Most of the buildings were instant infernos, flaming too fiercely for anyone to escape from them. But two huts, close to the outskirts of the village, burned less strongly than the others. Four people ran screaming out of them, two men and a woman dragging a young child.
I had worried about such a situation. It had made me sit and hold my head in my hands for many minutes. What was the correct action to take? Then I realized that the answer was obvious. This was a deity’s funeral pyre, and a god had a right to his servants. All his servants.
I lifted the rifle, set it to automatic, and fired. They fell into the flames.
No more people came out of the huts. After a few minutes I took my camera, came as close to the blaze as I could stand, and photographed the flames that licked around the great tunicate shell. The last mortal remains of Master Tunicate vanished as I watched. I stayed for two hours, but there was nothing more to see. Nothing at all. And at last I lay down on the hard black earth and cradled my head in my arms. For the first time since I came to Africa, I slept a deep and purifying sleep.
Dawn awoke me, dawn and heavy rain. It hissed down into the blackened ash of the village, quenching every ember. And it beat on my unprotected scalp to begin again the drumming inside my brain. I got to my feet, went to a thick-canopied tree a few yards away, and stood beneath it until the downpour was over. The rain sluiced down, hammering the soil and bouncing back two feet high in a white, seething spray. It did not last long. Within half an hour the clouds had gone, the sun was well up, the ground was steaming, and I could head back to our camp.
That was where it became unendurable. I had done what had to be done. I knew I would not like what came next, and I was weeping again. But it was far worse than I expected. You see, I had not known about the flies. The heat and the rain brought them out in their millions, more than I had ever seen before. They buzzed and swirled about my head all the way to the camp, clouds of them.
Within our camp itself there were more than you would ever believe. And I had to go into their midst. There was no choice. I couldn’t leave Jane and Walter and Wendy like that, of course I couldn’t. They were dearer to me than anyone else in the world, my friends and my true beloved. But they lay so thickly covered by flies that all I saw through my haze of tears were three humming, purple-black mounds, their outlines indistinct and wavering.
It sickened me to go near, and it took all my strength to dig places for the three of them. Gasoline drove the flies away temporarily. After I put Wendy in I took the wedding band from my finger and placed it on her hand. We had noted this fact long ago: my third finger is just as thick as her thumb’s second joint.
Jane and Walter would lie together. It was what they wanted, and it was only right. I took off Walter’s glasses, and smoothed the hair back from his brow, the way he had done so often. He looked peaceful and very young.
The army officers I did not bury. I splashed gasoline over them and lit it. Then I struck camp for a final time, climbed into the mini-bus, and headed west. I did not bother with the trailer at all.
Within seventy-two hours I was back at Boyoma Falls. The boat was where we had left it, moored close to the bank. The solitary crew member was nowhere to be found, but he appeared eventually from the bush.
He looked at me and tried to run. I caught him easily. He screamed, and fell to the ground in a fetal position, covering his eyes with his hands. I lifted him with one hand, and his teeth chattered in his head.
It took a little while to make him understand my French. But I did it. He is just a few feet from me now, steering the boat, working as hard as any man ever worked. Like me, he has not slept since we left Boyoma Falls—five days. I do not think he will sleep until we reach Kinshasa and I leave for home.
I am almost done. That is good. This writing must be finished before we reach the city, before the President’s office asks what has become of the army men.
The writing will be done, but it cannot end here. I know that. Even if the Zaire authorities can be satisfied easily, there will be questions in Washington. It was an expedition conducted with government grant monies—there have to be written reports.
That is good. I will report everything. I did the right thing—the only possible thing. Yet I know that I will be punished.
I can stand that. What I cannot stand is the loss of Wendy, Jane, and Walter.
I promised proof. You will find it on the outskirts of Kintongo, near the roots of the old, fire-scarred tree. There is the roll of film that I took, encased in a sealed plastic box that should withstand many tropical seasons. There are small fragments of the ship, the one that bore him here. There are the two rifles, the two shallow graves, the wedding ring. They will give you all the proof that I have written the exact truth.
For me, that type of evidence is unnecessary. I know what I am: a servant. I am a servant of the Living God.
This is the message he gave me when I visited his home: Protect my children.
Master Tunicate will come again. When he does, you will all be as I am.
afterword: tunicate, tunicate,
wilt thou be mine?
This story began life in the strangest possible way. During lunch with a writer friend in Atlanta, the conversation somehow ranged from marine animals, including tunicates, to English nursery songs. At that point, with nothing in my head, I thought of the old rhyme that begins, “Curlylocks, curlylocks, wilt thou be mine.” So of course I said, “Tunicate, tunicate, wilt thou be mine,” and sat there feeling pleased with myself.
But I had at the time no thought of writing a story; in fact, I said to my friend, “There’s a title for you. You can have it. You should write a story called that.”
She didn’t. But after another year or so, I did. I sold it, and I published it. And then I learned that the English nursery rhyme, “Curlylocks, curlylocks, wilt thou be mine,” is not a widely known and standard American nursery rhyme. And without that context, the story title is gibberish. One reviewer, though he was writing in May, gave it the prize for worst title of the year. In this collection I therefore include what was not printed in the original publication, namely, the rest of the nursery rhyme that gives point to the title.
This story is, for me, awfully unpleasant and highly disturbing; the more so, because three of the characters who die in it are based on real people. Two are friends of mine, a husband and wife who are genuinely knowledgeable of and fascinated by Africa (they were once stranded for a week in Timbuktu, when the one Air Mali jumbo jet was preempted by the country president).
The third person was at the time my wife. Since she is now my ex-wife, I hate to think what dark subterranean notions may have been lurking within my brain while I was writing this story.
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article: counting up
1.THE THINGS THAT COUNT
The ancient Greeks gave a good deal of thought to the definition of Man (Woman, back in the fourth century B.C., was not considered a suitable subject for philosophy). The Greeks wanted to know what it is that makes humans different from all the animals. Early attempted definitions, e.g., the Platonist one: “Man is a featherless animal with two feet,” were not well-received. Kangaroos were unknown in ancient Greece, but Diogenes offered a plucked chicken and asked if it were human (he was not serious, and this must be the original chicken joke). Even when amended to “a featherless animal with two feet without claws,” the Platonist definition did not exactly sparkle. The Greeks never did come up with a good answer.
By the nineteenth century, definitions were more oriented to function than to appearance. “Man is a tool-using animal,” declared Thomas Carlyle, in Sartor Resartus, following an idea of Ben Franklin, but neither man knew, as we know now, that some chimpanzees make tools out of sticks and rocks.
“Mankind is the only animal wit
h language,” say the linguists, but not as loudly as they once did. The extent to which Washoe, the famous chimpanzee with a symbol vocabulary of 180 words, and her primate friends are capable of working with a real language remains a matter of hot debate; and the complex sounds produced by the humpback whale may not be messages at all; but in any case, the claim to uniqueness by humans is in question.
Man is the only animal that laughs.
Man is the only animal that cries.
Man is the only animal that blushes—or needs to.
Man is a thinking reed.
Man is Nature’s sole mistake.
All interesting, and all suspect, especially as more and more of the things that we once thought uniquely human, including murder and rape, are found to be shared with our animal relatives.
“Man is the fire-using animal” still sounds good, but my own favorite definition is this one: “Mankind is a counting animal.” If I substitute “entity” for “animal” in this sentence it becomes far less persuasive, since if computers do one thing well, they count—but only with human direction. Also, certain birds are supposed to be able to count the eggs in their nests, or at least to know when one is missing. However, it is certainly true that we know of no animal able to count to more than ten. Humans can, or at least, most humans can. (It was George Gamow, in his book, One Two Three…Infinity, who told the story of two Hungarian aristocrats who challenged each other to think of the bigger number. “Well,” said one of them, “you name your number first.”
“Three,” said the second one, after a little hard thought.
“Right,” said the other. “You win.”)
Humans are “the things that count.” Counting is something most of us learn early and learn well, to the point where there is something unnatural about the sequence 1, 2, 3, 5, 6, 7… Counting is the subject of this article, but we will be concerned intimately with counting as it applies to physics and the natural sciences, rather than to the role of counting and numbers in mathematics. I hope to start with things familiar to everyone, and end with novelties.
2.NUMBERS
The most familiar numbers are the integers, the whole numbers that we use whenever we start counting, 1, 2, 3. They are basic to everything else, to the point where the nineteenth century German mathematician Leopold Kronecker proclaimed: “God made the integers; all the rest is the work of man.” Personally, I think Kronecker had it backwards. Humans were quite capable of creating the integers, from the need to enumerate everything from cabbages to kings. It is the other sorts of number, the ones we need to describe lengths and times and weights, that may need divine intervention to explain them. We will get to them in a moment.
First, let us ask an odd question. Can we define the biggest number that we normally need for counting? The conventional answer is, no, there is no such biggest number. We start counting at 1, but we never reach a “last number.” No matter how far we go, there will always be a number bigger than the one we have just reached. However, there is another way to look at this, because when we get numbers larger than about a hundred, we don’t think of them as made up of a whole lot of ones. We think of them in groups. A thousand is ten hundreds, a hundred is ten tens. We need that structure, to allow us to work with anything more than we can count conveniently on our fingers and toes.
The largest number of objects that I ever have to deal with in my everyday life, where I am not able to group them into subsets in a useful way, are the bits of a jigsaw puzzle. Even here, with a 1,000-piece puzzle, I try to impose some sort of system that will help me to put the pieces into their structured form of the actual picture. I pick out all the edge pieces and do those first, and I organize pieces according to their colors. And even with this help, I take many hours to assemble a hard puzzle of more than 500 pieces. Given a puzzle of ten thousand pieces, I’m not sure I’d ever put it together; and yet ten thousand is not a particularly big number. We work with much larger ones all the time, merely by organizing them into sets of smaller ones. Ten thousand dollars is not a massive stack of singles, it’s a hundred one-hundred dollar bills.
The same principle works when we have to deal with something that’s too small to be handled conveniently by straight counting. For example, the heights of people, or the size of a room, are not usually an exact number of feet or meters. So we say that a foot is a group of smaller objects, inches; and if we have to, we say that each inch is also a group of still smaller objects, tenths of inches. Since we can define as many levels of subgroups as we like, we can describe anyone’s height, or the size of any room, as closely as we like, just by counting.
The Greeks had reached this conclusion by about four hundred B.C. They believed that they had a system that would allow them to define any given number. It was a horrible intellectual shock for them to discover that there were certain numbers that cannot be described in this way. The result seemed to fly in the face of common sense—to be irrational. And the name “irrational numbers” is used to this day, to describe numbers that can’t be written exactly as whole numbers, and subsets of whole numbers. Numbers that can be so written are called rational numbers.
The original example of an irrational number, discovered to be so by Pythagoras and his followers, is the square root of two. It’s easy to write this number approximately, as something a little bigger than 1.4 and less than 1.5. We can even specify very easily a value that is as good as we are ever likely to need for practical calculation, 1.41421356237309. In terms of our whole numbers and subsets of whole numbers, this is just 1, plus 4 one-tenths, plus 1 one-hundredth, plus 4 one-thousandths, and so on. (I should add that the Greeks themselves did not have a decimal notation. That came much later, introduced in Europe in 1586, and useful as it is, it is not popular with everyone. Jerome K. Jerome wrote of visiting the city of Bruges, where he “had the pleasure at throwing a stone at the statue of Simon Stevin, the man who invented decimals.”)
The problem comes when we ask when the sequence of numbers occurring in the square root of 2, namely, 1 + 4/10 + 1/100 + 4/1,000 + 2/10,000…, stops.
It doesn’t. The first thought might be that this is a problem created because we are using the decimal system. After all, 1/3 is described very nicely by dividing a unit into threes, but when we write it as a decimal, 0.3333…, it goes on forever.
The Greeks were able to show that this was not the cause of the problem, using a very simple and elegant proof, as follows:
Suppose that the square root of 2 can be written as a fraction in the form P/q, where p and q are whole numbers with no common divisors, i.e., there are no whole numbers other than 1 that divide both p and q. For example, 1.414 would be written as 1414/1,000 = 707/500.
Then (p/q)2 = 2, so p2 = 2q2.
Now if p is an odd number, then so is p2. But p2 is even, so p must be even, and can be written as 2r, where r is also a whole number. Then since 4r2 = (2r)2 = 2q2, we must have 2r2 = q2, and so q must be even. But before we began, we agreed that p and q would have no common divisors, and now we have decided they are both divisible by two. Thus our assumption that the square root of 2 can be written as a ratio of whole numbers must be wrong.
This is such an easy proof that the reader may feel anyone could find it, and that mathematics must therefore be simple stuff. If so, here’s another problem to try your teeth on. It can also be proved by very simple arguments:
Let m and n be whole numbers with no common divisors, and n have no divisors other than itself and 1 (i.e., n is a prime number). Then when mn is divided by n, it will leave the same remainder as when m is divided by n. For example, if m is 15 and n is 7, then 157 = 170,859,375. Divide by 7, and we get the remainder 1—the same remainder as when we divide 15 by 7.
If you can prove this, without assistance or looking up the proof in a book on number theory, I would like to hear from you.
The implications of the fact that not all numbers can
be written as fractions were disturbing to the Greeks, and they ought to be equally disturbing to us. I mentioned Kronecker’s idea of divine influence in connection with the whole numbers, but many people struggling over the years with irrational numbers would give credit for them to the devil.
Let’s look at one of the other peculiar facts about irrationals. If we imagine a long ruler, marked off in inches, then in any inch there will be an infinite number of points that can be marked on the ruler as rational numbers, of the form P/q; but we have found that not all points within the inch can be marked that way. There are points corresponding to irrational numbers, sandwiched in among the rationals. Thus, if you want to label all the points on the line, you must include the irrational numbers. Worse than that, and most surprising, it can be shown that there are many more irrational numbers than there are rational ones. The rational numbers, in mathematical terminology, constitute a set of measure zero on the line—which means they account for zero percent of the line’s length. Almost every number is irrational.
3.INFINITY
That last paragraph may sound odd, even crazy. We have already agreed that there is an infinite number of whole numbers, so there is certainly an infinite number of rational numbers, since the rational numbers include the whole numbers as a subset. Now we are saying that there are even more irrational numbers than rational ones—more than infinity. How can anything be more than infinity?
Until a hundred years ago, the answer to that question would have been simple: it can’t. Then another German mathematician, Georg Cantor, suggested a different way of looking at things relating to infinity. (Cantor died in a lunatic asylum, and that may be more than coincidence.)
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