TOOLS FOR BRAINS
A science-article on the art
of thinking by machinery. A machine can
have memory, logic, number sense, and an ability to do superhuman
calculus--but it lacks the ability to say "That's plain cockeyed.!"
By Leo Vernon
Illustrated by Schneeman
CAN machines think?
The question keeps coming up every time a new kind of calculating machine
is invented, or a new attachment is put on to the almost unbelievable machines
that are now being manufactured. Unfortunately for the mathematician
and physicist, the question must be answered, "No."
There is a big difference between a machine that can think and the awe-inspiring new analyzer being built at the Massachusetts Institute of Technicology [sic]. This new machine can correct errors, will even flash a light to point out the place where an incorrect adjustment has been made, but it cannot by itself decide which is the better of two figures or do anything except what the operator has decided for it to do.
A calculating machine does not know the answer to 3 + 5, as a thinking machine would. The human operator puts the number three in the machine, and then puts in the number five. The machine adds the number 1 five different times, and stops. A more complex machine might add something other than unity, such as trigonometric functions, but essentially it makes use of only those things which were put in the machine and shows no choice.
The important thing for scientists is that the machine can add 1 five times, or it can add 14,689,378 some 8,564 times, and do it almost as quickly as the operator can push a button. In more advanced machines it can repeat such an operation a thousand times in an hour for different sets of numbers. It is truly a labor-saving device.
The history of calculating machines shows that labor saving is the important factor. As commerce and science developed and required more and more arithmetic to be done, new machines and methods were invented. In the earliest days all the arithmetic that was needed could be done on fingers and toes, the first calculators. If more than twenty twenties were required, nothing could be done, until one of our cavemen ancestors thought of using pebbles in addition to his toes. Not long after this another labor-saving person learned to string beads on twigs, and the first abacus came into existence.
The abacus is our first truly mechanical device for doing arithmetic. When divided into separate wires for units, tens, hundreds, and so on, addition and subtraction can be done very rapidly. Even multiplication can be done by repeated addition, and a very crude division is possible, though not simple.
There were no real understanding of arithmetic as long as the old Roman and Greek numerals were used. The next improvement came with the introduction of the Arabic number system, and the invention of the number zero. When this system was first learned in Europe, the men who knew it went around and gave exhibitions of how rapidly they could add, subtract or multiply without an abacus.
The invention of the arithmetical symbols +, -, and x gave additional impetus for mathematical advances in the sixteenth and seventeenth centuries. By the year 1600 it was absolutely necessary to find aids for science and commerce. Mathematicians were spending years computing problems that can now be done in days. L. van Ceulen spent his entire lifetime computing the ratio p to 35 places of decimals, completing the job in 1610. The invention of calculus, the discovery of infinite series, meant more and more computing for overworked mathematicians. Increasing commerce and the complicated rates of exchange meant worse and worse problems for the merchants, since coinage was not on the decimal system.
THE FIRST aid came from
India by way of Persia, with blocks of wood laid out in the form of multiplication
tables. By shifting these tables around, multiplication could be
done more accurately, and in some cases more speedily. Then John
Napier made his two biggest contributions toward reducing wear and tear
on mathematicians. In 1594 he invented logarithms, and in 1617 improved
the Persian tables by applying his logarithms to the blocks of wood that
were in use.
The logarithms were devices which reduced multiplication and division to addition and subtraction. By applying them to blocks of wood shaped like rulers, two rulers could be laid side by side and the answers read off directly. These rods of wood, called "Napier's Bones," were improved rapidly. Samuel Pepys, author of the famous diaries, had a job requiring him to measure the number of cubic feet in large lots of lumber, a real problem without some devices to help him. Pepys used Napier's Bones, and quickly got the idea of fastening them together in a frame so they could be slid back and forth conveniently--the first form of the modern slide rule.
The slide rule has continued in use, with considerable improvement, up to the present day. For much engineering work and some commercial work it is unbeatable, but for the scientist it helps only in getting a rough idea. The difficulty is that it is accurate to only four figures at the most. With a slide rule it is possible to multiply 1024 x 1728, but instead of getting the correct answer of 1,769,472, the slide rule simply says that the first four figures of the answer are 1769 or 1770. That is, there is an accuracy of a little better than one part in a thousand.
That was not enough for scientists or for merchants. They had to know their answers more accurately. Logarithms helped greatly, especially after Briggs invented the modern form based on the decimal system in 1617 and computed ten place tables for all numbers between 1 and 100,000. More was needed, and it came soon.
Blaise Pascal, in 1641, made one of the great contributions of his weirdly mixed-up life. In order to help his merchant father, he invented an adding machine. As France at that time had a coinage system of 12 deniers equal to 1 sou, 20 sous 1 livre, imagine the problems of a merchant trading with England and other countries with similarly jumbled coinage systems, and with varying rates of exchange.
Pascal did not use the modern style of key-driven machine, but he invented the mechanism which is used in modern speedometers and calculators. One complete revolution of a wheel causes the wheel next to it to turn over one-tenth of a revolution, and so on for a series of gears. More, he did them at a time when machinery was not known for cutting gear teeth accurately. He had to use pin gearing. Each wheel had ten holes drilled in its outer rim, and little pins were carefully handfiled to size, inserted in the holes and soldered in place. He even invented the cams necessary to go on the wheel's axle to cause the number to remain in place on the dial until the time arrived for the next number to shift into position.
For years Pascal tried desperately to sell the machine and its basic idea, but without success. It was still cheaper to hire a lot of clerks, instead of mechanics to build machines. A few of his machines were used by scientists, with improvements added by men such as Moreland in 1666 and the great Liebnitz who worked from 1674 to 1691 building the modern stepped reckoner and pin wheel, making the key-driven machine possible.
With these advances of a single century, scientists caught up on their computations and commenced discovering new theorems and equations. By the beginning of the nineteenth century, the problems were again becoming complex. Some men who had a genius for doing arithmetical work, such as Gauss and Euler, devised new methods of computing, simplified formulas, but still work was piling up.
THE NEXT advance
came by way of industry, from a weaver, Joseph Marie Jacquard, inventor
of the jacquard loom. When power-driven looms first were made, they
were used almost exclusively for straight weaving with the warp and woof
alternating very simply. Designs still had to be made by hand, or
the power-driven machinery stopped so that the adjustment for lifting the
warp could be changed.
Jacquard invented a device to weave any pattern without stopping the machinery or using hands at all. He used something like a player-piano roll. As this tape rolled through the loom, wires attached to the warp would poke up through the holes in the pattern and cause the warp to be lifted, changing the pattern automatically. The design for the cloth could be planned beforehand, holes punched in the roll of paper to correspond to the design, and the machines would take care of it automatically. This method is still in use, as in Jacquard satins.
Charles Babbage got the idea that instead of designs for cloth, equations could be punched in the roll of paper and instead of looms, the roll could be made to actuate computing machines. In 1823, Babbage started construction of his difference engine, but money ran out in 1833 and the machine was never finished.
No new ideas were developed for over half a century. The key-driven machines were improved, and with the addition of electric power became almost magically efficient and speedy. Anybody who has operated a modern electric calculating machine knows the tremendous power it gives in calculating, but even this is not enough.
The sheer volume of computing that must be done in business and science has forced all scientists to think of new developments, and yet better mechanisms. It is hard to appreciate the amount of arithmetic that has to be done. In a single big bank, there are dozens, or hundreds, of machines in use constantly. Something has to be discovered to run the machines still more rapidly, and to run several machines at once. It took too many people and too long hours to punch the keys by hand and make records of the results. In a big nation-wide industry, accounting has become more and more complex. Hand-operated machines with typists to tabulate the results cannot handle the volume of work. Machines are needed which take the data, do the necessary calculations, print the results, and store these results to go into other machines.
In scientific work the problem is as big. One of the best-known problems is the motion of the moon under the influence of the earth's gravitation, the problem which even Newton said "has given me a headache." This problem had to be solved if there was to be accurate navigation and prediction of tides. Using modern types of key-driven machinery, this problem was computed under the direction of Professor Ernest W. Brown of Yale. It was finished in 1923, after thirty years of work.
Cosmic-ray problems, ionosphere research, quantum theory, require the solution of equations so long and complex that entire sheets of paper are required to write a single one. And it is necessary to solve hundreds, or thousands, of such equations. In quantum theory, it is common for one man to spend most of his time for a year using high-speed electric key-driven machines to get the answer to one small problem. Many atomic problems require ten or fifteen significant figures. They have to be checked to avoid any error, since an extremely small difference between a computed and a measured value may be of immense meaning in scientific and industrial development.
There are problems now which scientists hesitate to try to solve, just because it would take a lifetime to get the answer. They are forced to use short cuts and do the work less accurately, holding back development in radio communication, television, thermionics, atomic structure (and possibly atomic power) as well as astronomy.
The key-driven machines run as fast as the fingers can be moved. But it is necessary to make a record of numbers and put them back in the machine by way of the keyboard to do another operation. The machine does not reset itself, or record the results obtained, or correct errors due to punching the wrong key.
Scientists need machines which will punch the keys, record the results, and use these tabulated answers to punch the keys again and still again, doing what mathematicians call iteration automatically. Or else they need machines in which mechanisms imitate complex mathematical operations. These two types of machines have been built within the last few years. The new analyzer at Massachusetts Tech combines principles of both with some of its own, the greatest advance in calculating machinery yet seen.
We can look over the two methods of development and see what machines will do. Then in conclusion, take a brief look at what the new machine will be like when it is finished a year from now, and what a scientist would like to see.
THE first machine
is called the punched-card type, since it was found more convenient to
use cards instead of long rolls of tape. A card has holes punched
in it to represent a single number, or groups of numbers, or even complex
functions or other data. These can be fed into the machine which
in turn punches other cards. The present form of this machine was
first developed by Herman Hollerith to take care of the problems of the
census of 1890.
A more recent use of this type of machine with which almost everybody is familiar is in the famous Federal Bureau of Investigation. A card is made up for every individual listed in the files. Along the sides of the card holes are punched in different positions for different types of information, as color of hair and eyes, shape of nose and fingerprint classification. Cards are placed in a machine adjusted to remove all cards with holes punched in certain spots. The cards go through the machine at the rate of several thousand an hour. At the end, there may be a dozen cards left, from which to pick the desired identification.
If, instead of sorting, the machine is equipped to take cards in which holes represent numbers, with mechanisms for adding, subtracting, multiplying or dividing the numbers, you have the modern type of accounting machine. There could be a set of cards representing, say, accounts with holes punched to indicate the proper numbers. The machine is set to indicate the interest rate, by which the account is to be multiplied. Another set of cards is punched with withdrawals, and still another set for deposits. Feeding the three sets of cards into the machine, the withdrawals are subtracted from the account, deposits added, and the final result multiplied by the interest rate. The answers are punched on another set of cards ready to be fed into the machine at some later date. At the same time the answers are printed on a tape, with code letters to indicate whose account it is.
It is easy to think of variations that can be performed with straight accounting machines, which do only simple arithmetical operations, such as A+B, A-B, AB+C, AB+C+D, A+B+C, A-B-C, A+B-C. These are all done on present-day commercial machines, with the possibility of punching the results on new cards and using these as new values of A, B, C, and D.
|Fig. 1. The astronomical calculating machines used by the Columbia University Department of Astronomy. This apparatus is assembled from various International Business Machine calculating and bookkeeping units, slightly modified for the work of astronomers. The numbered units are (1.) the multiplier, (2.) tabulator, (3.) summary punch, (4.) card sorter, (5.) high speed reproducer, and (6.) blackboard for notes and instructions for workers on the next shift. To this machine, the cards are memory, the sorter the ability to select a given fact from the mass of knowledge in memory, et cetera.|
An extension of these
machines has been built at Columbia University by Professor Wallace J.
Eckhart to solve the moon problem mentioned before--the problem that took
thirty years to calculate, using ordinary key-driven machines. It
took six machines, each as large as a piano. Holes representing the
date, or information obtained by observations, were pushed by hand
in some five thousand cards to start with.
Cards were fed into tabulating machines at a rate of seven to eight thousand an hour. Then, from sorting machines they went to adding, subtracting and multiplying machines which punched new cards which in turn went through the machines. In all, about 250,000 cards were punched by the machine. The problem was solved in two years' time, compared to the original thirty years. It was more accurate, and the machine printed the results.
|Fig. 2. Typical punched-card used in Columbia's astronomical mathematical machine. This card gives relevant data on a certain star, as follows: Star Number 120-588, an arbitrary designation of the star in the catalog. Since this is a star well below naked-eye visibility, it has never been named. Magnitude, 7.3, Right Ascension: 02 hrs, 43 mins, 12.2 secs. Precession, 3.0784 secs. (of time). Secular Variation, .085 secs. (of time). Declination, -00 deg. 24' 0.45".|
That is what punched-card
machines can do now, but they need expansion and extension to do problems
in algebra. In addition to the four actions of simple arithmetic,
the machine should be built to use positive and negative numbers.
It must be able to solve problems with brackets and parentheses, as (A+B)(C+[B-D])
which means doing a problem in parts, storing the results, and then making
use of the stored results. It should be able to raise numbers to
any power, or take any root. It should store and have available logarithms
to the base 10 and exponential functions to at least ten figures on both
sides of decimal. There should be available in the machine all trigonometric
and hyperbolic functions as well as the more complex elliptic functions,
probability integrals, Bessel functions, gamma functions and others which
are commonly used. That is, the machine should have stored in it
the equivalent of many volumes of tables.
It will be necessary to use the machine itself to compute some of these functions, since they are not tabulated accurately any place now to the extent that would be necessary. But think of what can be done with such a machine. Algebraic equations of almost any order can be solved. Any second order differential equation, integrals, complex formulas, can be evaluated and tabulated neatly. The thought of what can be done is enough to make any mathematician or theoretical physicist overjoyed.
There are types of problems, though, which would be awkward even for a super machine such as this. There are two machines now in use at Massachusetts Tech which will solve some of these other problems, as well as many of those that can be solved on the suggested punched-card machine. They are the network analyzer and the differential analyzer, both built on the second principle of use of machinery to simulate mathematical problems.
THE NETWORK analyzer,
the more complicated and more difficult to understand, is yet based on
more easily understood principles. Everybody who has worked with
electricity is familiar with the Wheatsone bridge, a setup for measuring
resistance. With an ordinary Wheatstone bridge, it is possible to
solve two equations for two unknowns, putting in the constants in the equations
as ohms of resistance. Imagine this multiplied many times, with bridges
connected and interconnected and capacitances and inductances included.
Much more complicated problems can be solved and the answers read directly
from electrical instruments. The machine occupies a fair-sized room,
with panels on all the walls and a space two or three feet deep back of
the panels filled up with wiring and connections. The machine is
usually used to solve electrical problems, because it is possible to repeat
in miniature all the connections and resistances and power inputs and so
forth that occur in problems of power lines. All that is necessary
is to plug into any part of the line and read the current, or resistance,
or whatever else is required.
The differential analyzer, though, is more general. Particularly, it is designed to solve differential equations, problems that involve rates of change. It does it by duplicating these rates of change in the moving parts.
The analyzer itself is a machine about the side of the body of an ore-car. Essentially it consists of a set of adjustable shafts and gears with integrators and input apparatus connected to a printing attachment. The rates at which the different shafts rotate represent different quantities in the problem to be solved. It is because of this that the machine is best suited for problems that involve rates of change.
There are connections between the different shafts, gear trains which will cause one shaft to turn at a rate which is exactly equal to the sum of the rates of two other shafts, or perhaps the product of two or three rates or rotation. There are even special gears which will multiply the rotation of a shaft by some special constant such as p or e.
Since backlash in any of the gear trains would cause error, a mechanism had to be invented which would apply negative backlash between any two connections, removing possibility of error in that form.
Integrators, which had been used before for very simple problems, had to be developed for high-speed operation. Each one consists of a flat glass rotating at fairly high speed, and a small metal disk resting with its edge on the plate. The small disk is turned simply by the frictional contact between the two, the rate of rotation changing as it is moved by another shaft back and forth across the diameter of the glass plate.
The little disk has to be light in weight, to prevent inertial lag in changing speed. Its bearings must be smooth enough to prevent slippage when the speed changes quickly, or is high. Yet this little wheel, barely an inch in diameter and mounted in jewel bearings, must drive a heavy half-inch steel shaft with several long gear trains and be able to reverse direction of rotation. This required the development of what is called a torque amplifier, a device which amplifies the torque or twisting power of this light little wheel ten thousand fold.*
*The machine itself is described in detail by Dean V. Bush, who directed
building of it, in the Journal of the Franklin Institute, Vol. 212 (1931), page 447.
There are, then,
three shafts connected to each integrator; one controlling the rotating
plate, one the position of the disk on the plate, and the third driven
by the little disk. The mathematical connection between these shafts
is u = ƒw dv. Roughly, this says that the rate of rotation of the
shaft w at any instant of time is multiplied by the amount the v-shaft
turn during that instant of time. Further, these products for every
instant of time since the machine was set, are added together. The
sum of all these is the rate of rotation of the u-shaft.
There are six of the integrators which may be connected and interconnected.
IN THE differential
analyzer, instead of giving numbers to the machine, a graph is used.
The data is very carefully plotted, preferably on a metal plate which will
not change with humidity changes (the room is close enough to constant
temperature to avoid heat-expansion errors). The graph is put on
one of the input tables, looking like a drafting board, and a cross hair,
with magnifier attached to an accurately machined screw, is placed over
the beginning of the curve. When the machine starts, the arm carrying
the cross hair starts moving at constant speed across the graph.
A hand-crank moves the cross hair vertically, through the action of another
screw-and-worm gear. This screw controls the rate of rotation of
one of the main shafts of the machine. It is the job of an operator
to keep the cross hairs accurately on the curve. As there are five
input tables, it is possible to feed in five different sets of data at
the same time, integrating, multiplying, adding all of these five sets
through the six integrators in a bewildering complexity of solutions.
To make the operations easier, each input table is arranged with connections for starting and stopping the machine, and with gear shift levers for three speeds forward and one reverse, so that if one operator loses his place on the curve, or can't turn the crank fast enough, he can always shift. There is also an automatic speed control on the machine itself, and a control board panel with a fascinating array of lights to show which tables and shafts are in operation, which directions they are going, and their speed, with warning lights flashing as speeds approach the maximum allowed.
At one end of the machine is a set of dials like those on a speedometer connected to various of the shafts to register their rotation, recording the various parts of the solution of the problem. These dials can be connected to any desired shaft. In addition, they can be adjusted to print the results of all dials simultaneously at regular intervals, or at any time an operator desires. Still more, the machine can plot curves of two of the results, on the same paper.
In full operation, with five input operators, one person on the printing mechanism, a general captain of the team, and a mechanic standing by, the full complexity of the machine and its great power become apparent.
Just one example of what the machine can do may be cited from problems in atomic theory. To work out the numerical solution of a wave equation for a moderately simple atom, with the equations already written down, may take as much as six weeks steady work with a key-driven, electrically operated calculating machine. On the differential analyzer, once the machine is set and ready to operate, the same results may be turned out in an hour or less with three people working on it.
Still, there are objections to this machine. It is accurate, at best, to one part in ten thousand. This is sufficient for a great many of the problems given it, since the preliminary data has been computed to greater accuracy and the final results of the problem require less. But one more figure would be better. Again, there is a strict limit to the accuracy possible in plotting a graph of data. Also, it sometimes takes several day to adjust the machine and get all the gears and shafts arranged in proper order to satisfy the factors involved in the problem. Then, despite the tremendous saving in time, it is still too slow. Problems have been put on it that, even with its tremendous calculating power, required months to finish. Then these results require more months of study before they are recombined and fed into the machine again. There are so many problems needing solution that it is necessary to wait months to use the machine.
WITH THESE difficulties
in mind, and the knowledge gained from building the machine, work was started
some two or three years ago on a new differential analyzer--one which will
be as far ahead of the present one as that was ahead of anything else.
In another year the machine should be completed.
The new machine will be a sight worth seeing. As part of the equipment, a complete automatic dial exchange from the telephone company was brought in. The whole machine will weigh tons, and fill a large room. There will be hundreds of vacuum tubes, new and baffling systems of power communication. Instead of a mere six integrators, there will be nearly twenty, with space for several more to be hooked in. They will be really high speed, running 5000 rpm. or more. Mechanical transmission of the torque is impossible now without slippage. The little disk is extremely light weight, and runs balanced on fine jeweled bearings. The torque will be transmitted by a new tele-torque amplified. A specially designed segment on the axle of the disk cuts an electro-static field, transmitting by complex circuits and vacuum tube amplification the rotation of the small shaft to a large shaft with regular gear train. Electro-magnetic fields had to be avoided because of hysteresis and heating effects.
In the new machine, instead of plotting curves, the actual numerical data can be used, adding an advantage of the punched-card machines. In this case the data will be punched on a continuous tape like a player-piano roll. Since this will be the same type of data that would be used to plot a curve, there will be blank spots between separate entries. A curve could be drawn smoothly between these points. The new machine, though, will automatically compute the intermediate points, making use of fifth differences, providing an accuracy greater than could be obtained manually by drawing the curves.
In addition, instead of having to set the integrators and other parts to fit the initial conditions of the problem, all that can be put on a piece of tape, one section of tape for each integrator. The tape will feed through the machine, pausing for about twenty seconds to allow the machinery to start moving to set each integrator. After the tape has fed through, the machine automatically starts the tape through again, just to check that everything is set correctly. If one of the integrators is wrong, the machine checks that again, and if it still is wrong, a bright light flashes on at that one integrator. Dean Bush, in describing the machine, once said that there had been discussion of having the machine call out "Yoo-hoo!" if something went wrong, but the light was thought sufficient.
Despite all this complexity, despite everything the machine can do in solving complex problems, still it is not a thinking machine. Operators are necessary to decide the way in which the problem goes in, and to decide for the machine what it will do. An atomic wave equation may be put into the machine, and the machine set to print solutions for certain numerical values of the constants. But the machine will not pay any attention to whether the results become physically absurd for an atom because of wrong constants. The machine does not think. It merely duplicates mathematical formulas with mechanical precision. Or if it is of the punched-card type, it picks out and uses only those cards which it has been set to use. Even if the displacement of one card should be come advisable as a consequence of results already obtained, the machine goes blindly, unthinkingly, ahead.
THE NEW differential
analyzer is in process of construction. Men are working and planning
for a new and greater punched-card machine. When these are in operation,
tremendous advance is possible; new knowledge in atomic structure, new
theories on the origin of cosmic rays, improvement in the methods of radio
and television transmission, better power transmission and new design of
transformers and vacuum tubes. Even, perhaps, new designs for new
types of computing machines based on different principles will come.
Scientists look ahead and dream of new calculators. They will not
be as compact and as effective as a human brain, or even portable.
The dream machine may fill an entire building. It will be operated from a central control room made up entirely of switchboard panels, operated by trained mathematicians, and an automatic printer giving back the results.
A physicist has spent months getting a problem ready for solution. He has long tables of numbers, experimental data. These numbers have to be combined and recombined with still more numbers, producing hundreds of thousands or millions of numbers--far ahead of the simple moon problem.
His tables are typed by a stenographer on a special machine which punches them on a tape. The constants for his equations are punched on another tape. The tapes and the directions giving the order of the use of the numbers are turned over to the operator. The tapes are fed into a slot, a switch pulled, and in a few minutes every number has been multiplied by all the other necessary numbers. Perhaps they must all be multiplied by a series of Bessel functions. A switch is turned to feed in Bessel functions, and a key punched to allow only one class of these to operate.
A dream machine! Not a thinking machine, but a powerful tool for the scientific brain; to do work that can never be done otherwise without wasting lifetimes on drudgery. A dream, but possible right now with what we know of mechanical principles.
Return to The Geek History of the Universe
Return to Trogholm