Books: “Here’s Looking at Euclid”

November 24, 2010

Overall, I don’t think my father was disappointed in me. He didn’t set himself up for disappointment, because he didn’t pressure me to pursue any particular career. When I said I wanted to be a priest, that was all right with him. When I became a newspaper journalist instead, that was all right, too. He was both a practicing Catholic and a newspaper reader, so he was in a good position for success.

There was, however, one thing that he might have found frustrating about the younger of his sons — Tony’s brother, as it were — and that was my inability to learn how to add several columns of figures without carrying numbers.

NACD photo

At slow moments in my family’s grocery store — where adding columns of figures was a frequent chore — Dad would try to show me how to add three or four columns at once, rather than starting with the right-hand column (the pennies column) and carrying the excess to the top of the column to the left. “Put down the two, and carry the four” — that was how I had learned arithmetic. I couldn’t understand the alternate method Dad tried to teach me, which annoyed me, because he could add columns of figures with his technique nearly twice as fast as I could do it with mine.

Many years later, the dawn broke in my clouded mind while I was reading a book on math. There, for Pete’s sake, was Dad’s method — explained just as I remembered Dad explaining it — but somehow I finally understood it and have used it ever since.

Like many people, I suppose, I regarded math at best as a necessary evil in elementary and high school. I didn’t go near the subject in college or graduate school. When I was in my 30s, however, I inexplicably chose to read a book on math written by Bertrand Russell, and was surprised to find that the subject was attractive. As I result, I have read many books about math, the most recent one being “Here’s Looking at Euclid” by the British journalist Alex Bellos.

In fewer than 300 pages, Bellos covers a remarkably wide range of topics. He explains the origins of mathematical concepts that we take for granted — the sixty-minute hour and sixty-second minute, for example — and how mathematical understanding has evolved since some Sumerian in the fourth millennium B.C. first pressed a stylus  into a clay tablet. He writes about pi and infinity and probability (including its role in gambling), and the bell curve.


Bellos begins his book with an account of the Munduruku people of Brazil, who have a number system that goes only from one to five. Moreover, the Munduruku use only the numbers one and two to count precisely, using three, four and five more as estimates. In fact, Bellos explains, the Munduruku are baffled by others’ compulsion to enumerate people or objects and either cannot or will not answer if asked how many children they have. They know who their children are; that’s enough for them. It’s healthy, I think, to be reminded from time to time that everyone doesn’t look at the world through the lens we use.

Ventura College

What I particularly like reading about is the mystery and elegance that many people find in numbers. One example is the “golden proportion” or “golden ratio,” to which Bellos devotes a chapter. The definition of this term, known to mathematicians as phi, might be off-putting at first. Here it is as Bellos explains it: “The golden mean is the number that describes the ratio when a line is cut in two sections in such a way that the proportion of the entire line to the larger section is equal to the proportion of the larger section to the smaller section.” That number begins as 1.61803 and, like pi, goes on forever. It appears in many familiar geometric figures, including the five-pointed star. The 16th century mathematician Luca Pacioli, Bellos reports, “concluded that the number was a message from God, a source of secret knowledge about the inner beauty of things.” That notion may seem remote until Bellos explains how a retired dentist discovered that the golden proportion was the key to designing dentures that give an individual patient a proper smile – now a widely accepted principle in dentistry.

I’m sorry now that I once thought of math only as a nuisance, but books like this one have helped me make up for a misspent youth.


3 Responses to “Books: “Here’s Looking at Euclid””

  1. bronxboy55 Says:

    I have a couple of dozen books on mathematics, and would likely buy this one based on the great title alone. If only they taught math this way in schools.

    In a recent conversation, I wondered if animals could count. If a mother duck has nine ducklings and you secretly snatch one, she notices that one of her babies is missing. Is she counting, or demonstrating a relationship more similar to the one illustrated by the Munduruku?

    You’re probably familiar with The Teaching Company. I’d recommend “The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas.” It explores many of the same ideas discussed in “Here’s Looking at Euclid,” always drilling deeper for underlying meaning and also roaming far and wide in search of unexpected applications. Endlessly fascinating — much like your blog.

    Thank you for another excellent post.

  2. shoreacres Says:

    Any book about math that can come up with a glorious pun like that title deserves a second look.

    Unfortunately, math really has been a closed book for me my entire life. Once I began sailing and navigating, some things became clear (the usefulness of triangles, for example, and those speed/time/distance formulae) but mostly my eyes glaze over. There was a time in my life when changing banks in order to balance a checkbook seemed perfectly reasonable.

    Your mention of the Munduruku and the different lenses we use to view the world did remind me of something not precisely mathematical, but perhaps related. When I lived in Liberia, my houseboy considered my kitchen non-funtional. It made no sense to him that the plates were here, the glasses there, the pots in another place and the utensils in yet another.

    He organized by function rather than by abstract category. The mortar to pound the palm nuts, the pot to cook them in, the peppers and fish that went in the concoction all were kept together.
    Rice bowls went with the cwt of rice, and so on.

    In the same way, I thought the machetes belonged out on the porch with the tools. He thought they belonged at each door. When I rolled my eyes and asked why, he just laughed and said, “Missie! That be where da snake come in!”

    Well, ok!

    • charlespaolino Says:

      That’s a very interesting story. I always remind my students that no two people can see the world from the same vantage point. I once handed a class sheets of paper with nothing on them but a blank circle. I asked them to draw the globe from memory, and, of course, they all drew it with North America and South America right in the middle. I taped the papers up on the board and asked the students why they had done that, and they got the point. Many years ago we became acquainted with Hildred Hobson, a native woman who lived on the island of Nevis in the Caribbean. For a while, she worked for a family that lived in Princeton, an old university town here in New Jersey, and she used to come up here to visit. We asked her one time how she liked Princeton, and she said she supposed it was all right, but that she didn’t approve of “touching houses.” We asked her if she thought she’d like to live there, and she said: “No. I couldn’t see the ocean from my window.”

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s