Reading Michael J. Bradley’s Age Of Genius: 1300 to 1800, I came across many interesting facts concerning the development of mathematics over the last five hundred years or so. Sadly, Bradley does not detail the process by which these developments were made, but relates them as isolated facts– interesting, but not very illuminative. I present below some of the mathematical trivia divulged in the book…
DOING MATH… WITH LETTERS?
We are all accustomed now to using letters of the alphabet to stand for unknown quantities in equations. But that particular brilliant idea didn’t occur to someone until the 16th century, when Francois Viete [vee ET] (1540 – 1603) first incorporated letters into equations. Viete’s system was specific about two things: vowels stood for unknown values… known values were represented by consonants.
According to Bradley, before Viete, mathematicians would sometimes use words or other symbols to stand for variable or unknown quantities, but not individual letters. When they wanted to describe some universally true mathematical relationship, they would often resort to describing, in one or several paragraphs, the relationships discovered, then go on to give several concrete examples. Viete offered mathematicians a much more efficient way to talk about entire classes of equations and, says Bradley, “prepared the way for the development of modern Algebra.”
Viete contributed something else that I consider at least as vital, and I would argue is certainly more fundamental. Ernst Mach was the man who taught me (from the grave, God rest his soul), that Mathematicians often accomplish their more complex and original work by taking equations that they haven’t clue how to solve directly, and finagling with them until they get them re-stated or simplified into equations which they DO know how to solve.
Viete was one of the first and best at taking difficult cubic equations (those equations containing terms “to the third power”) and substituting terms or otherwise reducing them until he got each equation into one of three standard, simpler equations which he then could solve easily. My formal mathematical education was long over before Mach clued me in to the fact that, actually, of the infinity of complex equations possible, mathematicians probably can not DIRECTLY solve MOST of them, but instead simplify them into more familiar patterns of relationships which they DO know how to solve. Just another reminder that the world is full of skills which awe us to see performed but which, once we know the tricks and put in the practice time, turn out not to be so impossible after all.
Also, you may find it interesting to know that as late as 1600, mathematicians were still not quite comfortable with negative numbers. In fact, they had not yet even settled on a name for them. Famous mathematician John Napier [NAY pee yur] (1550 – 1617) was still labeling those values we today call “negative” as “defective” numbers. Today’s positive numbers he called “abundant” numbers.
X STANDS FOR THE UNKNOWN
Rene Descartes (1596 – 1650) tweaked Viete’s system of using letters to stand for known and unknown quantities in equations. Instead of the consonant/vowel division Viete used, Descartes assigned letters (lowercase) from the first part of the alphabet to stand for known or unchanging quantities, and letters from the last part of the alphabet to stand for variable or unknown quantities. This is the tradition we still follow today.
I CAN GRAPH THAT!
Descartes is famous today for begin able to describe a given line or curve with an Algebraic equation– thus being the first– or at least one of the first– to combine the two disciplines of Geometry and Algebra. We are so accustomed today to the idea that many a graph can be described by an equation and many an equation by a graph that we can easily overlook what a bombshell discovery this was. People like Galileo were already describing the physical world in general terms using ratios, but no one had yet described the physical world with such versatility and mathematical precision as Descartes proved able to do with his new “analytic” Geometry… Unless it was Fermat…
Pierre de Fermat [fair MAH (or you’ll also hear “FER mat”)] lived 1601 to 1665, and he also was combining Geometry and Algebra at about the same time as Descartes. Similar to how Newton and Leibniz would argue over who invented Calculus a century or so later, Descartes and Fermat disputed who deserved more acclaim for linking Geometry and Algebra.
Fermat arrived at his method by combining the Algebra-work of Viete with the Geometry-work of Apollonius Of Perga (who lived in the 200s BC). Where Descartes used a quadrant system (which allowed him to graph negative as well as positive values), Fermat used a single horizontal axis to anchor his graphs. Fermat was able to graph the curves and lines produced by equations of up to two variables. [I’m not sure how complex Descartes was getting with his quadrant system, but his work on Geometry is coming up soon on my reading list, “The 300”].
One of the other major differences between the systems of Descartes and Fermat was that Descartes started with the geometric shape and then derived the equation describing it, whereas Fermat began with the equation first, and then derived its graph.
A CORNERSTONE EQUATION OF GRAPHING
As I said before, one of the tricks of the trade in mathematics is to re-express or simplify complex or perplexing equations into simpler equations we know how to solve. Fermat was able to see that he could graph any equation which he could get into some form of the equation…
ax^2 + by^2 + cxy + dx + ey + f = 0
You may recognize that equation. Different forms of it are taught to math students the world over when they are learning the equations which describe lines, circles, parabolas, hyperbolas, and ellipses.
At this point in time, even the greatest mathematicians could still not find the exact area bounded by a curve… they could get close– very very close– but not exact. That nut would be cracked by Newton and Leibniz and the Calculus they (separately) invented in the next generation of great mathematicians. However, even in the days of Descartes and Fermat, many mathematicians were already nibbling around the edges of what would become “the Calculus.” Fermat and others could already find the maximum and minimum values of a curve, as well as equations for finding lines tangent to curves.
Furthermore, Fermat was basically using a proto-Calculus form of Integration to solve for the area beneath curves. In the true Calculus as devised by Newton and Leibniz, areas beneath curves are found by summing the areas of uniformly wide rectangles contained inside the curve, whereas Fermat used a method of summing the areas of rectangles which got smaller and smaller. To me, Fermat’s method actually makes more intuitive sense.
Speaking of Calculus, as I hope to explore the field in later posts, I just want to gloss over a few contributions of Newton and Leibniz in the field which Bradley cites in his book. Actually, on second thought, I’m gonna skip Newton entirely, and mention a few things about Leibniz’s work…
INTERSTING WORK, BUT IT’S SO *DERIVATIVE*
According to Bradley, the “major breakthrough” of Gottfried Leibniz [LIEP nitz] (1646 – 1716)was his “discovery of the inverse relationship between the operations of Differentiation and Integration,” that they were sorta flip-sides of the same coin. This principle would come to be called the Fundamental Theorem Of Calculus, and I fully intend to speak of it more later in the year.
The thing Leibniz gave us that I’m personally most thankful for is the Power Rule For Derivatives. This little trick saved me countless hours of having to actually derive the derivatives of equations. Instead, I could simply use his trick, which is…
the derivative of (x^n) = n * x^(n-1)
The German Leibniz, like the Frenchman Descartes before him, approached math from a physical world or geometric perspective. So when he thought of a derivative, he pictured it as the slope of a line which is tangent to the curve described by the original equation. By this method, he realized that where-ever the tangent/aka/derivative was equal to zero, there a flat line (zero slope) was described which touched the curve at one and only one spot. Visualizing this, we can see with Leibniz that if the tangent-line is completely horizontal and yet only touches the curve once, then it must be touching it at either the curve’s highest or lowest point. Thus, Leibniz realized that the tangent of zero slope tells us where a curve’s maximum or minimum point is located. Unfortunately, we can’t tell which of those points, the maximum or minimum, the tangent touches by analyzing the derivative alone. However, Leibniz found that he could determine whether a maximum or a minimum was described by finding the value of the derivative of the derivative /aka/ the “second derivative.” I forget WHY this works, but I do remember using this little trick many times too.
Also, for those curious, Bradley implies that it was Leibniz –and not Newton– who derived the Product Rule for derivative, which is…
d (uv) – u*d(v) + v*d(u)
THE GEOMETRIC APPROACH VERSUS THE ABSTRACT
Reading Bradley, I was intrigued by the fact that some mathematicians seem to approach mathematics abstractly, while others come from a more geometrical or physical-world perspective. For myself, I am definitely of the physical-world sort. I don’t like any math I can’t picture. I already spoke of how Descartes started with curves and then derived the equations describing them, and that Fermat, on the other hand, came up with equations first, and then attempted to graph them. Thus, of the two, Fermat was the more abstract-oriented thinker.
Similarly, whereas Leibniz thought primarily of the geometrical implications of Calculus, Leonhard Euler [pronounced “oiler” Bradley tells me, though I’ll just have to take his word for that] approached Calculus primarily as a theory of functions. Euler (who lived 1707-1783 and was not yet a teenager when Leibniz died) saw Calculus not so much as the study of curves as a method of describing abstract relationships between quantities.