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\title{Paywalling the laws of the universe.}


 \author{The Authorea Team}


\date{}

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\begin{longtable}{l l r l}
        \textbf{Pythagoras' Theorem} & $a^2 + b^2 = c^2$ & \textit{Pythagoras, 530 BC} & \text{Clay tablet} \\
        \hline
        \textbf{Logarithms} & $\log{x y} = \log{x} + log{y}$ & \textit{John Napier, 1610} & \text{Book} \\
        \hline
        \textbf{Calculus} & $\frac{\mathrm{d} f}{\mathrm{d} t} = \lim_{h \to 0} \frac{f(t~+~h)~-~f(t)}{h}$ & \textit{Newton, 1668} & \text{Book} \\
        \hline
        \textbf{Law of Gravity} & $F = G \frac{m_1 m_2}{r^2}$ & \textit{Newton, 1687} & \text{Book} \\
        \hline
        \textbf{The Square Root of Minus One} & $\mathrm{i}^2 = -1$ & \textit{Euler, 1750} & \text{\href{http://eulerarchive.maa.org/docs/originals/E123.pdf}{Open}} \\
        \hline
        \textbf{Euler's Formula for Polyhedra} & $V - E + F = 2$ & \textit{Euler, 1751} & \text{\href{http://eulerarchive.maa.org//docs/originals/E231.pdf}{Open}} \\
        \hline
        \textbf{Normal Distribution} & $\psi(x) = \frac{1}{\sqrt{2 \pi \rho}} e^\frac{(x~-~\mu)^2}{2~\rho^2}$ & \textit{C. F. Gauss, 1810} & \text{\href{https://www.deutsche-digitale-bibliothek.de/item/7J3XQTJRDWCLEECZ77HEL72YMSFHGNXX}{Open}} \\
        \hline
        \textbf{Wave Equation} & $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ & \textit{J. D`Ambert, 1746} & \\
        \hline
        \textbf{Fourier Transform} & $f(\omega) = \int_{-\infty}^\infty f(x) e^{-2~\pi~\mathrm{i}~x~\omega} \mathrm{d}x$ & \textit{J. Fourier, 1822} & \text{Book} \\
        \hline
        \textbf{Navier-Stokes Equation} & $\rho \left ( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right ) = - \nabla p + \nabla \cdot T + f$ & \textit{C. Navier, G. Stokes, 1845} & \\
        \hline
        \textbf{Maxwell's Equations} & $\nabla \cdot E = 0$ & \textit{J. C. Maxwell, 1865} & \text{\href{http://rstl.royalsocietypublishing.org/content/155/459.full.pdf+html}{Open}}  \\
        & $\nabla \times E = - \frac{1}{e} \frac{\partial H}{\partial t}$ & \\
        & $\nabla \cdot H = 0$ & \\
        & $\nabla \times H = \frac{1}{e} \frac{\partial E}{\partial t}$ & \\
        \hline
        \textbf{Second Law of Thermodynamics} & $\mathrm{d}S \geq 0$ & \textit{L. Boltzmann, 1874} & \textbf{\href{http://dx.doi.org/10.1017/CBO9781139381420.029}{{PAYWALL}}} \\
        \hline
        \textbf{Relativity} & $E = m c^2$ & \textit{Einstein, 1905} & \textbf{\href{http://onlinelibrary.wiley.com/doi/10.1002/andp.200590007/abstract}{PAYWALL}} \\
        \hline
        \textbf{Schr\"{o}dinger's Equation} & $\mathrm{i} \hbar \frac{\partial}{\partial t} \psi = H \psi$ & \textit{E. Schr\"{o}dinger, 1927} & \textbf{\href{http://journals.aps.org/pr/abstract/10.1103/PhysRev.28.1049}{{PAYWALL}}} \\
        \hline
        \textbf{Information Theory} & $H = - \sum p(x) \log{p(x)}$ & \textit{C. Shannon, 1949} & \textbf{\href{http://onlinelibrary.wiley.com/doi/10.1002/j.1538-7305.1948.tb01338.x/abstract}{PAYWALL}} \\
        \hline
        \textbf{Chaos Theory} & $x_{t+1} = k~x_t (1 - x_t)$ & \textit{Robert May, 1975} & \textbf{\href{http://http://www.nature.com/nature/journal/v261/n5560/abs/261459a0.html}{{PAYWALL}}} \\
        \hline
        \textbf{Black-Scholes Equation} & $\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} + \frac{\partial V}{\partial t} - r V = 0$ & \textit{F. Black, M. Scholes, 1990} & \textbf{\href{http://www.jstor.org/stable/1831029?seq=1\#page\_scan\_tab\_contents}{PAYWALL}} \\
        \hline
%        \textbf{A Sum of Euler} & $\sum_{i = 1}^\infty (-1)^{1 + n}~n = - \frac{1}{4}$ & \textit{Euler, 1755} & \textbf{\href{http://www.springer.com/us/book/9780387985343}{PAYWALL}} \\
        \textbf{Euler's Transformation} & $\sum_{n = 0}^\infty (-1)^n a_n = \sum_{n=0}^\infty (-1)^n \frac{\Delta^n a_0}{2^{n+1}}$ & \textit{Euler, 1755} & \textbf{\href{http://www.springer.com/us/book/9780387985343}{PAYWALL}} \\
        \hline
        \textbf{Russell's Paradox} & Let $R = \{ x \mid x \not \in x \}$, then $R \in R \iff R \not \in R$ & \textit{Russell, 1902} & \text{Letter} \\
        \hline
        \textbf{G\"{o}del's Incompleteness Theorem} & $G(x) := \neg\text{Prov}(\text{sub}(x,x))\Rightarrow\text{PA}\vdash G(\ulcorner G\urcorner)\leftrightarrow\neg\text{Prov}(\ulcorner G(\ulcorner G\urcorner)\urcorner)$ & \textit{G\"{o}del, 1931} & \text{\href{https://dx.doi.org/10.1007/s00605-006-0423-7 }{{Open}}}
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