Supporting himself on his elbow, he sketched a diagram on a sheet of paper that gave a clear picture of the conditions of the problem. I can no longer recollect how he arrived at the terms of his equation. At any rate, the result soon came to hand in a time not much longer than I had taken to enunciate the problem to him. It was a so-called indeterminate (Diophantic) equation between two unknowns, that was to be satisfied by simple integers only. He showed that the desired position of the hands was possible 143 times in 12 hours, an equal interval separating each successive position; that is, starting from twelve o'clock, the two hands may be interchanged every 5 minutes ²⁄₁₄₃ seconds, and yet give a possible time.
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I mention this little episode, which is insignificant in itself, merely to give an example of how a great discoverer, too, finds amusement in such distractions. In Einstein's case this tendency to practise his ingenuity on unimportant trifles is so much the more pronounced from the fact that he requires an outlet for his virtuosity in calculation, and gratefully welcomes every suggestion that helps him to relieve his mental tension. Similar characteristics are reported of the great Euler, as well as of Fermat, whereas many another eminent mathematician feels decidedly unhappy if he drifts within reach of the realm of actual numerical calculation. In my mind's eye I still see Ernst Kummer, the splendid savant (who, in his time, conferred distinction on Berlin University by his very presence), suffering agonies whenever ordinary arithmetical tables threatened to appear in the working-out of his formulae. As a matter of fact, these two things, a mastery over mathematics and a talent for ingenious calculation, are to be considered as quite independent, even if we now and then find them present in the same person.
In the case of Einstein this tendency is a symptom of an incredible universality of spirit. It moreover presents itself in the pleasantest forms, and a character-sketch of Einstein would be incomplete if this trait were not mentioned. Every problem which is in any way amusing excites in him a willing interest and enthusiasm. I once directed our conversation to the so-called Scherenschnitte. These are made from long strips of paper or canvas, the ends of which are caused to overlap a little and then pasted together, but instead of being fixed so that a flat wheel results, which rolls on one side of the strip, the strip is twisted one or more times before the ends are fastened together. If now the strip is cut lengthwise right along its centre, various unexpected results occur, depending on the number of twists that have been made before pasting.
Some very complex geometrical difficulties are involved in these problems. This is shown by the fact that learned mathematicians have written extensive disquisitions on these curious constructions (for example, Dr. Dingeldey's book, published by Teubner, Leipzig). Einstein had never taken notice of these wonders of the scissors, but when I began to form these strips, to paste them, and to cut them, he immediately became interested in the underlying problem, and predicted in a flash what puzzling chain constructions would result in each case, with a certainty that would lead one to imagine that he had spent days at it. On another occasion a space-problem dealing with dress came up for discussion: Can a properly dressed man divest himself of his waistcoat without first taking off his coat? One would not have dared to confront Copernicus or Laplace with such a problem. Einstein at once attacked it with enthusiasm, as if it were an exercise in mechanics, the body being the object; he solved it in a trice, practically, with a little energetic manipulation, much to the amazement and joy of the beholder, who asked himself: Is this the same Einstein who developed the work of Copernicus and Newton? A little later, perhaps, the conversation centres around some serious point drawn from politics, political economy, sociology, or jurisprudence. Whatever it may be, he knows how to spin out the suggested thread, to establish contact with his partner in conversation, to open up his own perspectives without ever insisting on his point of view, always stimulating and showing a ready sympathy for the subject of discussion and for all the ideas which it crystallizes, the prototype of the scientist, in the mouth of whom Terence put the words: "I am a human being; nothing that is human is alien to me!"
CHAPTER IX
AN EXPERIMENTAL ANALOGY
Forms of Physical Laws.—Aids to Understanding.—Popular Descriptions.—Optical Signals.—Simultaneity.—Experiments in Similes.
"I WISH to ask you. Professor, to help me over a difficulty and to treat me as the spokesman of a great number who are similarly troubled. In most accounts of your theory of relativity, there is a dearth of definite, concrete, illustrative examples on which we can fix our minds whenever the theorem is to be applied generally without limitation. Let me express this more precisely: Your simplified picture of the structure of the universe is achieved in the theory of relativity by emancipating all observations from fixed co-ordinate systems, and by proclaiming the equivalence of all systems of reference. One of your earliest theorems states that physical laws describing how the states of physical systems alter, remain the same, no matter to which of two co-ordinate systems these states are referred, provided that the co-ordinate systems are moving rectilinearly and uniformly relatively to one another. This theorem entails the following statement. If we—erroneously—adopt a non-relativistic view, we shall come to the conclusion that physical laws depend on the particular system of reference chosen, and will thus assume a different form for each different system. At this point we experience a desire to hear definite examples. What varying forms may a certain given physical law, known under a definite form, assume, and how can we use this law to show that it must adapt itself to the postulate of relativity?"
Einstein explained that such examples cannot be given in special cases, but only in very general terms. If we were to suggest the elliptic orbits of the planets (at which I had hinted in my remarks), we should fall into error, for the law of elliptic orbits is no such law. For, from another point of view, the elliptic paths of the planets might be drawn out into wavy lines, or into spirals, and they would remain ellipses only as long as the lines of motion are referred to the central attracting body. But the constancy of the velocity of light is such a law, as also is the law of inertia, according to which a body that is left to itself moves uniformly in a straight line.
I confessed to him that this limitation to a few very general laws would be a painful matter for many an enthusiast of average attainments, who has great difficulty in distinguishing the laws that are generally valid from those that hold only within circumscribed limits. But if this were not so, we should have to alter our conception of what is conveyed by a popular exposition. For it is called popular, not because it now and then uses the patronizing words "dear reader," but because it anticipates the questions and doubts of the man of average sense, and examines them, proving some to be unjustified and others to be reasonable or unreasonable, as the case may be. "Then there is a further matter that troubles me," I continued. "Let us suppose an ordinary reader of such a popular account to get a first insight into the new conception of Time. He is glad to feel the ideas dawning in him, and, to get a more lasting view of the idea, he repeats the arguments through which he has just threaded his way, and, in doing so, again encounters the phrase 'uniform motion.' At the first reading he imagined that he understood the expression quite well, but the second time he pauses and considers. For now that he knows how much depends on it, he is anxious to find out the exact meaning of a 'uniform motion.' He looks for a definition, and if he cannot find one in the book he is perusing, he endeavours to reason it out for himself. With good luck he arrives at the usual statement: a body moves with 'uniform motion' if it traverses equal distances in equal intervals of time. But equal intervals of time are clearly those during which a body in uniform motion traverses equal distances. In other words, he explains A by means of B, and B by means of A, so that he has involved himself in a vicious circle from which he cannot escape. This is his hour of need, due to the difficulty of 'time.'