Don’t Say You Weren’t Warned

Yesterday I wrote something about how I studied calculus without having to read Newton’s notes, and I said I probably would not have understood them because only one bit of his mathematical notation survived. Today I decided to look it up, and I found out the story is not quite that simple.

Cambridge University has Newton’s papers. Some of them, at least. I am too lazy to check. Google and come back and look smarter than me. Cambridge has put images of the papers online, partly for reference purposes, but also partly to make the rest of us feel stupid.

I admit, the last phrase of the last paragraph is conjecture.

It turns out there are several reasons why I would not have understood his notes, and his notation is probably not among them. The biggest obstacle would have been his copious use of Latin and Greek. I was “educated” in American schools, so the only Latin I know is pig Latin. As for Greek, I’m halfway through The Iliad, and I use Windex as a topical antiseptic.

Actually, I took a semester of Latin in high school, but, following my strict policy, I did nearly nothing, so I got a D.

Latin is a horrible language. Like Russian, Latin is inflected. That means you have to change the endings of words depending on how you use them. If English worked this way, you might have to write things like, “I shot the cow,” and, “The coworum rolled over and died.”

Leo Tolstoy and his wife had different last names because he was male and she was female. Terrifying.

When I was studying physics, I got really handy with exponents. Typically, when you take the root of an expression (once you get past about the 21st grade, you have expressions instead of numbers), you put a radical around it.

It looks like I’m wrong. I checked, and it’s a “radical sign,” not a radical. It’s a little box sort of like a division box. A radical sign means “square root.”

Radical signs are stupid. A radical sign can’t mean anything other than “square root” unless you add a little superscript to the left of it. It’s sloppy and cumbersome. I didn’t like it.

“Square root” actually means “to the 1/2 power.” You can have any real number as an exponent. It doesn’t have to be 1/2 or an integer. It can be 2/3 or pi or whatever you like. It can be annoying, using a radical sign to express powers not involving multiples of 1/2.

I worked around that. I used to use parentheses with fractional exponents. For a square root, I used “1/2.”

This leads to all sorts of conveniences.

Say you want to put the square root of something in the denominator of a fraction. You just make the “1/2” negative. What if you want a cube root? Use “1/3.” What if you want the square root of the cube? Use “3/2.”

What if you have an equation with an expression with an exponent on one side, and you want to get rid of the exponent? You invert the exponent, squeeze it between the bars of the equal sign so it pops out on the other side, and apply it there. For example, if you have 4 = (16)^1/2, you turn it into 4^2 = 16. It works with any real number. For all I know it works with complex numbers and quaternions. Don’t ask me what a quaternion is.

Well…of COURSE it works for complex numbers. I’ve seen like a billion complex exponents.

This is exciting, isn’t it? This is what you come here for.

If you have a big string of expressions with exponents, multiplied by each other, you just add the exponents. If you want to move an expression from a denominator to a numerator, you multiply the exponent by -1.

This is really helpful, because it helps you put everything on the same line, and it turns multiplication and division into addition and subtraction.

Why tell you all this? Because Newton didn’t do it. If he wanted to take the sixth root of something, he put THREE radical signs around it. I would have lost my mind counting the brackets.

I guess if you’re Isaac Newton, you find all forms of notation unbearably simple, so it doesn’t matter what you do. Me, I get confused writing grocery lists.

If you look at Newton’s notes, you will see why he accomplished so much. Yes, he had a giant brain, but he also did tons of math. He filled page upon page. Much like the ancient Greeks, who put up with Homer’s endless droning and hour-long similes, Newton did not have TV. Also, when he invented calculus, he was hiding on a farm while the plague destroyed England. Talk about bored. Put me in Fargo, North Dakota with no broadband, a ream of paper, and a box of fresh pencils, and I might invent something, too.

But probably not.

I would definitely make a lot of paper airplanes.

Newton wrote out big, ponderous expressions, and they looked a lot alike, so he must have had tremendous powers of concentration in order to keep everything straight.

For tiny-brained mortals like me, clear notation matters.

I hated the way my old physics profs wrote things. For example, a professor talking about a function named A might call a variant of it A-prime, and then when he wanted to change it to make a point, he would call it A-double-prime, which was followed by A-triple-prime. Then he might combine them in one huge expression with about nine A’s in the numerator and fifteen in the denominator. I never understood that. It’s not like there’s a fee for using new letters.

I wrote a pretty insulting essay about it, claiming there actually was a fee, and that when professors used Greek letters, they had to pay royalties to the Greeks.

I was not popular in grad school.

If you feel like a thrill, Google and find Cambridge’s photos of Newton’s big notebook, known as the “Waste Book.” It’s a real page-turner, if you’re an alchemist and religious fanatic who reads ancient languages fluently.

Unfortunately, it’s math and science, so there are no Cliff’s Notes. Not that I’m saying the impossibility of writing Cliff’s Notes for math and science proves liberal arts subjects are a joke.

But I would not argue if you took it that way.