סיכומונה:ארגז החול

Archimedes was a Greek mathematician who is best known for the myriad mathematical notations that he invented, most of which are still in use today. His earliest work included devising simple inline equations such as [math]\displaystyle{ \sin x = \cos^2(y+t) }[/math] and [math]\displaystyle{ x^2 + y^2 = -e^{-\theta} }[/math]. He pioneered the use of greek symbols such as [math]\displaystyle{ \alpha }[/math] in English writing. While performing complicated calculations such as [math]\displaystyle{ \sum_{i=1}^3 i^2 = 47 }[/math], he noticed that despite the baseline of the equation lining up nicely with the surrounding text, the so-called displayed equation

[math]\displaystyle{ \displayed\sum_{i=1}^3 i = 46, \qquad \textrm{unless} 46 \not= 47 }[/math]

was probably better value. A similar effect occurred for integrals such as [math]\displaystyle{ \int_0^1 \sin^2 x \, dx }[/math]. He marked this up using the kludgy "\displayed" command, although he suspected that later and greater thinkers would come up with something better. When he couldn't make up his mind he would write

[math]\displaystyle{ \displayed F(x) = \begin{cases} \left\uparrow\frac{\partial^2 G}{\partial u \partial v}\right\} & \textrm{if the sky was \bf blue}, \\ A_0 + \cdots + A_k & \textit{if Troy was on the attack.} \end{cases} }[/math]

He also invented the polynomial rings [math]\displaystyle{ \mathbf{R}[x] }[/math], [math]\displaystyle{ \mathcal{C}[y] }[/math] and [math]\displaystyle{ \boldsymbol{\mathcal{C}[z]} }[/math], and being fluent in Chinese he was comfortable writing things like

[math]\displaystyle{ \displayed 钱 = \sqrt{不好}, }[/math]

although historians have debated whether his Chinese really was all that good.