circle'''. :480px
Definition:North Point of Horizon
The point on the celestial horizon through which the principal vertical circle passes is called the '''north point of the horizon'''.
Definition:South Point of Horizon
The point on the celestial horizon exactly opposite the '''north point of the horizon''' is called the '''south point of the horizon'''.
Definition:West Point of Horizon
Let an observer $O$ be facing the '''north point of the horizon''' $N$, facing directly away from the '''south point of the horizon''' $S$. Let the celestial equator meet the celestial horizon at the points $E$ and $W$. Let $W$ be the point on the left of $O$. This is called the '''west point of the horizon'''.
Definition:East Point of Horizon
Let an observer $O$ be facing the '''north point of the horizon''' $N$, facing directly away from the '''south point of the horizon''' $S$. Let the celestial equator meet the celestial horizon at the points $E$ and $W$. Let $E$ be the point on the right of $O$. This is called the '''east point of the horizon'''.
Definition:Cardinal Points (Celestial)
The '''cardinal points''' of the celestial sphere are the '''north point''', the '''sourth point''', the '''east point''' and the '''west point''': === North Point of Horizon === {{:Definition:North Point of Horizon}} === South Point of Horizon === {{:Definition:South Point of Horizon}} === East Point of Horizon === {{:Definition:East Point of Horizon}} === West Point of Horizon === {{:Definition:West Point of Horizon}}
Definition:Azimuth (Astronomy)
Let $X$ be a point on the celestial sphere. The spherical angle between the principal vertical circle and the vertical circle on which $X$ lies is the '''azimuth''' of $X$. The '''azimuth''' is usually measured in degrees, $0 \degrees$ to $180 \degrees$ either west or east, depending on whether $X$ lies on the eastern or western hemisphere of the celestial sphere.
Definition:Prime Vertical
The vertical circle on the celestial sphere which passes through both the east point and the west point is called the '''prime vertical'''.
Definition:Celestial Equator
Consider the celestial sphere with observer $O$. Let $P$ be the north celestial pole. The great circle whose plane is perpendicular to $OP$ is known as the '''celestial equator'''.
Definition:Nadir
Consider the celestial sphere $C$ with an observer $O$ at the center. Let $Z$ be the zenith of $C$ Let  the straight line $ZO$ be produced from $O$ to meet $C$ at $R$. The point $R$ is known as the '''nadir''' of $C$.
Definition:South Celestial Pole
Let $O$ be a celestial observer situated at the surface of Earth in the Southern hemisphere. Let $OQ$ be a straight line parallel to Earth's axis. Let $Q$ be the point in the visible hemisphere at which $OQ$ meets the celestial sphere. Then $Q$ is the '''south celestial pole'''. That is, the '''south celestial pole''' is the pole of the celestial equator which is the zenith of the South Pole.
Definition:Celestial Axis
Consider the celestial sphere with observer $O$. The '''celestial axis''' is the line which joins the north celestial pole and the south celestial pole.
Definition:Celestial Pole
Consider the celestial sphere $C$. The '''celestial poles''' of $C$ are the north celestial pole and the south celestial pole: === North Celestial Pole === {{:Definition:North Celestial Pole}} === South Celestial Pole === {{:Definition:South Celestial Pole}}
Definition:Declination
Consider the celestial sphere $C$ with observer $O$. Let $P$ and $Q$ be the north celestial pole and south celestial pole respectively. Let $X$ be a point on $C$. Let $PXQ$ be the vertical circle through $X$. Let $D$ be the point where the celestial equator intersects $PXQ$. The length of the arc $DX$ of $PXQ$ is known as the '''declination''' of $X$.
Definition:Declination/North
If $X$ is in the northern celestial hemisphere, $DX$ is '''north declination'''.
Definition:Declination/South
If $X$ is in the southern celestial hemisphere, $DX$ is '''south declination'''.
Definition:North Polar Distance
Consider the celestial sphere $C$ with observer $O$. Let $X$ be a point on $C$. Let $X$ have declination of $\delta$. The '''north polar distance''' of $X$, denoted $\operatorname {N.P.D.} x$, is defined as: :$\operatorname {N.P.D.} x = 90 \degrees - \delta$ That is, the '''north polar distance''' of $X$ length of the arc from the north celestial pole $P$ along the vertical circle through $X$. :500px
Definition:Observer's Meridian
Consider the celestial sphere whose observer is $O$. Let $P$ and $Q$ be the north celestial pole and south celestial pole respectively. Let $Z$ be the zenith. Let $S$ be the south point of the horizon. The '''observer's meridian''' is the semi-great circle $PZSQ$.
Definition:Babylonian Number System
The number system as used in the {{WP|First_Babylonian_Dynasty|Old Babylonian empire}} was a positional numeral system where the number base was a combination of decimal (base $10$) and sexagesimal (base $60$). The characters were written in {{WP|Cuneiform|cuneiform}} by a combination of: :a thin vertical wedge shape, to indicate the digit $1$ :a fat horizontal wedge shape, to indicate the digit $10$ arranged in groups to indicate the digits $2$ to $9$ and $20$ to $50$. :600px At $59$ the pattern stops, and the number $60$ is represented by the digit $1$ once again. Thus these groupings were placed side by side: :the rightmost grouping would indicate a number from $1$ to $59$ :the one to the left of that would indicate a number from $60 \times 1$ to $60 \times 59$ and so on, each grouping further to the left indicating another multiplication by $60$ For fractional numbers there was no actual radix point. Instead, the distinction was inferred by context. The fact that they had no symbol to indicate the zero digit means that this was not a true positional numeral system as such. For informal everyday arithmetic, they used a decimal system which was the decimal part of the full sexagesimal system.
Definition:Number System
A '''number system''' is a technique for representing numbers.
Definition:Numeral System
A '''numeral system''' is: :a set of symbols that is used to represent a specific subset of the set of numbers (usually natural numbers), referred to as numerals :a set of rules which define how to combine the numerals so as to be able to express other numbers.
Definition:Numeral
A '''numeral''' is a symbol which is used to identify a particular number (usually a natural number). A '''numeral''' is one of a small subset of the numbers. Other numbers can be created by combining '''numerals''' using specific well-defined rules.
Definition:Positional Numeral System
A '''positional number system''' is a number system with the following properties: :It has a set of numerals which represent a subset of the numbers. :The number being represented is written as a string of these numerals, which represent a different value according to their position in the numerals. The design of the '''positional number system''' is such that all numbers can be represented by such a string, which may or may not be infinite in length.
Definition:Musical Interval
A '''musical interval''' is a ratio of frequencies of musical notes which is recognised by convention as being particularly aesthetically pleasing or meaningful. === Octave === {{:Definition:Octave}} === Fifth === {{:Definition:Fifth (Music)}} === Fourth === {{:Definition:Fourth (Music)}} === Tone === {{:Definition:Tone (Music)}} === Semitone === {{:Definition:Semitone}}
Definition:Neusis
=== Neusis Ruler === {{:Definition:Neusis Ruler}} === Neusis Construction === {{:Definition:Neusis Construction}}
Definition:Decimal Expansion/Historical Note
The idea of representing fractional values by extending the decimal notation to the right appears to have been invented by {{AuthorRef|Simon Stevin}}, who published the influential book {{BookLink|De Thiende|Simon Stevin}}. The idea was borrowed from the Babylonian number system, but streamlined to base $10$ from the cumbersome sexagesimal. However, his notation was cumbersome: he would write, for example, $25 \bigcirc \! \! \! \! \! \! 0 \ \, 3 \bigcirc \! \! \! \! \! \! 1 \ \, 7 \bigcirc \! \! \! \! \! \! 2 \ \, 9 \bigcirc \! \! \! \! \! \! 3$ for what we now give as $25 \cdotp 379$. {{AuthorRef|John Napier}}, in the early $17$th century, appears to have been the first into print with the contemporary notation, although {{AuthorRef|Walter William Rouse Ball}} suggests that credit for this ought to be due to {{AuthorRef|Henry Briggs}}. It was not until a century later, however, that the decimal point came into general use.
Definition:Trigonometry/Linguistic Note
The word '''trigonometry''' derives from the Greek for '''measurement of triangles'''.
Definition:Epicycle/Historical Note
{{AuthorRef|Claudius Ptolemy}}'s {{BookLink|Almagest|Claudius Ptolemy}} modelled the orbits of the planets of the solar system as a system of epicycles.
Definition:Sextant
A '''sextant''' is a device for measuring the celestial altitude of a celestial body. It consists of a telescope and a system of adjustable reflectors for presenting the image of the celestial body to the eye so as to be next to an image of the horizon. The resulting reading on the scale of the '''sextant''' gives an indication of the altitude of the body.
Definition:Astrolabe
An '''astrolabe''' is a device for measuring the celestial altitude of a celestial body. It consists of a series of disks on which are shown the positions of prominent stars. These are aligned with their positions on what can be seen of the celestial sphere, and hence obtain a reading of their altitudes.
Definition:GPS
'''GPS''', or '''Global Positioning System''', is a technique using signals from satellites to identify the geographical coordinates of an appropriately configured receiver.
Definition:Noon
'''Noon''' is the point in time during the course of a day that the Sun reaches its highest celestial altitude.
Definition:Unknown
An '''unknown''' is an indeterminate variable in a system of equations or inequalities whose value the solver is invited to determine.
Definition:Diophantine Problem
A '''Diophantine problem''' is a system of Diophantine equations which has more unknowns than it has equations.
Definition:Binary Tuple
A '''binary tuple''' is an ordered tuple whose domain is the set $\set {0, 1}$. That is, it is an ordered $n$-tuple $\tuple {x_1, x_2, \ldots, x_n}$ such that each $x_i$ is either $0$ or $1$. The set of all binary $n$-tuples can often be seen denoted $V_n$: :$V_n := \set {\tuple {x_1, x_2, \ldots, x_n}: \forall i \in \set {1, 2, \ldots n}: x_i \in \set {0, 1} }$
Definition:Set of Strings of Given Length
Let $n \in \N$ be a natural number. Let $\Sigma$ be an alphabet. The set of all strings from $\Sigma$ of length $n$ is denoted $\Sigma^{\paren n}$.
Definition:Set of Finite Strings
Let $\Sigma$ be an alphabet. The set of all finite strings from $\Sigma$ is denoted $\Sigma^*$.
Definition:Symmetric Mapping
=== Mapping Theory === {{:Definition:Symmetric Mapping (Mapping Theory)}} === Linear Algebra === {{:Definition:Symmetric Mapping (Linear Algebra)}}
Definition:Symmetric Mapping (Mapping Theory)
Let $n \in \N$ be a natural number. Let $S^n$ be an $n$-dimensional cartesian space on a set $S$. Let $f: S^n \to T$ be a mapping from $S^n$ to a set $T$. Then $f$ is a '''symmetric mapping''' : :$\map f {x_1, x_2, \dotsc, x_n} = \map f {x_{\map \pi 1}, x_{\map \pi 2}, \dotsc x_{\map \pi n} }$ for all permutations $\pi$ on $\set {1, 2, \dotsc n}$. That is, a '''symmetric mapping''' is a mapping defined on a cartesian space whose values are preserved under permutation of its arguments.
Definition:Uncertainty
Let $X$ be a random variable. Let $X$ take a finite number of values with probabilities $p_1, p_2, \dotsc, p_n$. The '''uncertainty''' of $X$ is defined to be: :$\map H X = \displaystyle -\sum_k p_k \lg p_k$ where: :$\lg$ denotes logarithm base $2$ :the summation is over those $k$ where $p_k > 0$.