# The Authorea Team

 Pythagoras’ Theorem $$a^{2}+b^{2}=c^{2}$$ Pythagoras, 530 BC Clay tablet Logarithms $$\log{xy}=\log{x}+log{y}$$ John Napier, 1610 Book Calculus $$\frac{\mathrm{d}f}{\mathrm{d}t}=\lim_{h\to 0}\frac{f(t~{}+~{}h)~{}-~{}f(t)}{h}$$ Newton, 1668 Book Law of Gravity $$F=G\frac{m_{1}m_{2}}{r^{2}}$$ Newton, 1687 Book The Square Root of Minus One $$\mathrm{i}^{2}=-1$$ Euler, 1750 Open Euler’s Formula for Polyhedra $$V-E+F=2$$ Euler, 1751 Open Normal Distribution $$\psi(x)=\frac{1}{\sqrt{2\pi\rho}}e^{\frac{(x~{}-~{}\mu)^{2}}{2~{}\rho^{2}}}$$ C. F. Gauss, 1810 Open Wave Equation $$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial x^{2}}$$ J. D‘Ambert, 1746 Fourier Transform $$f(\omega)=\int_{-\infty}^{\infty}f(x)e^{-2~{}\pi~{}\mathrm{i}~{}x~{}\omega}\mathrm{d}x$$ J. Fourier, 1822 Book Navier-Stokes Equation $$\rho\left(\frac{\partial\mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\right)=-\nabla p+\nabla\cdot T+f$$ C. Navier, G. Stokes, 1845 Maxwell’s Equations $$\nabla\cdot E=0$$ J. C. Maxwell, 1865 Open $$\nabla\times E=-\frac{1}{e}\frac{\partial H}{\partial t}$$ $$\nabla\cdot H=0$$ $$\nabla\times H=\frac{1}{e}\frac{\partial E}{\partial t}$$ Second Law of Thermodynamics $$\mathrm{d}S\geq 0$$ L. Boltzmann, 1874 PAYWALL Relativity $$E=mc^{2}$$ Einstein, 1905 PAYWALL Schrödinger’s Equation $$\mathrm{i}\hbar\frac{\partial}{\partial t}\psi=H\psi$$ E. Schrödinger, 1927 PAYWALL Information Theory $$H=-\sum p(x)\log{p(x)}$$ C. Shannon, 1949 PAYWALL Chaos Theory $$x_{t+1}=k~{}x_{t}(1-x_{t})$$ Robert May, 1975 PAYWALL Black-Scholes Equation $$\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}+rS\frac{\partial V}{\partial S}+\frac{\partial V}{\partial t}-rV=0$$ F. Black, M. Scholes, 1990 PAYWALL Euler’s Transformation $$\sum_{n=0}^{\infty}(-1)^{n}a_{n}=\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta^{n}a_{0}}{2^{n+1}}$$ Euler, 1755 PAYWALL Russell’s Paradox Let $$R=\{x\mid x\not\in x\}$$, then $$R\in R\iff R\not\in R$$ Russell, 1902 Letter Gödel’s Incompleteness Theorem $$G(x):=\neg\text{Prov}(\text{sub}(x,x))\Rightarrow\text{PA}\vdash G(\ulcorner G\urcorner)\leftrightarrow\neg\text{Prov}(\ulcorner G(\ulcorner G\urcorner)\urcorner)$$ Gödel, 1931 Open