From the previous O.D.S. stories (#3, #4) on Quintic equations (degree 5) by Galois and Abel in the 19th century, we now trace back to the first breakthrough in the 16th century of the Cubic (degree 3) & Quartic (degree 4) equations with **radical** solution, i.e. expressed by 4 operations (+ – × /) and *radical* **roots ** { }.

**Example:** Since Babylonian time, and in 220 AD China’s Three Kingdoms Period by 趙爽 Zhao Shuang of the state of **Wu 吳**, we knew the radical solution of Quadratic equations of degree 2 :

can be expressed in radical form with the coefficients a, b, c:

Are there radical solutions for Cubic equation (degree 3) and Quartic equations (degree 4) ? We had to wait till the European Renaissance Period in the 16th century Italy.

If not for Cardano, the 16th century man who stole the secretive radical solution of cubic equation from Tartaglia, Algebra would not have had progressed so rapidly since then, especially the discovery of Complex number used in solving Cubic equations.

Although on moral ground Cardano’s act was wrong, but keeping any Mathematical truth as own asset is also wrong. In this case:

Wrong x Wrong = Right

[(-) × (-)= (+)]

(Note: As a side track, interestingly the same story goes to the Chinese Taiji widespread in the world today. It was in the 19th century stolen from its secretive inventor Master Chen 陈长兴 by Yang Luchan 杨露禅, who disguised himself as his mute servant for 10 years.)