The following article appeared in "Scripta Mathematica" September, 1932.
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HOWEVER, the process is imperfect, and for serious work, the original should be used.
ON the 19th of October a little group of mathematicians gathered
in the Burt Laboratories in Pasadena, California, around a
mysterious machine to watch it attack a problem in mathematics.
It was a simple enough problem to state. It had only to
find two numbers which when multiplied together would give
5,283,065,753,709,209. Any person with a few hundred years of
leisure time on his hands could work it out.
"Here we step out into the unknown!" said the young inventor as he threw the switch and set whirling a complicated mass of gears. It was of no use for the human eye to try to make anything out of the rapidly rotating wheels. One might as well try to identify the teeth on a buzz-saw. Besides it was quite unnecessary, for a fixed, unwinking eye was turned on the machine waiting for a ray of light to slip through certain holes in the wheels, which should be the signal for it to stop the motion and to gather in the solution of the baffling problem in the theory of numbers.
"Rapid calculation, all right," went on the inventor. "It has to try out in a certain formula about ten million numbers. Each number would take a man at least six minutes to examine, which comes to sixty million minutes, or about a million hours. A man could not work at this sort of thing more than ten hours a day, so that gives a hundred thousand days. One could do it in three hundred years if he did not get stale." "How fast is the machine working on this list of ten million numbers?" some one asked. "About a hundred thousand a minute," replied the young man. "It may take an hour and a half to clean up the problem. With a larger driving motor we could make it in twenty minutes. The electric eye would catch it if it were going five times as fast."
Suddenly, click! The power was shut off. "It must have seen something." The machine was turned slowly back till a tell-tale beam of light appeared through the little hole before which the electric eye had been watching. Then some reading of dials and a little grinding of a computing machine and two numbers were found such that the square of one of them plus seven times the square of the other were equal to the number under examination.
"Once more to the window, fair Rebecca!" and the eye was once more fixed on the tiny hole through which the ray of light must come, and the wheels were again set in motion. This time twenty-five minutes passed before anything was reported by the faithful watcher at the window. Had the fair Rebecca grown weary and fallen asleep? One of the mathematicians at least, the father of the young inventor, was in something of a state of nerves. But Rebecca was on the job. She had seen a light and had stopped the whirling wheels. Again the tell-tale ray of light was located and again the number 5,283,065,753,709,209 was given as a square plus seven times another square. The machine had done its duty. These two results were all that was necessary. A few minutes computation still remained, and thus it was, while coffee was being served on one of the working tables in the laboratory the big number was broken up into the factors 59,957 and 88,114,244,437. These are the two hidden numbers which when multiplied together will give the sixteen digit number under examination. It may seem to the man in the street an odd thing to get excited about, but on this occasion
And even the ranks of Tuscany
Could scarce forbear to cheer.
We are apt to stand aghast at the numbers which appear in the study of astronomy. The use of the "light-year" as a yardstick strikes one with a certain awe. This amounts to taking a distance of nearly 6,000,000,000,000 miles as the unit for the measurement of astronomical distances; and in some of his calculations which have to do with extra-galactic systems the astronomer has to apply this little measuring rod thousands of times. These vast distances and these vast numbers stagger the imagination, and yet the mathematician reaches out with his high-powered machines and his high-powered theorems and investigates the internal structure of his distant bodies much as the astronomer inquires into the structure of some distant star. If we con- sider his numbers as expressed in terms of the astronomer's light-year; and use his six trillions as our unit we shall find the vast numbers which are the playthings of the astronomer are only as inches to light-years. What measuring rod shall we use to describe the largest known prime 170,141,183,460,469,231,731,687,303,715,884,105,727? A light-year of light-years would have to be applied a light-year of times to reach this remote star. For over fifty years this number has stood on the outposts of the number system, a challenge to explorers in this field of thought. By a theorem proved as long ago as Euclid's day, we know that there are infinitely many primes beyond this monster. Who shall be the first to discover them and identify them?
The machine in Pasadena is the first attempt to apply the magic of the photo-electric cell to this problem of the study of remote numbers. Some five years ago a crude machine was constructed of bicycle sprocket-wheels with chains of different lengths running over them. The sprocket-wheels were mounted solidly on a rotating shaft, and on certain of the links of the chains pins were fastened, which when they ran over the highest point would lift a spring and break for an instant an electrical contact. When all of the pins were in a line at the top all the contacts would be broken and the machine would stop. This machine, crude as it was, and capable of only a few hundred revolutions a minute gave, after running two hours, the decom- position of the curious number 9,999,000,099,990,001 into the two factors 1,676,321 and 5,964,848,081. The method was seen to be a powerful one and a more delicate machine was planned which should be more dependable in its action and capable of higher speeds. With the generous cooperation of the Carnegie Institution of Washington the device was carried to perfection in spite of many difficulties.
The sprocket-wheels and chains have been replaced by steel gears of different radii meshing into cogs of other gears solidly mounted on a heavy steel shaft. The pins with their uncertain contacts have been replaced by a ray of light which slips through holes in the gears. For a given problem some of these holes will be opened and some closed. When there is an alignment of open holes (which event signalizes a solution) a ray of light slips through them and falls for the ten-thousandth part of a second on the sensitive eye of a photo-electric cell. This infinitesimal impulse is sent by the cell to an amplifier which magnifies it 729,000,000 times. This magnified impulse is strong enough to throw off the current which drives the machine and the whirling wheels come to a standstill. The number of revolutions is then read off on a counting device and the solution is in hand.
Simple as the scheme is there were many unforeseen difficulties that threatened to wreck it altogether. The ray of light was so feeble an impulse to start with, and the length of time it had to speak to the photo-cell in passing was so short that every detail of the construction had to be very delicately adjusted. Moreover the amplifier is so sensitive to outside disturbances that a slight jar, an infinitesimal variation in the electric current, the turning off of an electric appliance or even a hasty word from an attendant would send it off into a temperamental fit. The amplifier was placed in another room separated from the whirling gears by a heavy brick wall, and was enclosed in a coffin of celotex. This made things go a little better, but it was still subject to what seemed a perfectly unpredictable series of "jitters". It would run happily and sweetly for a few minutes and then suddenly become incoherent. Then it would as suddenly pull itself together again and behave in an entirely rational manner for another quarter of an hour before having another tantrum. And all without any appreciable change in the environment ! Doctors were called in, but nothing could be inferred from the symptoms. To be sure it was asking a good deal of a machine to work steadily with a magnification of over seven hundred million to one 1 It was like trying to write a smooth flowing hand with a pen ten thousand miles long, while all the time some mischievous imp was jogging at your elbow. And what was the imp, and how to lay hands on him?
Days of futile adjustments and readjustments went by. There seemed to be nothing organically wrong; just a case of nerves. Nevertheless it was an interesting and important case and the young doctor in charge was unwilling to give the patient up. At last it occurred to him to use a stethoscope. He installed a loud speaker and "listened in". Instantly the hiding place of the "imp" was discovered. There was a short-wave radio fan operating in a station in the immediate neighborhood. So long as he was quiet all went well. When he came on the air the amplifier went into a spasm; the electric eye saw red. There was no accounting for tastes and if the machine wanted to be temperamental on the subject of radio there was nothing to be done about it except to screen the amplifier from this interesting but undesirable disturbance, or to find a time when the radio operator was not playing with his machine. One could hope, of course, that the congruence machine had been giving him as much trouble as he had been sending out. At any rate, till the rival machine could be located, and some sort of truce patched up there was nothing to be done but to wait till the ether was quiet before asking the machine any important arithmetical questions.
On the 9th of October, just after the discovery of the "imp" the machine was set to do some real work in the theory of numbers. There was a large factor of 293 + 1, namely the number 1,537,228,672,093;301,419 which was known by a very powerful test to be composite, but the test would not furnish the factors. The smallest factor was known also to be larger than 300,000, and might be large enough to occupy the time of a skilled computer for over twenty-five years to find it. The machine was "set" to grope for the factors of this number. The amplifier woke up and threw off the power after only a few seconds of work, and it was thought, of course, that the "imp" was on the air again. On examination, however, it was found that the electric eye had been on the alert, and had really seen something. The immense "binary" was decomposed into the product of the two prime factors 529,510,939 and 2,903,110,321. The work of a quarter of a century had been done in less than a quarter of a minute. It was as if some watcher of the heavens had turned his instrument upon some star two million light-years distant, and had determined not only that it was a binary, but should give the exact weight of each of the component stars. But the astronomer is handicapped by the Einstein assumption that there is no higher velocity than the velocity of light. He would have to content himself with the fact that he is in this case some two million years behind the times. He is investigating what the star was like two million years ago. What it is like now, who can say? A million years is a long time and the sun might have smashed into some other sun in the meantime and spread itself out into a spiral nebula. Fortunately the mathematician deals with velocities greater than the speed of light. His news from his remote suns is up to the minute. "Simultaneity" is for him a real thing and not a matter for definition in terms of the velocity of light. Down the infinite vistas of his number system he goes with the speed of thought. His star-catalogue is complete as far as the first ten million numbers. Beyond this "solar system" his instruments can readily examine individual stars up to two billion. Beyond that, in certain particular regions of the heavens, he can reach out and lay his investigating hand on certain clusters. It is fitting that at last the slow moving ray of light should come to his aid. Other galaxies may now come under his scrutiny.
It may come as a surprise to many that the most compelling urge to the study of mathematics is not its practical application to the study of every day, bread-and-butter life, but lies in the romance and glamor surrounding its mysterious secrets. Romance and glamor in connection with mathematics seems absurd on the face of it. One recalls Macaulay's letter as a student to his mother. "I can scarcely bear to write on mathematics or mathematicians. Oh, for words to express my abomination of that science, if a name sacred to the useful and embellishing arts may be applied to the perception and recollection of certain properties in numbers and figures! Oh, that I had to learn astrology, or demonology, or school divinity; oh, that I were to pore over Thomas Aquinas, and adjust the relation of Entity and the Two Predicaments, so that I were exempted from this miserable study!" Such are the outbursts of a youthful mind impatient at being made to walk carefully along the slippery path that leads to the mountain peak whence may be had some vision of this universe of wonder and mystery and beauty. To young Macaulay mathematics was only a cross-word puzzle constructed for his mystification by his ingenious but misguided tutor.
But what shall we say of the problems which this arithmetical machine is constructed to cope with? Who set these problems? Who hid the factors of these vast numbers so cunningly? Behold, here is a number which has been waiting for me (since when?) to discover its secret composition! For me it has been reserved to find the answer to this riddle which has baffled the minds of men for centuries. What is a North Pole or a South Pole, or a darkest Africa compared with this unexplored country? And think what it has all come from! Here is a pure creation of the mind; this system of numbers; invented for the mere merchant, the trader, the man of business. It lends its aid willingly and effectively to the solution of the practical problems of every day life. It is merely a creation of the mind of man; surely he can bend it to his creative will. But it has sprung full armed from his forehead, and boast as he may that. he is the measure of all things he finds himself in the presence of a Sphinx whose secrets he is helpless to discover. Of some of the problems he can say, "I have found the answer to them;" of others he must say, "I have not solved them yet;" of still others he must say; "They cannot be solved." Cross-word puzzles? Whose is the cunning hand that set them up? A Puzzle Maker of infinite cunning and infinite resources! He gives nothing up without a struggle; "but He is never dishonest."
It will come as a shock also to some to be told that there is, so far as can be seen now, absolutely no "practical application" expected to develop out of this astonishing machine upon which so much thought and care has been expended. There is a cowardly and slinking sort of a scientist, no doubt, who is ashamed or afraid to take a walk in the country with the avowed purpose of enjoying the landscape. He must provide himself with a fishing rod or a collecting basket of some sort, so that if any one asks him why he is abroad he will be able to point to some "practical application" for his stroll in the hills. He is, no doubt, merely trying to avoid the odium that seems to have attached itself to the poet or to the musician who is hard put to it to produce a healthy, bread-and-butter reason for making a sonnet or a symphony. To listen to the apologists for the study of pure mathematics one would get the impression that this study is sustained, not by the Wonder and Beauty of the subject, but by its external utilities. But how little of the vast field of mathematics has to do with the study of the outside world! The theory of differential equations stretches far beyond its application to bridges and universes. Modern mathematics is of more importance in its philosophical than. in its physical implications. The same thing may be said even for the modern study of physics. The subtle and expensive determinations of the bending of a ray of light by a gravitational field, or the careful listing of the binary stars in the heavens, can have little application to the making of two squashes grow where only one grew before. Faraday, playing with wires in his laboratory, wrests from the hands of nature a torch that Edison uses to light the world, and Einstein to light the universe. Who can tell? Perhaps in some far distant century they may say, "Strange that those ingenious investigators into the secrets of the number system had so little conception of the fundamental discoveries that would later develop from them!"