History of Cubic & Quartic equations

High school

  • geometry
  • Trick
  • algebra
  • calculus

Today is about Algebra:

History

100CE：Heron

300CE: Dophmglvs

Constantie

The existence of Hindu Arabic numerals allow for Algebra to flourish.

You for the first time have access to a different set of symbols for numbers as opposed to unknown variables.

1494: Luca Pacioli : Summa di Arithmetica

give methods to solve linear equations, quadratic equations;

for a ax+b=0ax+b=0, you don’t need a linear formula like x=x=

but for a ax2+bx+c=0ax^2+bx+c=0, you shou have a

x=−b+−b2−4ac2ax=\frac{-b+-\sqrt{b^2-4ac}}{2a}

And it is completely different from : ax2+bx=cax^2 + bx = c or ax2+c=bxax^2 + c = bx

What is a x2x^2

parabola? Need a cartisine plane.

It means :

image-20260728211425904

There is not One quadratic formula, but many, because they’re using diagrams for squares and rectangles, and thinking of adding up areas

Negative numbers don’t exist, negative areas don’t make sense to them

Pacioli cubit equations are probably as unsolvable as something like squaring the circle. (Note: he doen’t yet know that the latter really IS unsolvable)

del Ferro: finds a method to solve “depressed” cubics

Culture : mathematical duels = jobs

Withholding the information = tenure

On del Ferro’s deathbed, he tells his method to : Fior

Fior goes and takes the jobs of mathematicians by stumping them with depressed cubics

Fior eventually challenges Tartaglia (1530)

Cardano learns the solution to the depresed cubic from Targalia

Ferrari (1535) figure out how to solve ALL cubics, and Quartic equations too!

Around 1535: Cardano writes a book Ars Magna

By 1550, all of Europe knows how to solve cubic, and quartic equations

Next big target : solution to quintic equation ???

1810: Ruffini + Abel genera quintic equation DOES NOT EXIST

image-20260728215220676