(In probability problems, all numbers are 1 or less than 1. Since 1 represents universality or certainty, a probability greater than 1 cannot be spoken of. What is more probable than the universal or certain?)

  Observe another thing about the multiplication of numbers less than 1. If you keep on multiplying them, the products keep on getting smaller. Suppose you multiplied 0.99984 by itself ten times. The answer would be 0.99816. That’s the arithmetical way of saying that if you pulled ten hydrogen atoms at a time out of your sack, the chances that all of them would be hydrogen-1 without exception is 99,816 out of 100,000. The chance of finding at least one hydrogen-2 atom in that group of ten is 184 out of 100,000.

  Hydrogen molecules are very simple. They contain only two atoms apiece. What if we took a more complicated molecule, such as ethyl alcohol? The molecule of ethyl alcohol is made up of two carbon atoms, six hydrogen atoms, and one oxygen atom.

  To find the frequency with which normal molecules of ethyl alcohol (those containing only normal isotopes) occur, we must multiply the fractional occurrence of carbon-12 by itself (two carbon atoms, you see), multiply that product by the fractional occurrence of hydrogen-1 six times (six hydrogen atoms) and multiply that by the fractional occurrence of oxygen-16 (one oxygen atom).

  The arithmetic would go like this: 0.9888 x 0.9888 x 0.99984 x 0.99984 x 0.99984 x 0.99984 x 0.99984 x 0.99984 x 0.9976 = 0.97432. Out of every 100,000 ethyl alcohol molecules, 97,432 are normal and 2,568 are abnormal.

  That’s a larger number of abnormal molecules than you expected perhaps, but let’s go on. Ethyl alcohol is still a small molecule. What if we take a molecule of table sugar which is made up of twelve carbons, twenty-two hydrogens, and eleven oxygens? We have to multiply 45 numbers together and once that is done, we find the probability of a normal molecule of table sugar to be 0.84748. Out of every 100,000 molecules of table sugar, 84,748 are normal and 15,252 molecules are abnormal.

  The normals still have it by a considerable majority, but it is nothing like the preponderance in the case of the smaller molecules. Interesting!

  What about larger molecules still? A typical fat molecule contains 57 carbon atoms, 104 hydrogen atoms and 6 oxygen atoms. Multiplying all the appropriate probabilities the appropriate number of times, we come up with a final value of 0.50901.

  The truth is. then, that just about half the fat molecules are normal, by the definition of normality we are using. The other half are abnormal.

  Now let’s pass on to the hemoglobin molecule, the red substance in the blood which absorbs oxygen in the lungs and carries it to the tissues. Its molecule is made up of 2,778 carbon atoms, 5,303 hydrogen atoms, 1,308 oxygen atoms, 749 nitrogen atoms, 9 sulfur atoms and 4 iron atoms. Now, we must really multiply and it is at such times that I am most grateful for the existence of logarithms and calculating machines.

  The answer to all these calculations is something smaller, as you ought to expect, then anything we’ve had so far. It is, in fact, 0.0000000000000001134. This means that about one hemoglobin molecule out of every ten million billion is ‘normal’.

  And let’s see what that means. In a single drop of blood, there are about 250,000,000 red blood corpuscles. In one single drop of blood, that is. Well, now, if six hundred men pool all their ‘normal’ hemoglobin molecules, they will have enough to fill exactly one (I repeat, one) of those corpuscles. That single corpuscle will contain hemoglobin completely free of abnormal isotopes. Every other red blood corpuscle in every drop of blood of all six hundred men will contain only hemoglobin molecules with one or more abnormal isotopes included.

  You see, then, that if we insist on considering a hemoglobin molecule to be normal only when it contains normal isotopes and nothing else, we are going to end up with a ‘normal’ molecule that is neither common, natural, ordinary, regular, typical nor usual. Anything but, in fact.

  What we have called a ‘normal’ molecule turns out, as you can now see, to be indeed a limiting case, one which can be reached but is not very likely to be except very rarely. A hemoglobin molecule can be made up of all normal atoms or, alternatively, of all abnormal atoms. Each is a limiting case. Or else, it can be made up of any combination of normal and abnormal atoms. Those are the inbetween cases.

  If the limiting case is so rare (the one where all the atoms are abnormal is many-and-many times rarer than the one we have just considered), are any of the inbetween cases more common? If so, which is most common, and how do we find out?

  Let’s simplify once again and take up a case where there are only two alternatives, each of exactly equal occurrence. The most convenient example involves coin-tossing. Here we have heads and tails, one of each, and we can play with those exclusively.

  If you throw a coin once (an honest coin, of course), your chance of throwing heads is 0.5 and your chance of throwing tails is 0.5. Fifty-fifty, in other words.

  If you throw a coin twice, you may get two heads (one limiting case) or two tails (the other limiting case) or one head and one tail (the inbetween case). The chance of getting two heads is 0.5 x 0.5 or 0.25. The chance of getting two tails is 0.5 x 0.5 or 0.25.

  So far, so good. However, the chance of getting one head and one tail is 0.5, twice as good as getting two heads or two tails.

  You may wonder why that is so. After all the chance of throwing a head is 0.5 and the chance of throwing a tail is 0.5 and multiplying them together leaves a 0.25 chance of throwing both. Ah, but you may throw the head-tail combination in either of two ways. You may throw the head first and then the tail, or the tail first, then the head. That gives you 0.25 x 2 or 0.5, as said. Two heads or two tails can only be thrown one way.

  The rule is that the probability of limiting cases (all heads or all tails) is obtained by multiplying the probability of one head or one tail by the number of tosses.

  For the inbetween cases, the probability obtained in this way must be further multiplied by the number of different ways (always greater than one) in which the particular inbetween case can occur.

  Thus, if you threw the coin eight times, the possible combinations would have the following probabilities:

  The most frequent combination occurring in eight throws is that of four heads and four tails. To be sure, even that would turn up only a little oftener than a quarter of the time so that it couldn’t really be said to be normal. Certainly, though, it is the least abnormal of the combinations.

  Now notice that the most common case is the one in which heads and tails are represented according to their comparative probabilities. The probability of throwing a head is 0.5 and that of throwing a tail is 0.5. Therefore in the set of eight throws, the most common combination is the one where 0.5 of the throws are heads and 0.5 are tails (four of each).

  Without going through any figuring at all. I’d know that the most common combination occurring in a hundred successive throws would be 50 heads and 50 tails. It would be less common (0.1115) than the most common case in the eight-throw problem, occurring only one-tenth of the time. As the number of throws increases, the number of possible combinations increases and the probabilities have to be spread continuously thinner to cover more and more combinations. Still, the fifty-fifty combination would be commoner than anything else.

  Furthermore, if for some reason the probability of throwing a head was 0.9 and that of throwing a tail was 0.1, then we can say confidently, without figuring, that in a total of a hundred throws the most common combination would be 90 heads and 10 tails.

  The situation may not always be as conveniently even as that. Suppose that the probabilities are 0.9 for heads and 0.1 for tails and you are interested in sets of 68 throws. Then you pick the whole number ratio that is nearest to the proportion of 0.9 to 0.1. In this case, your most frequently occurring combination would be 61 heads and 7 tails.

  Or suppose you tossed the coin twice. Your most frequently occurring combination would be 2 heads and no tails. (That’s closer to 0.9/0.1 than the next possible combination, 1 head and 1 tail, would be.)

  I’m going through all this for a specific reason. I’m going to determine the most frequently occurring combination in hemoglobin and I don’t want to have to use the binomial theorem with four-figure numbers. Logarithms, computing machines and all, it would still be tedious.

  But first, I must make one more point. You may have noticed that when two alternatives are of equal probability, as in coin-tossing, the inbetween cases (heads and tails mixed) are always more probable than the limiting cases (all heads or all tails).

  When one alternative is more probable than the other, however, sets made up of a small number of individual items will show one limiting case (that composed only of the more probable alternative) to be the most probable combination. We mentioned several such. For instance, ten hydrogen atoms drawn at random are all hydrogen-1 (a limiting case) 99,816 times out of 100,000.

  As the number of individual items making up a set increases, however, the inbetween cases gradually become more common than the limiting cases, however lopsided the two alternatives are. Hemoglobin, made up of more than 10,006 atoms, has reached this stage even though the probability of the occurrence of the normal isotopes (one alternative) is way and ahead of the probability of the occurrence of the abnormal isotopes (the other alternative).

  For instance, hemoglobin has 2,778 carbon atoms. The frequency of carbon-12 is 0.9888 and that of carbon-13 is 0.1112. Dividing the 2,778 carbon atoms in that ratio, we find that the most frequently occurring hemoglobin molecule is one with 2,747 carbon-12 atoms and 31 carbon-13 atoms. Using the same system for the other atoms, we find that the most frequently occurring hemoglobin molecule has also 3 oxygen-18 atoms, 1 hydrogen-2 atom and 1 nitrogen-15 atom. This makes for a total of 36 abnormal isotopes in the most frequently occurring hemoglobin molecule.

  Even this most frequently occurring combination occurs very infrequently. There are something like a hundred trillion possible combinations, so considerable room has to be left for most of the others. (Not for all, though. Some are so rare that they aren’t likely to occur even once anywhere on earth.)

  In going back to human beings, now, we have little need to dwell on any points. Normal plus normal plus normal-ever-so-many-times does not equal normal. It equals highly abnormal, and it is a limiting case.

  The number of individual factors—physical, mental, temperamental and emotional—making up a human being are so high that no combination can possibly be called normal in the dictionary meaning of the term. All combinations are tremendously abnormal, and if some combinations are a trifle less abnormal than others, the one the psychologists picked, their ‘normal man’, is definitely not among them.

  In fact, any statistical abstraction involving something as complex as the human being is suspect. However handy such may be in computing actuarial tables and predicting elections, it can give rise to great and unnecessary grief through misconstruction by ordinary people in the ordinary business of life.

  Still, as long as psychologists use the words ‘normal’ and ‘abnormal’ in the way that they do, we will always be able to make statements like: ‘It is normal to be a little abnormal’ and ‘It is highly abnormal to be completely normal.’

  And, after all, such statements, while confusing, are also comforting.

  CHAPTER SIX—PLANETS HAVE AN AIR ABOUT THEM

  Ever since it was recognized that other planets existed besides our own, there has been considerable speculation concerning the possibility of life on these planets and on the kind of life that could be possible on them. Intimately bound up with such speculation are considerations of the kind of atmosphere that might be expected to surround a given planet. What do we actually know, or what can we reasonably speculate concerning planetary atmospheres?

  Let’s go about it in a systematic way, by considering first the raw materials of which a planetary atmosphere may be constructed. The various elements, which are the building blocks of any substance, atmospheres included, are available to different degrees. Some are more common than others and this must be taken into account. Common elements get first consideration in atmosphere building; the commoner, the better. After all, if you were told that for some certain purpose you could use either water or liquid radium equally well, you would be a most unusual character if you went further than the nearest water-tap to accomplish your purpose. And this ‘principle of least action’ is as applicable to the Universe as to you.

  The comparative abundance of the more common elements in the universe as a whole (according to recent estimates) is given in Table XVI. The atoms of silicon are set arbitrarily equal to 10,000 and the quantities of atoms of other elements are given in proportion. What is at once obvious is that 90 per cent of the Universe is hydrogen (the simplest element) and 10 per cent is helium (the next simplest element). There is also about 1 /6 of 1 per cent of impurities—meaning all the other elements.

  It follows then that if you’re going to collect a sample of interstellar gas and dust and make a sun or planet out of it you’re likely to end up with a big ball of hydrogen and helium.

  That’s what the sun is made of, for instance. It is 85 per cent hydrogen and 15 per cent helium, plus a bit of impurity. (The shortage of hydrogen and excess of helium is due to the fact that for four billion years at least the Sun has been turning hydrogen into helium to keep shining.)

  It’s what Jupiter seems to be made of, too, if the most recent theories are more correct than previous theories have been.

  Now that we have a list of the available materials, the next question is: Which of these are suitable for use in atmosphere-making? To be a component of an atmosphere a substance must be a gas or a volatile liquid (or solid) at the temperature of the planet’s surface. (By a volatile liquid or solid I mean one which is in equilibrium with a substantial amount of its own vapor at the temperature being considered. For instance, at ordinary Earth temperatures, water is a volatile liquid and iodine a volatile solid. For that reason, water vapor is a normal component of Earth’s atmosphere and, if there were enough iodine lying around, iodine vapor would be.)

  Now we have quite a decent array of surface temperature in the planets of our own Solar System, and these are given for reference in Table XVII. The temperatures are given in degrees above absolute zero to avoid the complications of negative numbers.

  For comparison the boiling points of the common elements of the Universe are given in Table XVIII in degrees above absolute zero. Note to begin with that at no planetary temperature in the Solar System can carbon, iron, silicon or magnesium form part of any atmosphere. (The surface temperature of the sun is 6,000 degrees absolute and all these high-boiling elements are found in its atmosphere. This discussion, however, is concerned with planetary and not with stellar atmospheres.)

  Sulfur is not a gas at any planetary temperature either, but a substance often remains more or less volatile down to temperature 100 to 200 degrees below its boiling point and we can set 150 degrees below boiling as a kind of arbitrary limit for significant volatility. Sulfur would therefore be a volatile liquid at temperatures equalling Mercury at its hotter moments and sulfur vapor could then exist in the atmosphere.

  The other elements are more likely substances for atmosphere-making. Oxygen is a gas out to Saturn and nitrogen is a gas out to Uranus. Both are volatile liquids on Neptune and volatile solids on Pluto. Neon, hydrogen and helium are gases even on Pluto. And since hydrogen and helium are overwhelmingly preponderant in the Universe as a whole, any planetary atmosphere must, to begin with, consist almost entirely of hydrogen and helium.

  I say, to begin with.

  There’s a catch. In the gaseous state, the molecules of a substance don’t stick together as they do in the liquid and solid state. Each molecule in a gas goes its own way at various speeds and in various directions, including up. There is always a thin trickle of gas continually drifting up and up and some molecules inevitably succeed in escaping from planetary bondage altogether. Atmospheres leak, in other words.

  The size of the leak varies according to the size and temperature of the planet and is different for different gases. The smaller a planet is, the weaker its gravitational hold on the molecules, and the easier it is for the atmosphere to escape into space. The warmer a planetary surface is, the faster the molecules in its atmosphere move, and the more rapidly the atmosphere will escape into space. Smallness and warmth increase the atmospheric leak.

  In addition, the smaller the molecules of a particular gas, the faster the average velocity of the individual molecules of that gas, and the more likely it is to escape into space. Hydrogen has the smallest molecule and helium the next smallest molecule of all known substances. The atmospheric leak is therefore largest for hydrogen and only a little smaller for helium.

  Even a planet as large as Jupiter (317 times as massive as Earth and with a surface gravity 216 times as great), and as cold as Jupiter, may not be able to hold on to all its hydrogen. The hydrogen/helium ratio in Jupiter’s atmosphere is only 3:1 instead of the 10:1 it is in the Universe as a whole. This means that if Jupiter has held on to all its helium, it has lost 2/3 of its hydrogen. (There is an alternative here which I must point out. It may be that helium with its lower melting and boiling points has been squeezed out to some extent in the body of Jupiter and that more of it has been forced into Jupiter’s upper layers and atmosphere.)

  Now Saturn, Uranus and Neptune are all smaller than Jupiter but all are colder, too, and the two effects cancel one another out. We can guess that all these (I leave Pluto out as an unknown quantity) have similar hydrogen-helium atmospheres and that, in fact, so do all planets that are large and cold.