This five minutes, I am discovering new things about the world of Algebra and Geometry. The goal I set for myself initially was to work through the typical high school curriculum . Then revisit my early college courses. To what end though, am I chasing a Math degree? Do I have enough years left in me to do all of that again? Would I be able to know any one subject in any kind of depth? I think the answer is a big NO to each one of those questions.
The temptation is to not do something if it seems impractical, or beyond our ability. I mean, really, it’s not like I don’t have enough more important things to take care of every day. We fill our lives with so many things that distract us from what we enjoy.
I have had some success in tutoring by helping young people understand some small aspect of Algebra or Geometry. Helping others has always been at the center of my career because I’ve found it rewarding. There are opportunities to help people in any job. So the tutoring has been both challenging and fun.
If I hadn’t had the aspirations for myself, the tutoring would not have become a thing for me. So yes, it’s about the journey, not necessarily about the goal.
Back to what I am currently active with, Geometry and Algebra (oh and building bookshelves, more on that later).
I’ve happened onto three books that will probably take the rest of my life to get through:
- Oliver Byrne’s Elements of Euclid (the first 6 books of Euclid’s Elements)
- Classic book illustrating the proofs from Euclid’s Elements by using color diagrams.
- A College Algebra, by Henry Burchard Fine (copywrite 1901, began in 1895)
- The copy I have is a classic reprint from “Forgotten Books”.
- The Secret Formula, by Fabio Toscano
- Not a math book, but a book about the lives of men in the 16th century who came up with “The Secret Formula” (the general solution to the cubic equation), and how it all transpired through competition, and pride.
The subjects of Geometry and Algebra are so vast and rich in history that I can spend all of my time in these books alone. Although that might limit what I can offer as a tutor to students who are interesteed in more.
I have found Geometry even more fascinating these days since I’ve become familiar with Euclid’s Elements. I believe the first reference I saw to it was from a text that stated Abraham Lincoln read it periodically as a law student. He felt it was a valuable tool to train his mind for the logic and demonstrations needed in law practice. What fascinating people. I look forward to reading through it myself over time. Right now, I just know that it exists, starts with definitions, postulates, and axioms, and that I have a copy on my Kindle.
Then in a recent attempt to solve a certain third order equation I saw a reference to “The Secret Formula”. It’s a historical perspective of the art (Algebra) in the 16th century. Again, what fascinating minds, but in the end Mr Toscano shows how they were all still human after all. It’s a relatively short book and is filled with excerpts from letters between the principal characters that shed light on their personalities and ambitions. The author uses them to provide background for his story. That was a time of rebirth in algebra.
My most memorable parts of “The Secret Formula” include a geometric derivation of the quadratic equation, the solution of the cubic equation in the form of a poem, and the fact that math in those days was rhetorical instead of symbolic. Equations were expressed in word form. For example, “cubic and cenno and things equal to number”, which is known to us as \(x^3 + bx^2 + cx = d \). In fact, mathematicians didn’t work with negative numbers during that time. The equation, \( x^3 – bx^2 + cx = d \), was expressed as, \( x^3 + cx = bx^2 + d \), instead.
Continuing to chase the solution to a certain cubic equation I found the general solution discovered by Gerolamo Cardano and Niccollo Tartaglio discussed in “A College Algebra”. Enough so that it helped me find the irrational real solution to the equation at hand. (I was able to get the answer from a Casio calculator first, but that wasn’t any fun, then verified it by graphing the equation in Desmos, which was a little more satisfying). By the way, Cardano and Tartaglio were the main characters in “The Secret Formula”.
Henry Fine’s text of 589 pages is packed with just about everything you can think of. It starts by describing natural numbers, whole numbers, and integers (positive and negative whole numbers) from the notion of groups of things. The book takes you through the linear, quadratic, and cubic equations, theory of equations, logarithms, and ends with properties of continuous functions. I’ve only managed to navigate the parts I needed so far. It also gives us a window into what college text books were like in the early part of the 1900s. There are not as many examples and problems, but more in terms of explanations and derivations.
So off we go. This is my first post. I thought, why keep all of these little bits to myself. I know it’s not for everyone but I hope there are others out there who find things like this interesting. My wife and children grin and bear it when I go on about obscure topics. Partly because I’m not great with the spoken word so the blog might be a way to let me express my thoughts a little clearer. Everyone needs an audience, right?
I will never claim to be an expert just an interested party. And, if I can learn something from the readers of this blog, well, then that’s a bonus.