How did mathematicians interpret quartic equations and fourth powers before Descartes propose to perform elementary arithmetic on line segments?

I ask this because it seems strange to me that multiplication before him was essentially a geometric operation (as far as I know), and still mathematicians could study methods for finding roots of quartic equations, for instance.

Since I am new to History of Mathematics, I would also be glad to know about references in which I can learn more about it.

  • 3
    $\begingroup$ You can find commentary in classical Greek times that the fourth power is meaningless, since (in modern language) space is 3-dimensional. Quartic equations were solved in 1540 or so (Ferrari, Cardano, etc.) That was before Descartes was born. So quartic equations were certainly considered before Descartes. For quartic equation, a first place to look is en.wikipedia.org/wiki/Quartic_equation $\endgroup$ Mar 5 at 17:55
  • $\begingroup$ That's precisely what I believe is strange: how did they interpret quartic equations before fourth powers become meaningful? $\endgroup$ Mar 5 at 18:30
  • 3
    $\begingroup$ Higher powers of a variable and equations with them appear already in Diophantus's Arithmetica (c. 250 AD), over a millennium before Descartes. He does away with the geometric interpretation, values of a variable are taken as numbers and he gets powers by simply multiplying numbers by themselves. Arithmetica was translated into Arabic and influenced Islamic algebraists, who also considered higher order equations, and eventually Europeans like Cardano and Ferrari. $\endgroup$
    – Conifold
    Mar 5 at 18:59
  • $\begingroup$ Thus they only considered the (positive) integer roots as valid roots (in this 4th degree case)? $\endgroup$ Mar 5 at 20:48
  • 1
    $\begingroup$ Diophantus accepted rational solutions, but only positive ones. Negative roots did not become common until the 17th century. Curiously, Cardano would occasionally consider negative roots, but did not admit negative numbers as coefficients, which led to proliferation of cases for the cubic. Stifel did the opposite, Viete allowed neither. Stevin was probably the first to systematically admit negative and irrational roots and coefficients c. 1590, see History of negative numbers. $\endgroup$
    – Conifold
    Mar 6 at 23:11

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