hcju/proofwikips huggingface dataset:
Definition:New Moon
The '''new moon''' is the astronomical phenomenon of the Moon in its orbit coming to its closest point to the Sun: :800px Note that the above diagram is not to scale. At this point in its orbit, the Moon is between Earth and the Sun. Thus the hemisphere of the Moon which is illuminated by the Sun is facing directly away from Earth. Thus for an observer of the Moon from Earth, unless an actual solar eclipse is in progress, at this point it cannot be seen, because: :$(1): \quad$ It is dark :$(2): \quad$ The Sun is more or less behind it, outshining everything else in the sky.
Definition:Zero Mapping
Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$. Let $S$ be a set. Let $f_0: S \to \mathbb A$ denote the constant mapping: :$\forall x \in S: \map {f_0} x = 0$ Then $f_0$ is referred to as the '''zero mapping'''.
Definition:Non-Trivial Ring
A '''non-trivial ring''' is a ring which is not trivial. That is, a ring $R$ such that: :$\exists x, y \in R: x \circ y \ne 0_R$ where $0_R$ denotes the zero of $R$.
Definition:Convergent Sequence/P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\sequence {x_n} $ be a sequence in $\Q_p$.
Definition:Phase of Moon
The '''phase of the Moon''' is its apparent shape when viewed in the night sky of Earth. This shape is dependent upon the position of the Moon with respect to the Sun and Earth. :800px When the Moon is between the Sun and Earth, the sunlit side faces more or less away from Earth, and so less than half of it can be seen lit up. When Earth is between the Sun and the Moon, the sunlit side faces more or less away towards Earth, and so more than half of it can be seen lit up. When the Moon is directly between the Sun and Earth, the sunlit side is facing completely away from Earth, and in normal conditions cannot be seen. This is the '''phase''' known as '''new moon'''. When Earth is directly between the Sun and the Moon, the sunlit side is facing completely towards Earth, and in normal conditions appears as a brign disk. This is the '''phase''' known as '''full moon'''. At the point half way between new moon and full moon, the '''phase''' is known as '''first quarter'''. At the point half way between full moon and new moon, the '''phase''' is known as '''last quarter'''.
Definition:Safe Prime/Sequence
The sequence of safe primes begins: :$5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, \ldots$
Definition:Metric System/Scaling Prefixes/yocto-
'''yocto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-24}$. === Symbol === {{:Symbols:Y/yocto-}}
Definition:Metric System/Scaling Prefixes/zepto-
'''zepto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-21}$. === Symbol === {{:Symbols:Z/zepto-}}
Definition:Metric System/Scaling Prefixes/atto-
'''atto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-18}$. === Symbol === {{:Symbols:A/atto-}}
Definition:Metric System/Scaling Prefixes/femto-
'''femto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-15}$. === Symbol === {{:Symbols:F/femto-}}
Definition:Metric System/Scaling Prefixes/pico-
'''pico-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-12}$. === Symbol === {{:Symbols:P/pico-}}
Definition:Metric System/Scaling Prefixes/nano-
'''nano-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-9}$. === Symbol === {{:Symbols:N/nano-}}
Definition:Metric System/Scaling Prefixes/micro-
'''micro-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-6}$. === Symbol === {{:Symbols:Greek/Mu/micro-}}
Definition:Metric System/Scaling Prefixes/milli-
'''milli-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-3}$. === Symbol === {{:Symbols:M/milli-}}
Definition:Metric System/Scaling Prefixes/centi-
'''centi-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-2}$. === Symbol === {{:Symbols:C/centi-}}
Definition:Metric System/Scaling Prefixes/deci-
'''deci-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-1}$. === Symbol === {{:Symbols:D/deci-}}
Definition:Harmonic Potential Energy
Let $P$ be a physical particle. Let its position $\map x t$ be a real function, where $t$ is time. Let $k > 0$. Then the potential energy of the form :$\map U x = \frac 1 2 k x^2$ is called the '''harmonic potential energy'''.
Definition:Metric System/Scaling Prefixes/deka-
'''deka-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^1$. === Symbol === {{:Symbols:D/deka-}}
Definition:Harmonic Oscillator
Let $P$ be a physical particle. Let the potential energy of $P$ be that of the harmonic potential. Then $P$ is called the '''harmonic oscillator'''.
Definition:Metric System/Scaling Prefixes/hecto-
'''hecto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^2$. === Symbol === {{:Symbols:H/hecto-}}
Definition:Metric System/Scaling Prefixes/kilo-
'''kilo-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^3$. === Symbol === {{:Symbols:K/kilo-}}
Definition:Metric System/Scaling Prefixes/mega-
'''mega-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^6$. === Symbol === {{:Symbols:M/mega-}}
Definition:Metric System/Scaling Prefixes/giga-
'''giga-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^9$. === Symbol === {{:Symbols:G/giga-}}
Definition:Metric System/Scaling Prefixes/tera-
'''tera-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{12}$. === Symbol === {{:Symbols:T/tera-}}
Definition:Metric System/Scaling Prefixes/peta-
'''peta-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{15}$. === Symbol === {{:Symbols:P/peta-}}
Definition:Metric System/Scaling Prefixes/exa-
'''exa-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{18}$. === Symbol === {{:Symbols:E/exa-}}
Definition:Metric System/Scaling Prefixes/zetta-
'''zetta-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{21}$. === Symbol === {{:Symbols:Z/zetta-}}
Definition:Metric System/Scaling Prefixes/yotta-
'''yotta-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{24}$. === Symbol === {{:Symbols:Y/yotta-}}
Definition:Convergent Sequence/P-adic Numbers/Definition 1
The sequence $\sequence {x_n}$ '''converges to $x \in \Q_p$''' : :$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x}_p < \epsilon$
Definition:Convergent Sequence/P-adic Numbers/Definition 3
The sequence $\sequence {x_n}$ '''converges to $x \in \Q_p$''' : :$\sequence {x_n}$ converges to $x$ in the $p$-adic metric
Definition:Convergent Sequence/P-adic Numbers/Definition 4
The sequence $\sequence {x_n}$ '''converges to $x \in \Q_p$''' : :the real sequence $\sequence {\norm {x_n - x}_p }$ converges to $0$ in the reals $\R$
Definition:Limit of Sequence/P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\sequence {x_n} $ be a sequence in $\Q_p$. <section notitle="true" name="LimitDefinition"> Let $\sequence {x_n}$ converge to $x \in \Q_p$  Then $x$ is a '''limit of $\sequence {x_n}$ as $n$ tends to infinity''' which is usually written: :$\displaystyle x = \lim_{n \mathop \to \infty} x_n$ </section>
Definition:Convergent Sequence/P-adic Numbers/Definition 2
The sequence $\sequence {x_n}$ '''converges to $x \in \Q_p$''' : :$\sequence {x_n}$ converges to $x$ in the $p$-adic norm
Definition:Convergent Sequence/Normed Division Ring/Definition 1
The sequence $\sequence {x_n}$ '''converges to $x \in R$ in the norm $\norm {\, \cdot \,}$''' : :$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$
Definition:Convergent Sequence/Normed Division Ring/Definition 2
The sequence $\sequence {x_n}$ '''converges to $x \in R$ in the norm $\norm {\, \cdot \,}$''' : :$\sequence {x_n}$ converges to $x$ in the metric induced by the norm $\norm {\, \cdot \,}$
Definition:Convergent Sequence/Normed Division Ring/Definition 3
The sequence $\sequence {x_n}$ '''converges to $x \in R$ in the norm $\norm {\, \cdot \,}$''' : :the real sequence $\sequence {\norm {x_n - x} }$ converges to $0$ in the reals $\R$
Definition:Rounding/Treatment of Half/Round Up
The '''round up''' convention is that the '''larger''' of those two integers is used: :$y = 10^n \floor {\dfrac x {10^n} + \dfrac 1 2}$
Definition:Rounding/Treatment of Half/Round Down
The '''round down''' convention is that the '''smaller''' of those two integers is used: :$y = 10^n \ceiling {\dfrac x {10^n} - \dfrac 1 2}$
Definition:Rounding/Treatment of Half/Round to Even
The '''round to even''' convention is that the '''nearest even integer''' to $\dfrac x {10^n}$ is used: :$y = \begin {cases} 10^n \floor {\dfrac x {10^n} + \dfrac 1 2} & : \text {$\floor {\dfrac x {10^n} + \dfrac 1 2}$ even} \\ 10^n \ceiling {\dfrac x {10^n} - \dfrac 1 2} & : \text {$\floor {\dfrac x {10^n} + \dfrac 1 2}$ odd} \end {cases}$
Definition:Cumulative Rounding Error
Let $S$ be a set of continuous data which is to be rounded to the nearest $n$th power of $10$. The '''cumulative rounding error''' of $S$ is defined as: :$R = \displaystyle \sum_{x \mathop \in S} \paren {x - \overline x}$ where $\overline x$ denotes the rounded value of a given $x$.
Definition:Scientific Notation/Exponent
The number $e$ is known as the '''exponent'''.
Definition:Significant Figures
Let $n$ be a number expressed in decimal notation. The number of digits to which $n$ is rounded, apart from any digits needed to locate the decimal point, are referred to as the '''significant figures''' of $n$.
Definition:Graph (Statistics)
In the context of statistics, a '''graph''' is a word that loosely means a pictorial or diagrammatic presentation of a set of data.
Definition:Bar Graph
A '''bar graph''' is a form of graph which consists of a finite set of (usually) vertical  bars whose length determines the statistic being communicated.
Definition:Pie Graph
A '''pie graph''' is a form of graph which consists of a circle divided into sectors whose areas represent the proportion of the corresponding statistic relative to the whole.
Definition:Pictograph
A '''pictograph''' is a variant of a bar graph whose  bars consist of a number of icons or other pictorial symbols arranged in a (usually) line.
Definition:Inequality/Member
Let $\RR$ be an inequality. Let $a \mathrel \RR b$. Then both $a$ and $b$ are referred to as '''members''' of the inequality $\RR$.
Definition:General Logarithm/Common/Notation for Negative Logarithm
Let $n \in \R$ be a real number such that $0 < n < 1$. Let $n$ be presented (possibly approximated) in scientific notation as: :$a \times 10^{-d}$ where $d \in \Z_{>0}$ is a (strictly) positive integer. Let $\log_{10} n$ denote the common logarithm of $n$. Then it is the standard convention to express $\log_{10} n$ in the form: :$\log_{10} n = \overline d \cdotp m$ where $m := \log_{10} a$ is the mantissa of $\log_{10} n$.
Definition:General Logarithm/Common/Mantissa
Let $\log_{10} n$ be expressed in the form: :$\log_{10} n = \begin {cases} c \cdotp m & : d \ge 0 \\ \overline c \cdotp m & : d < 0 \end {cases}$ where: : $c = \size d$ is the absolute value of $d$ : $m := \log_{10} a$ $\log_{10} a$ is the '''mantissa''' of $\log_{10} n$.
Definition:General Logarithm/Common/Characteristic
$c$ is the '''characteristic''' of $\log_{10} n$.
Definition:Antilogarithm
Let $x \in \R_{>0}$ be a strictly positive real number. Let $b \in \R_{>1}$ be a real number which is greater than $1$. Let $y = \log_b x$ be the logarithm of $x$ base $b$. Then the '''antilogarithm of $y$ base $b$''' is: :$\operatorname {alog}_b y = x$
Definition:Sample Statistic/Raw
'''Raw data''' are collected data which have not been organized or analysed in any way.
Definition:Spherical Astronomy
'''Spherical astronomy''' is a branch of astronomy which studies the angular positions of the celestial bodies, with no concern for their distance from Earth. Its mathematical aspects can be considered a subfield of spherical geometry.
Definition:Great Circle
Let $S$ be a sphere. A '''great circle''' of $S$ is defined as the intersection of $S$ with a plane passing through the center of $S$. :400px
Definition:Small Circle
Let $S$ be a sphere. A '''small circle''' on $S$ is defined as the intersection of $S$ with a plane which does not pass through the center of $S$. :400px
Definition:Pole of Great Circle
Let $C$ be a great circle of $S$. Let $AB$ be the diameter of $S$ situated perpendicular to the plane of $C$. :700px The points $A$ and $B$, where the diameter intersects $S$, are the '''poles''' of the great circle $C$.
Definition:Pole of Circle
Let $S$ be a sphere whose center is $O$. Let $AB$ be a diameter of $S$, such that $A$ and $B$ are the points of intersection $S$ Let $C$ be a (small) circle of $S$ embedded in a plane perpendicular to $AB$.   :700px The points $A$ and $B$ are the '''poles''' of the circle $C$. === Special Case: Pole of Great Circle === {{:Definition:Pole of Great Circle}}
Definition:Geographical Equator
The '''(geographical) equator''' is the great circle described on the surface of Earth whose plane is perpendicular to Earth's axis of rotation. :500px
Definition:Earth's Poles
Let it be assumed to be established that planet Earth rotates in space about its axis. The '''poles''' of Earth are the endpoints of Earth's axis, where it intersects the surface of Earth. That is, the '''poles''' of Earth are the poles of Earth's equator. :500px
Definition:Earth's Poles/North
Let an observer $A$ be situated on Earth's equator. Let $A$ be facing the direction of Earth's rotation in space about its axis. Then the '''North Pole''' is the particular one of Earth's poles which is on the left of $A$.
Definition:Earth's Poles/South
Let an observer $A$ be situated on Earth's equator. Let $A$ be facing the direction of Earth's rotation in space about its axis. Then the '''South Pole''' is the particular one of Earth's poles which is on the right of $A$.
Definition:Earth's Axis
Let it be assumed to be established that planet Earth rotates in space about a more-or-less fixed diameter $D$. Then $D$ is known as '''Earth's axis'''. :500px
Definition:Closed Ball/P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''closed $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p }$''' is defined as: :$\map { {B_\epsilon}^-} a = \set {x \in R: \norm {x - a}_p \le \epsilon}$
Definition:Closed Ball/P-adic Numbers/Radius
In $\map { {B_\epsilon}^-} a$, the value $\epsilon$ is referred to as the '''radius''' of the closed $\epsilon$-ball.
Definition:Closed Ball/P-adic Numbers/Center
In $\map { {B_\epsilon}^-} a$, the value $a$ is referred to as the '''center''' of the closed $\epsilon$-ball.
Definition:Open Ball/P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''open $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$''' is defined as: :$\map {B_\epsilon} a = \set {x \in \Q_p: \norm{x - a}_p < \epsilon}$
Definition:Open Ball/P-adic Numbers/Radius
In $\map {B_\epsilon} a$, the value $\epsilon$ is referred to as the '''radius''' of the open $\epsilon$-ball.
Definition:Open Ball/P-adic Numbers/Center
In $\map {B_\epsilon} a$, the value $a$ is referred to as the '''center''' of the open $\epsilon$-ball.
Definition:Meridian (Terrestrial)
A '''(terrestrial) meridian''' is a semi-great circle on Earth's surface whose endpoints are Earth's poles. :600px
Definition:Meridian (Terrestrial)/Principal
The '''principal meridian''' is the meridian of Earth which passes through the {{WP|Royal_Observatory,_Greenwich|Royal Observatory}} in {{WP|Greenwich|Greenwich}}, {{WP|Greater_London|Greater London}}, {{WP|England|England}}, which is used as the internationally-recognised $0 \degrees$ reference longitude meridian.
Definition:North (Terrestrial)
'''North''' is the direction on (or near) Earth's surface along the meridian directly towards the North Pole.
Definition:South (Terrestrial)
'''South''' is the direction on (or near) Earth's surface along the meridian directly towards the South Pole.
Definition:East (Terrestrial)
'''East''' is the direction on (or near) Earth's surface along the small circle in the direction of Earth's rotation in space about its axis.
Definition:West (Terrestrial)
'''West''' is the direction on (or near) Earth's surface along the small circle in the opposite direction from Earth's rotation in space about its axis.
Definition:Cardinal Directions (Terrestrial)
The $4$ '''cardinal directions''' are the directions on (or near) Earth's surface defined as follows: === East === {{:Definition:East (Terrestrial)}} === West === {{:Definition:West (Terrestrial)}} === North === {{:Definition:North (Terrestrial)}} === South === {{:Definition:South (Terrestrial)}} :600px
Definition:Cardinal Directions
The $4$ cardinal directions: :North :South :East :West are the basis of navigation and geographical coordinates.
Definition:East
=== East (Terrestrial) === {{:Definition:East (Terrestrial)}} === East (Celestial) === {{:Definition:East (Celestial)}}
Definition:West
=== West (Terrestrial) === {{:Definition:West (Terrestrial)}} === West (Celestial) === {{:Definition:West (Celestial)}}
Definition:South
=== South (Terrestrial) === {{:Definition:South (Terrestrial)}} === South (Celestial) === {{:Definition:South (Celestial)}}
Definition:North
=== North (Terrestrial) === {{:Definition:North (Terrestrial)}} === North (Celestial) === {{:Definition:North (Celestial)}}
Definition:Intercardinal Directions
The $4$ '''intercardinal directions''' are the directions on (or near) Earth's surface halfway between the cardinal directions: === Northeast === {{:Definition:Northeast}} === Northwest === {{:Definition:Northwest}} === Southeast === {{:Definition:Southeast}} === Southwest === {{:Definition:Southwest}}
Definition:Northeast
'''Northeast''' is the direction on (or near) Earth's surface halfway between '''north''' and '''east'''.
Definition:Pole of Arc
Let $S$ be a sphere whose center is $O$. Let $C$ be the arc of a circle on $S$, either a small circle or a great circle. The '''poles''' of $C$ are the poles of the circle on $S$ of which $C$ is a part.
Definition:Spherical Angle
Let $S$ be a sphere. Let $PQ$ be a diameter of $S$. Let $PAQ$ and $PBQ$ be great circles on $S$ both passing through the points $P$ and $Q$. Let tangents $S$ and $T$ to $PAQ$ and $PBQ$ respectively be drawn through $P$. :500px Then the angle between $S$ and $T$ is known as the '''spherical angle''' between $PAQ$ and $PBQ$. Thus a '''spherical angle''' is defined with respect only to $2$ great circles.
Definition:Hemisphere
Let $S$ be a sphere. Let $S$ be bisected by a plane which passes through the center of $S$. Each of the halves of $S$ into which $S$ is divided is called a '''hemisphere'''.
Definition:Spherical Triangle/Side/Length
Let $ABC$ be a spherical triangle on a sphere $S$. Let $AB$ be a side of $ABC$. The '''length''' of $AB$ is defined as the angle subtended at the center of $S$ by $AB$.
Definition:Sphere/P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''$\epsilon$-sphere of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$''' is defined as: :$\map {S_\epsilon} a = \set {x \in \Q_p: \norm {x - a} = \epsilon}$
Definition:Sphere/P-adic Numbers/Radius
In $\map {S_\epsilon} a$, the value $\epsilon$ is referred to as the '''radius''' of the $\epsilon$-sphere.
Definition:Sphere/P-adic Numbers/Center
In $\map {S_\epsilon} a$, the value $a$ is referred to as the '''center''' of the $\epsilon$-sphere.
Definition:Earth/Shape/First Approximation
To a first approximation, the shape of '''Earth''' can be considered to be spherical.
Definition:Earth/Shape/Second Approximation
To a second approximation, '''Earth''' is an oblate spheroid.
Definition:Spheroid/Oblate
An '''oblate spheroid''' is an ellipsoid which is the solid of revolution formed by rotating an ellipse about its minor axis.
Definition:Spheroid/Prolate
A '''prolate spheroid''' is an ellipsoid which is the solid of revolution formed by rotating an ellipse about its major axis.
Definition:Spheroid
A '''spheroid''' is an ellipsoid which is the solid of revolution formed by rotating an ellipse about one of its axes. === Oblate Spheroid === {{:Definition:Oblate Spheroid}} === Prolate Spheroid === {{:Definition:Prolate Spheroid}}
Definition:Ellipsoid
An '''ellipsoid''' is a solid figure whose plane sections are circles or ellipses.
Definition:Ellipse/Axis
Let $K$ be an ellipse whose foci are at $F_1$ and $F_2$. === Major Axis === {{:Definition:Ellipse/Major Axis}} === Minor Axis === {{:Definition:Ellipse/Minor Axis}}
Definition:Longitude (Terrestrial)
Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $\bigcirc NJS$ be a meridian passing through $J$, whose endpoints are by definition $N$ and $S$. The '''longitude''' of $J$, and of the meridian $\bigcirc NJS$ itself, is the (spherical) angle that $\bigcirc NJS$ makes with the principal meridian $\bigcirc NKS$. :500px If $\bigcirc NJS$ is on the eastern hemisphere, the '''longitude''' is defined as '''longitude $n \degrees$ east''', where $n \degrees$ denotes $n$ degrees (of arc), written $n \degrees \, \mathrm E$. If $\bigcirc NJS$ is on the western hemisphere, the '''longitude''' is defined as '''longitude $n \degrees$ west''', written $n \degrees \, \mathrm W$. If $\bigcirc NJS$ is the principal meridian itself, the '''longitude''' is defined as '''$0 \degrees$ longitude'''. If $\bigcirc NJS$ is the other half of the great circle that contains the principal meridian, the '''longitude''' is defined as '''$180 \degrees$ longitude'''.
Definition:Longitude
=== Terrestrial Longitude === {{:Definition:Longitude (Terrestrial)}} === Celestial Longitude === {{:Definition:Longitude (Celestial)}} === Galactic Longitude === {{:Definition:Longitude (Galactic)}}
Definition:Latitude (Terrestrial)
Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $\bigcirc NJS$ be a meridian passing through $J$, whose endpoints are by definition $N$ and $S$. Let $\bigcirc NJS$ pass through the equator at $L$. The '''latitude''' of $J$ is the (spherical) angle $\sphericalangle LOJ$ , where $O$ is the center of Earth. :500px If $J$ is in the northern hemisphere of Earth, the '''latitude''' is defined as '''latitude $n \degrees$ north''', where $n \degrees$ denotes $n$ degrees (of arc), written $n \degrees \, \mathrm N$. If $J$ is in the southern hemisphere of Earth, the '''latitude''' is defined as '''latitude $n \degrees$ south''', written $n \degrees \, \mathrm S$. At the North Pole, the '''latitude''' is $90 \degrees \, \mathrm N$. At the South Pole, the '''latitude''' is $90 \degrees \, \mathrm S$.
Definition:Semi-Great Circle
Let $S$ be a sphere. A '''semi-great circle''' on $S$ is a semicircle consisting of half of a great circle on $S$.
Definition:Northern Hemisphere
The '''northern hemisphere''' of Earth is the hemisphere between the equator and the North Pole. Points in the northern hemisphere have latitude between $0 \degrees \, \mathrm N$ (the equator itself) and $90 \degrees \, \mathrm N$ (the North Pole).