Mr. Bowditch: Brilliance and Populism

I was looking through something the other day and I came across a reference to someone named Bowditch. Immediately, my mind was flooded with images from the life of Nathaniel Bowditch, a mathematical and navigational savant in the late 1700's and early 1800's, whom I read about at age 12 or so in a wonderful biographical novel for middle-graders called Carry On, Mr. Bowditch.

Bowditch was the son of a cooper in Salem, Massachusetts who left school at the age of ten to work for his father. At twelve he was indentured for nine years as a bookkeeping apprentice to a ship chandler. They were kind to him there and he was given access to the library; being immersed all day in numbers, and having a sharp brain, he became interested in some of the more complex ideas surrounding math, spending his days learning all about what it takes to outfit a ship and his nights studying.

"In 1787, aged fourteen, Bowditch began to study algebra and two years later he taught himself calculus. He also taught himself Latin in 1790 and French in 1792 so he was able to read mathematical works such as Isaac Newton's Philosophiae Naturalis Principia Mathematica. At seventeen, he wrote a letter to a Harvard University professor pointing out an error in the Principia... Serendipity aided Bowditch's self-directed study in as much as he found himself able to use the eminent Irish chemist Richard Kirwan's library: a privateer from Salem had intercepted the ship carrying the library between Ireland and England and brought the library back to Salem in June 1791." [wiki]

When he was twenty-two, Mr.Bowditch went to sea as a ship's clerk and captain's writer. Working from the error-prone navigational books of the day was so frustrating to him that by his fifth voyage, of which he was master and part owner of the ship, he "decided to write his own book, and to 'put down in the book nothing I can't teach the crew.' On that trip, it is said that every man of the crew of 12, including the ship's cook, became competent to take and calculate lunar observations and to plot the correct position of the ship." So in 1802 The American Practical Navigator, was published. It was such an accurate and useful book that it immediately revolutionized navigation, and is still carried onboard every commissioned U.S. Naval vessel today. The first edition of Bowditch's American Practical Navigator, which became the western hemisphere shipping industry standard for the next century and a half, and in 1866, the United States Hydrographic Office purchased the copyright. Since then the book, with appropriate revisions, has been in continuous publication, and to this day it is simply known as Bowditch.


Reading about him when I was a young person, I was struck mostly by the variety and extraordinariness of his experiences, and the fascinating idea that math had such concrete applications. It had never occurred to me to think of math the way Mr. Bowditch did, as a conceptual thing and a way of looking at the world. I never thought of myself as someone who was good at math, even though later in life I became adept at pattern-making (making 3D objects from 2D patterns) and had a keen mechanical understanding of math-related concepts, based more on intuition than education, which allowed me to do well in all sorts of arenas.

Nowadays, of course, children learn the beginnings of algebra in third and fourth grade. The more conceptual elements of math begin much earlier, and along with the usual times tables school children are taught to see math as a schema, a place to play around with numbers; the basic elements of math's abstractions are set in place in more malleable brains. Add to this the basic tenets of binary and hexidecimal systems, and you get some seriously young minds working on much more abstract levels than of old. ...At least, for those minds who are actually getting an education.

Which brings me to another point: this Bowditch person was pretty much self-taught. Remember, too, that calculus as a systemic study had only been fully developed a hundred years before, so when Mr. Bowditch found an error in Newton's masterpiece only three years into his study of higher math, it showed considerable brainpower. This is something I have only come to appreciate as an adult: both in terms of Isaac Newton and in terms of understanding Bowditch's learning rate. These, and the fact that such a brilliant man should go on to become an insurance actuary, a job which he kept until his death at the age of sixty-three, strike me strongly now that I have some perspective on them.

But what strikes me particularly, and what endears me most to the man, is his deep-seated belief that every man on a ship can and should be able to navigate, despite the mores of the day which would have it that men on a ship are expendable, and only the navigator has the knowledge (and only he should have the knowledge) to find the ship on a map - to navigate. His stubbornness on this point is legendary, and his willingness to prove it comes not only from a man who taught himself to be who he was, but from a time and a place: Massachussetts, in post-Revolutionary America. Best of all, he put his belief to work in writing, too - which is why his book is still relevant today, and why we have such people as Dee Caffari out there on the seas.

Well done, Mr. Bowditch.

The Wonder of the Golden Proportions

Ever hear of the Golden Mean? Neither had I, until I was supposed to teach a graphic design course, and started (you know me) to do research on what, exactly, I should be teaching. It's one thing to be able to design things, and quite another to have to teach it to others.

A lot of what I found, gestalt theory and the principles of visual weight, and so on, were really interesting; but the Golden Mean was what really caught my fancy.

(NB: Math following. Don't be scared, all will be clear [and un-mathlike] in the end, I promise. I hope)

The Golden Mean, also known as the Golden Ratio, was developed as a proportional measurement by the ancient Greeks, as a way of making the most pleasing artworks. It was felt to be semi-divine, in that it seemed to show up in Nature as well. The ratio, an irrational number, began as 1.6180339887... and continued onward, pretty much forever. It is found by working out the following algebraic equation:

Essentially, if you take a line that is 1 long, and go from there, you will find that the above equation will work out to that same irrational number, which, when used as a proportional device, allows you to produce varying lengths of lines that are smaller and larger. But I'm not going to explain how, because though I really, really love math, I don't do well with equations, which are difficult for me: at least, to express the near-mystical magic that shows up in numbers.

So: now you have a bunch of varying line lengths. So what?

Well, let's see: take one of these lines and make a square out of it (putting four of them at right angles to each other, remember? I sometimes blank on these little leaps of logic). Then, starting at the center of one side, measure to one corner and draw an arc downward:


Aha! Now we start to have something. If the length of the square is 1, then the length of the rectangle (shown as the Greek letter phi here) is, of course, 1.6180339887... well, you get the point: the Greeks were smart. We call this shape the Golden Rectangle, and you can find it everywhere in Greek and Renaissance art (and elsewhere! Stage height proportions, window shapes, chair backs, believe me, they're everywhere. They are quite pleasing to the eye).

(If you really want to know the equation looks like this:)
(but don't ask me to explain that part).

Golden rectangles in the proportions of the Parthenon

Okay, onwards. I know this is looking like a lot of math, but bear with me here. Now we come to this guy named Fibonacci, born c.1170, who is considered "one of the most talented mathematicians of the Middle Ages" [wiki]. This is the man responsible for the introduction of the Hindu-Arabic numeral system when all of Europe was doing math in Roman numerals (I strongly urge you to read his link, above: it's fascinating). He used what is now known as the Fibonacci sequence - actually a pre-6th-century Indian concept - as an example in his famous and apparently brilliant book about math, Liber Abaci, or Book of Calculation, which is why we have it now. In any case, the Fibonacci Sequence, as it is now known, leads to all kinds of interesting events.

Try this. Draw a square, measuring one unit across (make it a small unit, like a centimeter, or perhaps the distance between binder-paper lines - otherwise you will need big paper). Now draw another square exactly the same right up alongside it (so they are sharing a side). Now, say you take the line along the top of both squares and use that to draw a bigger square, which of course is two units on a side - right? Okay, now moving clockwise (or counter-clockwise, like the picture below) around this construction, draw another square along the side where the edge of the big square and one small square align. Keep going clockwise and keep drawing bigger and bigger squares. The length of each consecutive square should make a sequence, like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... (by now you've probably run out of paper). Viola! Your own personal Fibonacci sequence, right there in your own home! And...lookie there. It sure looks kind of like...is it? Pretty close to a Golden Rectangle, isn't it?


Okay, okay, you say. That's kind of neat. But aren't we just wanking with numbers?

Well, perhaps. But check this out. You can draw an arc, starting with a point at the middle of the two first squares where they touch the third square. The arc goes from corner to corner of each square, so that the two squares together make a semicircle. Then, by expanding the arc and drawing one in each square, working around the structure, you can build a lovely spiral:

This Fibonacci spiral does not have the two original squares visible

This spiral is one that closely mimics the Golden Spiral, based on the Golden Mean, above. The main difference is that the Golden Mean goes in both directions, both up and down, whereas the Fibonacci spiral only goes upwards from 1 (though you could take it down if you were into math, I'm sure). Both are considered logarithmic spirals, which are found everywhere in nature. Jakob Bernouli, a mathematician from a great family of brilliant people, called the logarithmic spiral spira mirabilis, or "the Miraculous Spiral," so called because the size increases but its shape is unaltered with each successive curve. This kind of spiral shows up in shells, in hurricanes, in the shape of a cat's claw or a wave; galaxies and flowers all work with logarithmic spirals. The Fibonacci sequence can be found many places as well, such as in the ancestry patterns of bees, the branching of trees, the whorls of a sunflower and the fruitlets of a pineapple.

Technically, though, the Fibonacci spiral has a slight wobble; it is not perfect, so not really a proper logarithmic spiral.

Which brings me to something which I find absolutely wonderful: if you chart that wobble on a graph, it begins to look as if it is ocillating around something, some specific number. Guess which number?

You guessed it. Now tell me there's no mystery in numbers.



(For a nice, step by step, even better mathematical explanation of this, check out this site. It certainly got me going.)