hcju/proofwikips huggingface dataset:
Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE
Consider the Cauchy-Euler equation: :$(1): \quad x^2 \dfrac {\d^2 y} {\d x^2} + p x \dfrac {\d y} {\d x} + q y = 0$ By making the substitution: :$x = e^t$ it is possible to convert $(1)$ into a constant coefficient homogeneous linear second order ODE: :$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfrac {\d y} {\d t} + q y = 0$
Linear Second Order ODE/x^2 y'' + 2 x y' - 12 y = 0
The second order ODE: :$(1): \quad x^2 y'' + 2 x y' - 12 y = 0$ has the general solution: :$y = C_1 x^3 + C_2 x^{-4}$
Cardinality of Image of Set not greater than Cardinality of Set
Let $X, Y$ be sets. Let $f:X \to Y$ be a mapping. Let $A$ be a subset of $X$. Then $\left\vert{f^\to\left({A}\right)}\right\vert \le \left\vert{A}\right\vert$ where $\left\vert{A}\right\vert$ denotes the cardinality of $A$.
Projector has Norm 1
An idempotent operator $P$ is a projector on the Hilbert Space $H$  $P$ has norm $1$: :$\displaystyle \norm P \equiv \sup_{x \mathop \in H} \frac {\norm P} {\norm x} = 1$
Linear Second Order ODE/y'' + 3 y' - 10 y = 0
The second order ODE: :$(1): \quad y'' + 3 y' - 10 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-5 x}$
Linear Second Order ODE/y'' + 4 y = 0
The second order ODE: :$(1): \quad y'' + 4 y = 0$ has the general solution: :$y = C_1 \cos 2 x + C_2 \sin 2 x$
Linear Second Order ODE/y'' + 4 y = 3 sine x
The second order ODE: :$(1): \quad y'' + 4 y = 3 \sin x$ has the general solution: :$y = C_1 \cos 2 x + C_2 \sin 2 x + \sin x$
Linear Second Order ODE/y'' + 10 y' + 25 y = 0
The second order ODE: :$(1): \quad y'' + 10 y' + 25 y = 0$ has the general solution: :$y = C_1 e^{-5 x} + C_2 x e^{-5 x}$
Linear Second Order ODE/y'' + 10 y' + 25 y = 14 exp -5 x
The second order ODE: :$(1): \quad y'' + 10 y' + 25 y = 14 e^{-5 x}$ has the general solution: :$y = C_1 \cos 2 x + C_2 \sin 2 x + \sin x$
Linear Second Order ODE/y'' - 2 y' + 5 y = 0
The second order ODE: :$(1): \quad y'' - 2 y' + 5 y = 0$ has the general solution: :$y = e^x \paren {C_1 \cos 2 x + C_2 \sin 2 x}$
Linear Second Order ODE/y'' + y' - 12 y = 0
The second order ODE: :$(1): \quad y'' + y' - 12 y = 0$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-4 x}$
Linear Second Order ODE/y'' + 2 y' + 10 y = 0
The second order ODE: :$(1): \quad y'' + 2 y' + 10 y = 0$ has the general solution: :$y = e^{-x} \paren {C_1 \cos 3 x + C_2 \sin 3 x}$
Linear Second Order ODE/y'' + 4 y' + 4 y = 0
The second order ODE: :$(1): \quad y'' + 4 y' + 4 y = 0$ has the general solution: :$y = \paren {C_1 + C_2 x} e^{-2 x}$
Linear Second Order ODE/y'' + y' - 2 y = 0
The second order ODE: :$(1): \quad y'' + y' - 2 y = 0$ has the general solution: :$y = C_1 e^x + C_2 e^{-2 x}$
Linear Second Order ODE/y'' - 4 y = 0
The second order ODE: :$(1): \quad y'' - 4 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-2 x}$
Linear Second Order ODE/y'' - y' - 6 y = 0
The second order ODE: :$(1): \quad y'' - y' - 6 y = 0$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-2 x}$
Linear Second Order ODE/y'' - y' - 6 y = 20 exp -2 x
The second order ODE: :$(1): \quad y'' - y' - 6 y = 20 e^{-2 x}$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-2 x} - 4 x e^{-2 x}$
Linear Second Order ODE/y'' - 2 y' = 0
The second order ODE: :$(1): \quad y'' - 2 y' = 0$ has the general solution: :$y = C_1 + C_2 e^{2 x}$
Directed iff Finite Subsets have Upper Bounds
Let $\left({S, \precsim}\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is directed  :for every a finite subset $A$ of $H$ ::$\exists h \in H: \forall a \in A: a \precsim h$
Linear Second Order ODE/y'' + k^2 y = sine b x
The second order ODE: :$(1): \quad y'' + k^2 y = \sin b x$ has the general solution: :$y = \begin{cases} C_1 \sin k x + C_2 \cos k x + \dfrac {\sin b x} {k^2 - b^2} & : b \ne k \\ C_1 \sin k x + C_2 \cos k x - \dfrac {x \cos k x} {2 k} & : b = k  \end{cases}$
Combination of Solutions to Non-Homogeneous LSOODE with same Homogeneous Part
Let $\map {y_1} x$ be a particular solution of the linear second order ODE: :$(1): \quad y'' + \map P x y' + \map Q x y = \map {R_1} x$ Let $\map {y_2} x$ be a particular solution of the linear second order ODE: :$(2): \quad y'' + \map P x y' + \map Q x y = \map {R_2} x$ Then $\map y x = \map {y_1} x + \map {y_2} x$ is a particular solution of the linear second order ODE: :$(3): \quad y'' + \map P x y' + \map Q x y = \map {R_1} x + \map {R_2} x$
Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x + 6 cosine x + 8 x^2 - 4 x
The second order ODE: :$(1): \quad y'' + 4 y = 4 \cos 2 x + 6 \cos x + 8 x^2 - 4 x$ has the general solution: :$y = C_1 \sin 2 x + C_2 \cos 2 x + x \sin 2 x + 2 \cos x - 1 - x + 2 x^2$
Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x
The second order ODE: :$(1): \quad y'' + 4 y = 4 \cos 2 x$ has the general solution: :$y = C_1 \sin 2 x + C_2 \cos 2 x + x \sin 2 x$
Linear Second Order ODE/y'' + 4 y = 6 cosine x
The second order ODE: :$(1): \quad y'' + 4 y = 6 \cos x$ has the general solution: :$y = C_1 \sin 2 x + C_2 \cos 2 x + 2 \cos x$
Linear Second Order ODE/y'' + 4 y = 8 x^2 - 4 x
The second order ODE: :$(1): \quad y'' + 4 y = 8 x^2 - 4 x$ has the general solution: :$y = C_1 \sin 2 x + C_2 \cos 2 x - 1 - x + 2 x^2$
Linear Second Order ODE/y'' - 2 y' + y = 0
The second order ODE: :$(1): \quad y'' - 2 y' + y = 0$ has the general solution: :$y = C_1 e^x + C_2 x e^x$
Linear Second Order ODE/y'' - 2 y' + y = 2 x
The second order ODE: :$(1): \quad y'' - 2 y' + y = 2 x$ has the general solution: :$y = C_1 e^x + C_2 x e^x + 2 x + 4$
Linear Second Order ODE/y'' - y' - 6 y = exp -x
The second order ODE: :$(1): \quad y'' - y' - 6 y = e^{-x}$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-2 x} - \dfrac {e^{-x} } 4$
Filtered iff Finite Subsets have Lower Bounds
Let $\struct {S, \precsim}$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is filtered : :for every a finite subset $A$ of $H$ ::$\exists h \in H: \forall a \in A: h \precsim a$
Tangent of Half Angle plus Quarter Pi
:$\map \tan {\dfrac x 2 + \dfrac \pi 4} = \tan x + \sec x$
Linear Second Order ODE/y'' + 4 y = tangent 2 x
The second order ODE: :$(1): \quad y'' + 4 y = \tan 2 x$ has the general solution: :$y = C_1 \cos 2 x + C_2 \sin 2 x - \dfrac 1 4 \cos 2 x \, \map \ln {\sec 2 x + \tan 2 x}$
Linear Second Order ODE/y'' + 2 y' + y = exp -x log x
The second order ODE: :$(1): \quad y'' + 2 y' + y = e^{-x} \ln x$ has the general solution: :$y = C_1 e^{-x} + C_2 x e^{-x} - \dfrac {x^2 e^{-x} \ln x} 2 - \dfrac 3 4 x^2 e^{-x}$
Linear Second Order ODE/y'' - 2 y' - 3 y = 0
The second order ODE: :$(1): \quad y'' - 2 y' - 3 y = 0$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-x}$
Linear Second Order ODE/y'' + 2 y' + 5 y = 0
The second order ODE: :$(1): \quad y'' + 2 y' + 5 y = 0$ has the general solution: :$y = e^{-x} \paren {C_1 \cos 2 x + C_2 \sin 2 x}$
Linear Second Order ODE/y'' + 2 y' + 5 y = exp -x secant 2 x
The second order ODE: :$(1): \quad y'' + 2 y' + 5 y = e^{-x} \sec 2 x$ has the general solution: :$y = e^{-x} \paren {C_1 \cos 2 x + C_2 \sin 2 x} + \dfrac {x e^{-x} \sin 2 x} 2 + \dfrac {e^{-x} \cos 2 x \ln \cos 2 x} 4$
Singleton is Directed and Filtered Subset
Let $\struct {S, \precsim}$ be a preordered set. Let $x$ be an element of $S$. Then $\set x$ is directed and filtered subset of $S$.
Second Order ODE/(x^2 - 1) y'' - 2 x y' + 2 y = 0
The second order ODE: :$(1): \quad \paren {x^2 - 1} y'' - 2 x y' + 2 y = 0$ has the general solution: :$y = C_1 x + C_2 \paren {x^2 + 1}$
Linear Second Order ODE/(x^2 - 1) y'' - 2 x y' + 2 y = (x^2 - 1)^2
The second order ODE: :$(1): \quad \paren {x^2 - 1} y'' - 2 x y' + 2 y = \paren {x^2 - 1}^2$ has the general solution: :$y = C_1 x + C_2 \paren {x^2 + 1} + \dfrac {x^4} 6 - \dfrac {x^2} 2$
Directed iff Lower Closure Directed
Let $\left({S, \precsim}\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is directed  $H^\precsim$ is directed where $H^\precsim$ denotes the lower closure of set.
Filtered iff Upper Closure Filtered
Let $\left({S, \precsim}\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is filtered  $H^\succsim$ is filtered where $H^\succsim$ denotes the upper closure of set.
Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0
The second order ODE: :$(1): \quad \paren {x^2 + x} y'' + \paren {2 - x^2} y' - \paren {2 + x} y = 0$ has the general solution: :$y = C_1 e^x + \dfrac {C_2} x$
Position of Cart attached to Wall by Spring/x = x0 at t = 0
Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the horizontal position of $C$ at time $t$ can be expressed as: :$x = x_0 \cos \alpha t$
Directed in Join Semilattice
Let $\left({S, \preceq}\right)$ be a join semilattice. Let $H$ be a non-empty lower subset of $S$. Then $H$ is directed  :$\forall x, y \in H: x \vee y \in H$
Position of Cart attached to Wall by Spring under Damping/Underdamped/x = x0 at t = 0
Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the horizontal position of $C$ at time $t$ can be expressed as: :$x = \dfrac {x_0} \alpha e^{-b t} \left({\alpha \cos \alpha t + b \sin \alpha t}\right)$
Canonical Form of Underdamped Oscillatory System
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the value of $x$ can be expressed in the form: :$x = \dfrac {x_0 \, a} \alpha e^{-b t} \, \map \cos {\alpha t - \theta}$ where: :$\alpha = \sqrt {a^2 - b^2}$ :$\theta = \map \arctan {\dfrac b \alpha}$ This can be referred to as the '''canonical form''' of the solution of $(1)$.
Period of Oscillation of Underdamped System is Regular
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$\dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the period of its movement is well-defined, in the sense that its zeroes are regularly spaced, and given by: :$T = \dfrac {2 \pi} {\sqrt {a^2 - b^2} }$
Period of Oscillation of Underdamped Cart attached to Wall by Spring
Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the period of oscillation of $C$ can be expressed as: :$T = \dfrac {2 \pi} {\sqrt {\dfrac k m - \dfrac {c^2} {4 m^2} } }$
Natural Frequency of Underdamped System
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$\dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the natural frequency of $S$ is given by: :$\nu = \dfrac {\sqrt {a^2 - b^2} } {2 \pi}$
Natural Frequency of Underdamped Cart attached to Wall by Spring
Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the natural frequency of $C$ can be expressed as: :$\nu = \dfrac 1 {2 \pi} \sqrt {\dfrac k m - \dfrac {c^2} {4 m^2} }$
Linear Second Order ODE/y'' + 2 b y' + a^2 y = 0/b less than a
The second order ODE: :$(1): \quad y'' + 2 b y' + a^2 y = 0$ where $b^2 < a^2$ has the general solution: :$y = e^{-b x} \paren {C_1 \, \map \cos {\sqrt {a^2 - b^2} } x + C_2 \, \map \sin {\sqrt {a^2 - b^2} } x}$
Linear Second Order ODE/y'' + 2 b y' + a^2 y = K cosine omega x/b less than a
The second order ODE: :$(1): \quad y'' + 2 b y' + a^2 y = K \cos \omega x$ where $b^2 < a^2$ has the general solution: :$y = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x} + \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$ where: :$\alpha = \sqrt {a^2 - b^2}$ :$\phi = \map \arctan {\dfrac {2 b \omega} {a^2 - \omega^2} }$
Condition for Resonance in Forced Vibration of Underdamped System
Consider a physical system $S$ whose behaviour is defined by the second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$ where: :$K \in \R: k > 0$ :$a, b \in \R_{>0}: b < a$ Then $S$ is in resonance when: :$\omega = \sqrt {a^2 - 2 b^2}$ and thus the resonance frequency is: :$\nu_R = \dfrac {\sqrt {a^2 - 2 b^2} } {2 \pi}$ This resonance frequency exists only when $a^2 - 2 b^2 > 0$.
Filtered in Meet Semilattice
Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $H$ be a non-empty upper subset of $S$. Then $H$ is filtered  :$\forall x, y \in H: x \wedge y \in H$
Directed in Join Semilattice with Finite Suprema
Let $\left({S, \preceq}\right)$ be a join semilattice. Let $H$ be a non-empty lower subset of $S$. Then $H$ is directed  :for every non-empty finite subset $A$ of $H$, $\sup A \in H$
Filtered in Meet Semilattice with Finite Infima
Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $H$ be a non-empty lower subset of $S$. Then $H$ is filtered  :for every non-empty finite subset $A$ of $H$, $\inf A \in H$
Interval between Local Maxima for Underdamped Free Vibration
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\mathrm d^2 x} {\mathrm d t^2} + 2 b \dfrac {\mathrm d x} {\mathrm d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px Let $T$ be the period of oscillation of $S$. Then the successive local maxima of $x$ occur for $t = 0, T, 2T, \ldots$
Ratio of Successive Local Maxima for Underdamped Free Vibration
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\mathrm d^2 x} {\mathrm d t^2} + 2 b \dfrac {\mathrm d x} {\mathrm d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px Let $T$ be the period of oscillation of $S$. Let $x_1$ and $x_2$ be successive local maxima of $x$. Then: :$\dfrac {x_1} {x_2} = e^{b T}$
Motion of Particle in Polar Coordinates
Consider a particle $p$ of mass $m$ moving in the plane under the influence of a force $\mathbf F$. Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$. Let $\mathbf F$ be expressed as: :$\mathbf F = F_r \mathbf u_r + F_\theta \mathbf u_\theta$ where: :$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$ :$F_r$ and $F_\theta$ are the magnitudes of the components of $\mathbf F$ in the directions of $\mathbf u_r$ and $\mathbf u_\theta$ respectively. Then the second order ordinary differential equations governing the motion of $m$ under the force $\mathbf F$ are: {{begin-eqn}} {{eqn | l = F_\theta       | r = m \left({r \dfrac {\mathrm d^2 \theta} {\mathrm d t^2} + 2 \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \theta} {\mathrm d t} }\right)       | c =  }} {{eqn | l = F_r       | r = m \left({\dfrac {\mathrm d^2 r} {\mathrm d t^2} - r \left({\dfrac {\mathrm d \theta} {\mathrm d t} }\right)^2}\right)       | c =  }} {{end-eqn}}
Derivative of Angular Component under Central Force
Let a point mass $p$ of mass $m$ be under the influence of a central force $\mathbf F$. Let the position of $p$ at time $t$ be given in polar coordinates as $\polar {r, \theta}$. Let $\mathbf r$ be the radius vector from the origin to $p$. Then the rate of change of the angular coordinate of $p$ is inversely proportional to the square of the radial coordinate of $p$.
Linear Second Order ODE/y'' + y = K
The second order ODE: :$(1): \quad y'' + y = K$ has the general solution: :$y = C_1 \sin x + C_2 \cos x + K$
Velocity Vector in Polar Coordinates
Consider a particle $p$ moving in the plane. Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$. Then the velocity $\mathbf v$ of $p$ can be expressed as: :$\mathbf v = r \dfrac {\mathrm d \theta} {\mathrm d t} \mathbf u_\theta + \dfrac {\mathrm d r} {\mathrm d t} \mathbf u_r$ where: :$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$
Derivatives of Unit Vectors in Polar Coordinates
Consider a particle $p$ moving in the plane. Let the position of $p$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$. Let: :$\mathbf u_r$ be the unit vector in the direction of the radial coordinate of $p$ :$\mathbf u_\theta$ be the unit vector in the direction of the angular coordinate of $p$ Then the derivative of $\mathbf u_r$ and $\mathbf u_\theta$  $\theta$ can be expressed as: {{begin-eqn}} {{eqn | l = \dfrac {\mathrm d \mathbf u_r} {\mathrm d \theta}       | r = \mathbf u_\theta }} {{eqn | l = \dfrac {\mathrm d \mathbf u_\theta} {\mathrm d \theta}       | r = -\mathbf u_r }} {{end-eqn}}
Acceleration Vector in Polar Coordinates
Consider a particle $p$ moving in the plane. Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$. Then the acceleration $\mathbf a$ of $p$ can be expressed as: :$\mathbf a = \left({r \dfrac {\mathrm d^2 \theta} {\mathrm d t^2} + 2 \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \theta} {\mathrm d t} }\right) \mathbf u_\theta + \left({\dfrac {\mathrm d^2 r} {\mathrm d t^2} - r \left({\dfrac {\mathrm d \theta} {\mathrm d t} }\right)^2}\right) \mathbf u_r$ where: :$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$
Existence of Non-Empty Finite Suprema in Join Semilattice
Let $\left({S, \preceq}\right)$ be a join semilattice. Let $A$ be a non-empty finite subset of $S$. Then $A$ admits a supremum in $\left({S, \preceq}\right)$.
Euler's Product form of Riemann Zeta Function
Let $s \in \R: s > 1$. Then: :$\displaystyle \sum_{k \mathop \in \N_{>0} } \dfrac 1 {k^s} = \prod_{p \mathop \in \Bbb P} \dfrac 1 {1 - 1 / p^s}$ where $\Bbb P$ denotes the set of all prime numbers.
Variance of Exponential Distribution
Let $X$ be a continuous random variable with the exponential distribution with parameter $\beta$. Then the variance of $X$ is: :$\var X = \beta^2$
Cardinality of Singleton
Let $A$ be a set. Then $\left\vert{A}\right\vert = 1$  $\exists a: A = \left\{ {a}\right\}$ where $\left\vert{A}\right\vert$ denotes the cardinality of $A$.
Existence of Non-Empty Finite Infima in Meet Semilattice
Let $\struct {S, \preceq}$ be a meet semilattice. Let $A$ be a non-empty finite subset of $S$. Then $A$ admits a infimum in $\struct {S, \preceq}$.
Thales' Theorem/Historical Note
This result was known by the ancient Babylonians as early as $2000$ BCE. The proof of what is now known as Thales' Theorem that {{AuthorRef|Thales of Miletus}} actually used is unknown.  Whether he was aware of the Interior Angles Theorem and used it in his proof cannot be decided. The Interior Angles Theorem is traditionally ascribed to {{AuthorRef|Pythagoras of Samos|Pythagoras}}, who lived near {{AuthorRef|Thales of Miletus|Thales}}, and may have met him. Thus it is possible that {{AuthorRef|Pythagoras of Samos|Pythagoras}} learned of the Interior Angles Theorem either directly or indirectly from {{AuthorRef|Thales of Miletus|Thales}} himself.
External Angle of Triangle equals Sum of other Internal Angles
The external angle of a triangle equals the sum of the other two internal angles. {{:Euclid:Proposition/I/32}}
Pappus's Centroid Theorem
=== Pappus's Centroid Theorem for Surface Area === {{:Pappus's Centroid Theorem/Surface Area}} === Pappus's Centroid Theorem for Volume === {{:Pappus's Centroid Theorem/Volume}}
Pappus's Centroid Theorem/Volume
Let $C$ be a plane figure that lies entirely on one side of a straight line $L$. Let $S$ be the solid of revolution generated by $C$ around $L$. Then the volume of $S$ is equal to the area of $C$ multiplied by the distance travelled by the centroid of $C$ around $L$ when generating $S$.
Pappus's Centroid Theorem/Surface Area
Let $C$ be a plane figure that lies entirely on one side of a straight line $L$. Let $S$ be the solid of revolution generated by $C$ around $L$. Then the surface area of $S$ is equal to the perimeter length of $C$ multiplied by the distance travelled by the centroid of $C$ around $L$ when generating $S$.
Pythagoras's Theorem for Parallelograms
Let $\triangle ABC$ be a triangle. Let $ACDE$ and $BCFG$ be parallelograms constructed on the sides $AC$ and $BC$ of $\triangle ABC$. Let $DE$ and $FG$ be produced to intersect at $H$. Let $AJ$ and $BI$ be constructed on $A$ and $B$ parallel to and equal to $HC$. Then the area of the parallelogram $ABIJ$ equals the sum of the areas of the parallelograms $ACDE$ and $BCFG$.
Mapping Preserves Finite and Filtered Infima
Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be meet semilattices. Let $f: S_1 \to S_2$ be a mapping. Let $f$ preserve finite infima and preserve filtered infima. Then $f$ also preserves all infima
Volume of Sphere from Surface Area
The volume $V$ of a sphere of radius $r$ is given by: :$V = \dfrac {r A} 3$ where $A$ is the surface area of the sphere.
Volume of Solid of Revolution/Historical Note
The technique of finding the Volume of Solid of Revolution by dividing up the solid of revolution into many thin disks and approximating them to cylinders was devised by {{AuthorRef|Johannes Kepler}} sometime around or after $1612$, reportedly on the occasion of his wedding in $1613$. His inspiration was in the problem of finding the volume of wine barrels accurately. He published his technique in his $1615$ work {{BookLink|Nova Stereometria Doliorum Vinariorum|Johannes Kepler}} (New Stereometry of Wine Barrels). {{AuthorRef|Gottfried Wilhelm von Leibniz}} redefined the problem by applying the techniques of integral calculus around $1680$.
Factors of Mersenne Number M67
The Mersenne number $M_{67}$ has the factors: :$193 \, 707 \, 721$ :$761 \, 838 \, 257 \, 287$
Lower Bound is Lower Bound for Subset
Let $\struct {S, \preceq}$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$B \subseteq A$ Let $L$ be an element of $S$. Let $L$ be a lower bound for $A$. Then $L$ is a lower bound for $B$.
Upper Bound is Upper Bound for Subset
Let $\left({S, \preceq}\right)$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$B \subseteq A$ Let $U$ be an upper bound for $A$. Then $U$ is an upper bound for $B$.
Largest Rectangle with Given Perimeter is Square
Let $\SS$ be the set of all rectangles with a given perimeter $L$. The element of $\SS$ with the largest area is the square with length of side $\dfrac L 4$.
Mapping Preserves Finite and Directed Suprema
Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be join semilattices. Let $f: S_1 \to S_2$ be a mapping. Let $f$ preserve finite suprema and preserve directed suprema. Then $f$ also preserves all suprema
Integral of Power/Historical Note
The conventional proof of Integral of Power of course holds for all real $n \ne -1$, not just where $n$ is a strictly positive rational. However, the real point of this page is Fermat's proof, which demonstrates how integration was achieved before the full machinery of calculus had been thoroughly constructed. {{AuthorRef|Bonaventura Francesco Cavalieri}} had previously made progress with this problem, proving it for integral $1 \le n \le 9$ but the algebra for the proof of each power was more difficult than the previous one, and he found $10$ too much hard work. The clear beauty of {{AuthorRef|Pierre de Fermat|Fermat}}'s approach was that it works for ''all'' $n$, rational as well as integral.
Integer as Sum of Polygonal Numbers
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then $n$ is: :$(1): \quad$ Either triangular or the sum of $2$ or $3$ triangular numbers :$(2): \quad$ Either square or the sum of $2$, $3$ or $4$ square numbers :$(3): \quad$ Either pentagonal or the sum of $2$, $3$, $4$ or $5$ pentagonal numbers :and so on. That is: :for all $k \ge 3$, $n$ is the sum of no more than $k$ polygonal numbers of order $k$.