of course, that if one bought General Motors stock three months ago, the same thing would happen to it if he bought it now! We have to watch out for geographical differences too, for there course, variations in the characteristics of the earth's surface. So, for example if we measure the magnetic field in a certain region and move the apparatus to some other region, it may not work in precisely the same way because the magnet field is different, but we say that is because the magnetic field is associated with the earth. We can imagine that if we move the whole earth and the equipment, it would make no diference in the operation of the apparat nother thing that we discussed in considerable detail was rotation in space if we turn an apparatus at an angle it works just as well, provided we turn every thing else that is relevant along with it. In fact, we discussed the problem of sym metry under rotation in space in some detail in Chapter 11, and we invented a mathematical system called vector analysis to handle it as neatly as possible On a more advanced level we had another symmetry-the symmetry under uniform velocity in a straight line. That is to say-a rather remarkable effect-that if we have a piece of apparatus working a certain way and then take the same ap- paratus and put it in a car, and move the whole car, plus all the relevant surround ngs,at a uniform velocity in a straight line, then so far as the phenomena inside the car are concerned there is no difference: all the laws of physics appear the same We even know how to express this more technically, and that is that the mathe- matical equations of the physical laws must be unchanged under a Lorentz trans formation. As a matter of fact, it was a study of the relativity problem that concen trated physicists'attention most sharply on symmetry in physical laws Now the above-mentioned symmetries have all been of a geometrical nature time and space being more or less the same, but there are other symmetries of a different kind. For example, there is a symmetry which describes the fact that we can replace one atom by another of the same kind to put it differently, there atoms of the same kind. It is possible to find groups of atoms such that if we change a pair around, it makes no difference-the atoms are identical. Whatever one atom of oxygen of a certain type will do, another atom of oxygen of that type will do. One may say, That is ridiculous, that is the definition of equal types!"That may be merely the definition, but then we still do not know whether there are any atoms of the same type"; the fact is that there are many, many atoms of the same type. Thus it does mean something to say that it makes no difference if we replace one atom by another of the same type. The so-called elementary particles of which the atoms are made are also identical particles in the above sense-all electrons are the same; all protons are the same; all positive pions are the same; and After such a long list of things that can be done without changing the phe nomena, one might think we could do practically anything; so let us give some examples to the contrary, just to see the difference. Suppose that we ask: "Are the physical laws symmetrical under a change of scale? Suppose we build a certain piece of apparatus, and then build another apparatus five times bigger in every part, will it work exactly the same way? The answer is, in this case, no The wavelength of light emitted, for example, by the atoms inside one box of sodium atoms and the wavelength of light emitted by a gas of sodium atoms five times in volume is not five times longer, but is in fact exactly the same as the other So the ratio of the wavelength to the size of the emitter will change Another example: we see in the newspaper, every once in a while pictures of a great cathedral made with little matchsticks-a tremendous work of art by some retired fellow who keeps gluing matchsticks together. It is much more elaborate and wonderful than any real cathedral. If we imagine that this wooden cathedral were actually built on the scale of a real cathedral, we see where the trouble is it would not last-the whole thing would collapse because of the fact that scaled-up matchsticks are just not strong enough. "Yes, " one might say, "but we also know that when there is an influence from the outside, it also must be changed in pro- portion! We are talking about the ability of the object to withstand gravitation So what we should do is first to take the model cathedral of real matchsticks and 52-2