Up to the highest limit that has actually been tested, it turns out that the actual number of primes that exist is somewhat less than is predicted by the formula. In 1914, however, the British mathematician John Edensor Littlewood demonstrated that if one length- ened the string of numbers one investigated for primes, one would find that up to some limits there would indeed be less than the formula predicted, but that up to other limits there would be more than the formula predicted. If that were not so, Littlewood demonstrated, there would be a contradiction in the mathematical structure and that, of course, cannot be allowed. The number of primes is always less than the formula would indicate. The higher one goes, the longer it takes to test num- bers for primehood.
However, it might be possible to do some theoreti- cal work and determine some number below which the first switch from less than the prediction to more than the prediction must take place- That will at least set a limit to the work required. Some shorthand device must be used and the device used is the excellent one of exponential nota- tion. Since 10,000,000,000 is written as a 1 followed by 10 zeros, it can be written exponentially as 101″ (ten to the tenth power). But since ten billion is itself 1010, lo10. Writing exponentials is always a strain when an arti- cle is being written for a nonspecialized outlet- This is especially so when one is forced to place exponents on exponents. To avoid driving the Noble Printer crazy and to make the notation look prettier, I have in- vented a notation of my own.
I mention it here only because I am speaking confidentially. However, “classic” has a secondary meaning that dis- pleases me. The word came into its own when the literary men of the Renaissance used it to refer to those works of the ancient Greeks and Romans on which they were model- ing their own efforts.
Consequently, “classic” has come to mean not only good, but also old. Now 1, Robot first appeared a number of years -ago and some of the material in it was written. The point is that I have decided to feel a little hurt at being considered old enough to have written a classic, and therefore I will devote this chapter to the one field where “classic” is rather a term of insult. Naturally, that field must be one where to be old is, almost automatically, to be wrong and incomplete. One may talk about Modem Art or Modern Literature or Modem Furniture and sneer as one speaks, comparing each, to their disadvantage, with the greater work of earlier ages. In physics, particularly, this is the case. There is Modern 174 Physics and there is (with an offhand, patronizing half- smile) Classical Physics. To put it into Modern Terrninol- ogy, Modern Physics is in, man, in, and Classical Physics is like squaresvhle.
Quite a heavenly zoo! Odd, though, considering that most of the constellations were invented by an agricultural society, that not one represents a member of the plant kingdom. Or can,that be used to argue that the early star-gazers were herdsmen and not farmers? At two points, then, the ecliptic crosses the celestial equator and those two crossing points are the “equinoxes” (“equal nights”).
When the Sun is at those crossing points, it shines directly over the equator and days and nights are equal (twelve hours each) the world over. One of the equinoxes is reached when the Sun, in its path along the ecliptic, moves from the southern celestial hemisphere into the northern. It is rising higher in the sky (to us in the Northern Hemisphere) and spring is on its way. That, therefore, is the “vernal equinox,” and it is on March 20. On that day (at least in ancient Greek times) the Sun entered the constellation of Aries.
A good supernova at its height is releasing energy at nearly 10,000,000,000 times the rate of our Sun. An object five light-years away would pick up a tenth as much energy per second as the Earth picks up from the Sun. At half a light-year from the supernova it would pick up ten times as much energy per second as Earth picks up from the Sun. If a supernova let go five light-years from us we would have a year of bad heat problems. If it were half a light-year away I suspect there would be little left of earthly life. There is only one star-system within five light-years of us and it is not the kind that can go supemova. If our Sun were in the neighborhood of a supernova it would be subjected to a batb of energy and its own temperature would have to go up. After the supernova is done, the Sun would seek its own equilibrium again and be as good as before (though life on its planets may not be).
However, in the process, it would have increased its fuel consump- tion in proportion to the fourth power of its absolute tem- perature. Even a small rise in temperature might lead to a surprisingly large consumption of fuel. When the fuel supply shrinks low enough, the star expands into a red giant or explodes into a supernova.
By the time he got back to the telescope the planet was too close to the Sun to be observ- able. It would take months for the slow-moving planet to get to the other side of the Sun and into observable position, and without a calculated orbit it might easily take years to rediscover it. Fortunately, a young German mathematician, Karl Friedrich Gauss, was just blazing his way upward into the mathematical firmament. He had worked out something called the “method of least squares,” which made it possible to calculate a reasonably good orbit from no more than three good observations of a planetary position. Gauss was right and, on January 1, 1802, Olbers found it. To be sure, the new planet (named “Ceres”) was a peculiar one, for it turned out to be less than 500 miles in diameter. It was far smaller than any other known planet and smaller than at least six of the satellites known at that time. Could Ceres be all that existed between Mars and Jupiter?
The German astronomers continued looking (it 113 would be a shame to waste all that preparation) and sure enough, three more planets between Mars and Jupiter were soon discovered. Two of them, Pallas and Vesta, were dis- covered by Olbers. All Olbers got out of it was the name of a planetoid. The thou- sandth planetoid between Mars and Jupiter was named “Piazzia,” the thousand and first “Gaussia,” and the thou- sand and second (hold your breath, now) “Olberia.
An elec- tron and ia proton therefore approach closely and then maintain themselves at a wary distance, forming the hy- drogen atom. Individual protons can cling together despite electro- magnetic repulsion because of the existence of a very short-range nuclear strong interaction force that sets up an attraction between neighboring protons that far over- balances the electromagnetic repulsion. This makes atoms other than hydrogen possible. The weakness of the gravitational force is a source of frustration to physicists. The different forces, you see, make themselves felt by transfers of particles. The nuclear strong interaction force, the strongest of all, makes itself evident by transfers of pions (pi-mesons), while the electromagnetic force (next strongest) does it by the transfer of photons. An analogous particle involved in weak interactions (third strongest) has recently been reported.
It is called the “W particle” and as yet the report is a tentative one. It seems, then, that if gravitation is a force in the same sense that the others are, it should make itself evident by transfers of particles. Physicists have given this particle a name, the “graviton. It is electrically neutral and without mass. The graviton and the neutrino differ in some respects, however.
Or put it another way. Sup- pose each of the cards can be any card at all. Suppose the deck can have two tens of diamonds or three aces of clubs, or, for that matter, fifty-two threes of hearts.
The total number of orders of such a chameleonic deck could be calculated by imagining that the first card could be any one of fifty-two, and the second card could be any one of fifty-two, and so on for all fifty-two. To calculate the number of different orders, you would have to take the product of 52 X 52 X 52 X… Superfactorials are immensely larger than factorials. Well just have to forget playing cards. This number is arrived at if we suppose that the sun is an average star, that there are about a hundred billion stars in the average gal- axy, and that there are a hundred billion galaxies in the universe. In addition to electrons, protons, and neutrons, of course, there are numbers of unstable particles un- known to Eddington, but their numbers are compara- tively few.
You will find that most congress- people will not be willing to set the precedent, no matter how meaningless such a precedent might be. Martin, you have my sympathy, but I cannot tell you to hope. Doing away with you could turn out to be the easiest way of re- solving the dilemma. Consider that before deciding to push matters. Or if they do, it will be remembered against you. It will be said you did it only for yourself. You have never been part of a political hate campaign, Mr.
Martin, let your life be. Andrew said, “If I decide to fight for my humanity, will you be on my side? If at any time such a stand would appear to threaten my political future,-1 may have to abandon you, since it is not an issue I feel to be at the very root of my beliefs. I am trying to be honest with you. I intend to fight this through whatever the consequences, and I will ask you for your help only for as long as you can give it. Feingold and Martin coun- seled patience and Andrew muttered grimly that he had an endless supply of that.
Every 400 years there are 4 such century years, so why not keep 3 of them ordinary years, and allow onl one of them (the one that is divisible by 400) to be a leap year? In that same 400 years, the Gregorian calendar allows only 97 leap years for a total of 146,097 days. Compare these lengths with that of 400 tropical years, which comes to 146,096. Whereas, in that stretch of time, the Julian year had gained 3.
Had it done the job a century earlier, all western Europe would have changed calendars without trouble. These nations would far sooner remain out of step with the Sun in accordance with the dictates of the pagan Caesar, than consent to be corrected by the Pope. Therefore they kept the Julian year. The year 1600 introduced no crisis. It was a century year but one that was divisible by 400.
Yet, en masse, the Andromeda galaxy is faintly visible to the naked eye. He began by making several assumptions: 1. That the universe is infinite in extent. That the stars are infinite in number and evenly spread throughout the universe. That the stars are of uniform average brightness through all of space. Remember that the amount of light that reaches us from individual stars of equal luminosity varies inversely as the square of the distance from us. This holds for our shells. But as you work your way outward, each succeeding shell is more voluminous than the one before.
Since each shell is thin enough to be considered, without appreciable error, to be the surface of the sphere made up of all the shells within, we can see that the volume of the shells in- creases as the surface of the spheres would-that is, as the square of the radius. The 2000-light-year shell would have four times the volume of the 1000-light-year shell. The 3000-light-year shell would have nine times the volume of the 1000-light-year shell, and so on. If we consider the stars to be evenly distributed through space (Assumption 2), then the number of stars in any given shell is proportional to the volume of the shell. If the 2000-light-year shell is four times as voluminous as the 1000-light-year shell, it contains four times as many stars.