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## ca. 1850 BC: Babylonian Mathematics

As the Babylonians subdued the Sumers in the 18th century B.C, they have taken over their sexagesimal system and refined the underlying mathematics from arithmetics to algebra. Babylonians were able to solve linear and quadratic equations with up to two variables and also cubic equations of the form $$x^3+x^2=a$$ for a given $a.$ Geometrical problems were often transformed into their algebraic forms. Babylonians were in the possession of formulae for the areas of simple right-angled figures, including the so-called Pythagorean theorem. Square roots have been approximated using the formula $$a=\sqrt{A}=\frac 12\left(a+\frac Aa\right).$$

Babylonian mathematics allowed to compute compound interest, for instance, problems like how long would it take to double a sum of money if the interest was $20$%. This results in the equation $$\left(1\frac 15\right)^x=2.$$ Although they did not know logarithms, they solved such equations by finding that a solution $x$ must be between $> 3$ and $< 4$ and further approximating the result by a linear interpolation.

Babylonians also studied astronomy, including the movements of the Sun, the Moon and the Planets. Especially calculations related to astronomy had to be more precise than those related to every-day measurements or taxes. Some of them have been performed with the precision up to several sexagesimal digits after the comma.

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