Francois Viete, in the late 1500s, “started a revolution in mathematical notation.” Says Suzuki. Viete allowed vowels to stand for unknowns– which was not revolutionary in itself, for this had been here and there before… but what WAS revolutionary, says Suzuki, “was Viete’s next step: he allowed even the constants to be represented by letters.” … “Viete was the first mathematician to whom the idea of a GENERAL Equation such as ax + by = c would make sense.” Before Viete, mathematicians had been forced to explain their grand, overarching mathematical principles in long narratives and numerous examples. After Viete, they could write a general equation which could stand for the concept in any particular application.
Suzuki says that it is Oreseme (who lived in the 1300s), not Descartes, who deserves credit for coming up with the idea that mathematical relations were GRAPHABLE. Oresme believed that all qualities could be measured QUANTITATIVELY– and that once you had quantitative measurements, you could represent the situation geometrically. This could even be done when the Quality was changing in its INTENSITY. Oresme instructed that in such cases, one could represent the “intensity” of the quality on one line, and the “extension” (often being Time) of the Quality on another line perpendicular to the first. He thus basically described an x-y axes system.
If we are graphing a moving object using Oresme’s instructions, then we can imagine Time on the horizontal x-axis and the “intensity” of movement (the velocity) measured on what we would today call the y-axis– the vertical direction perpendicular to the horizontal direction. If the velocity is constant, all the lines representing “intensity” would be the same height, with what Suzuki calls the “quantity of the quality” represented by the rectangle outlining all these lines. In this case, the Quantity Of The Quality would be the distance traveled.
Around 825, al Kwhwarizmi wrote a treatise on how to solve equations, called “The Condensed Book Of Completion And Restoration.” In Arabic “completion” is “al-jabr,” and from this we get our word, “Algebra.”
As my mentor Mach first taught me, most relationships in the world produce mathematical equations which are NOT easily solvable by even the greatest mathematicians. However, the trick that mathy types discovered– and the value of this trick cannot be overstated– was learning how to transform or “reduce” difficult or impossible mathematical equations into one of a few fundamental types of equations that mathematicians CAN solve.
Al Kwarizmi, Suzuki tells us, was one of the first mathematicians to approach problem-solving in this way. After him, the emphasis changed from solving equations to transforming equations.
To tell the truth, until Mach, and then Suzuki, discussed this ubiquitous strategy, I had never realized I was being bamboozled. Mathematicians, which seemed so omnipotent to me in their field, were simply taking equations that even they couldn’t solve and rearranging and rearranging them until they got equations which they COULD solve.
Another Arab, Al Karaji (c. 1000), figured out that you could line up terms of the same power and add or subtract their coefficients… “This seemingly simple but crucial step,” writes Suzuki, “meant polynomial expressions could be added, subtracted, multiplied, and divided using the same algorithms that had been developed for working with numbers.”
Suzuki credits Samaw’al (c. 1150) with completing “the arithmetization of Algebra by giving procedures for finding polynomial quotients, as well as explaining how to handle the subtraction of two NEGATIVE numbers.”