Europe (Scientific Revolution)
Gerolamo Cardano
An Italian gambler, scientist, philosopher, and mathematician, Cardano is most known for the Ars Magna – a book that contains knowledge on solving cubic and quartic equations. Specifically, he published the formulas for solving third and fourth-degree polynomials and their proofs. All of the work he published was rhetorical algebra, though some symbols were used as a shorthand for specific words (his solutions were still very descriptive and wordy). His work also began discussions and debates on using negative and imaginary numbers, which became much more critical in the future.
Francoise Viete
Viete's work began the foundation of what is now instrumental to solving algebraic equations: manipulating both sides of an equation. Only until Viete's work did many mathematicians embrace that you could manipulate both sides of an equation to solve a problem. In addition, Viete also used vowels for unknowns and consonants for knowns to differentiate between the two, which made syncopated algebra easier to grasp.
Simon Stevin
Formerly an engineer, Stevin was instrumental in the start of number theory. Europe during that time had no rigorous definition of a number. Once Stevin explained that a number represented a unit/magnitude and didn't have to be an integer (it could have a decimal/fractional part), mathematicians shifted away from solely focusing on geometric algebra and fully embraced syncopated algebra as a new field in mathematics.