Calculator

“ In the case of the output arrangements we have the full power of the calculator behind us, i.e. we can do the conversion of the information into the required form as an ITO. In the case of the input organ we must go more warily. If we are putting the instruction tables into delay lines, then when the power has been turned off all memory will have been effaced, including the instruction tables. We cannot use instruction tables to get the information back, because the instruction tables are not there. ”

“ To understand the use of this abacus, the calculator must be supposed sitting before one of its long sides, and putting counters into the spaces between the marked lines. Each space represents an order of numerals, the space on the right hand being intended for units, the next space for tens, the next for hundreds, and so on. The numbers belonging to the first four of each series are put on that side of the bisecting line which is nearest the calculator; those over 5 are put beyond it. ”

“ I have an immense respect for a man of talents plus “the mathematics.” But the calculating power alone should seem to be the least human of qualities, and to have the smallest amount of reason in it; since a machine can be made to do the work of three or four calculators, and better than any one of them. Sometimes I have been troubled that I had not a deeper intuitive apprehension of the relations of numbers. But the triumph of the ciphering hand-organ has consoled me. I always fancy I can hear the wheels clicking in a calculator’s brain. ”

“ That the human operator, doing his arithmetic without mechanical aid, would not take more than a hundred thousand times the time available on the calculator, this figure representing the ratio of the speeds of calculation by the two methods. ”

“ An adding machine may perform one of the forms of calculation, but to [Pg 44] call it a calculating machine when it has no capacity for division, subtraction or multiplication, is an error; and yet we find the U. S. Patent Office records stuffed full of patents granted on machines thus erroneously named. The term calculating is the broad term covering all forms of calculation, and machines performing less should be designated according to their specific capacities.
It is true that adding is calculating, and under these circumstances, why then may not an adding machine be called a calculator? ”

“ Electronic computing is easily 100 times as fast as relay computing; nearly every future calculator will do its computing electronically. Many other improvements will be made. For example, in this calculator, there are 72 addition-subtraction mechanisms, yet only one of these can be used at a time. Also, the machine has only one combined multiply-divide unit. ”

“ In Dodson's Calculator we find a table of multiplication extending as far as 10 times 1000. [2] In 1775, a still more extensive table was published to 10 times 10,000. The Board of Longitude subsequently employed the late Dr Hutton to calculate and print various numerical tables, and among others, a multiplication table extending as far as 100 times 1000; tables of the squares of numbers, as far as 25,400; tables of cubes, and of the first ten powers of numbers, as far as 100. ”

“ Again, men of this class, and even those in charge of large departments, took the mere suggestion that they had use for a calculator or recording-adder as an insult to their efficiency, and would almost throw the salesman out. Others would very politely look the machine over and tell the salesman what a wonderful machine it was, but when asked to give the machine a trial, they would immediately back up and say that they had absolutely no use for such a machine ”

“ A copy of Vlacq’s Arithmetica logarithmica (1628 or 1631) , with the errors in numbers, logarithms, and differences corrected, is still the best table for a calculator who has to perform work requiring ten-figure logarithms of numbers, but the book is not easy to procure, and Vega’s Thesaurus has the advantage of having log sines, &c., in the same volume. ”

“ In Fig. 2 of the illustration of Pascal’s machine, the machine has been reversed, and the bottom of the casing, which is hinged, thrown back, showing the numeral wheels and gearing of the different orders and the transfer levers for the carry of the tens.
Increased capacity of modern calculator
The Art of the modern machines is far removed from the older Art by its greatly increased capacity for rapid calculation which is found emanating from the provision of keys as the means of manipulation. ”

“ I showed him my machine. At once he saw it possibilities. We joined hands and set about designing the machine to make it as nearly perfect as possible in its adaptation to the needs of modern business.
The result of that work and our later association is the Monroe High Speed Adding-Calculator which is filling no small part in faithful, economic service in the realm of business. ”

“ History shows that Europe and other foreign countries have been using calculating machines for centuries. Like that of Pascal’s, they were all multiple-order machines, and, although not key-driven, they were capable of adding a number of columns or items of six to eight places at once without the extra manipulation described as necessary with single-order digit adding machines. A number of such machines were made in the United States prior to the first practical multiple-order key-driven calculator. ”

“ Baldwin, who writes and recites Pi to 128 places’ Professor Newcomb was very much interested in the calculator and complimented me highly.
All this time I was holding demonstrations in the evening at the hotel or in private houses with the machine, as most people had never heard of such a thing as ‘metal brains,’ and some, thinking I had manipulated certain figures, would insist on giving me problems of their own, often in their haste making a mistake and having to retire in confusion. ”

“ The mathematicians have many methods, totally different from each other, of arriving at one and the same result, their celebrated approximation to the circumference of the circle. An intrepid calculator has, in our own time, carried his approximation to what they call 607 decimal places: this has been done by Mr. Shanks, [204] of Houghton-le-Spring, and Dr. Rutherford [205] has verified 441 of these places. But though 607 looks large, the general public will form but a hazy notion of the extent of accuracy acquired. ”

“ Request some person to place in one of his hands a bronze coin and in the other a silver one, and to let no one know which hand contains either particular coin. This may be ascertained by the following calculation:—The calculator should assign an even number, say 4, to the bronze coin, and an odd number, say 7, to the silver coin. The person holding the coins should be requested to multiply the number assigned to the coin held in his right hand by an even number, and that assigned to the coin held in the left hand by an odd number. ”

“ Calculators also save space and weight, a valuable consideration on many craft. One small calculator can replace several heavy and expensive volumes of tables, and is inexpensive enough that there is little reason not to carry a spare for backup use should the primary one fail. The pre-programmed calculators are at least as robust in construction, probably more so, than the sextant itself, and properly cared for, will last a lifetime with no maintenance except new batteries from time to time. ”

“ The law of errors admitted by all calculators is Gauss's law, which is represented by a certain transcendental curve known under the name of 'the bell.'
But first it is proper to recall the classic distinction between systematic and accidental errors. If we measure a length with too long a meter, we shall always find too small a number, and it will be of no use to measure several times; this is a systematic error. If we measure with an accurate meter, we may, however, make a mistake ”

“ See "The Black Star Passes", Ace Books, F-346.
I
Three men sat around a table which was littered with graphs, sketches of mathematical functions, and books of tensor formulae. Beside the table stood a Munson-Bradley integraph calculator which one of the men was using to check some of the equations he had already derived. The results they were getting seemed to indicate something well above and beyond what they had expected. ”

“ Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors, but was unable to explain the method by which he made his marvellous mental computation. ”

“ For there is no limit to their number: and though the methods of which we have spoken do theoretically enable us to calculate as many terms as we please, the labor and the complexity of the operations are so serious that common calculators are stopped by them. None but very great mathematicians have been able to walk safely any considerable distance into this avenue,—so rapidly does it darken as we proceed. ”

“ UNBURY (General Description) . The minor cycle whose position is given in TS 31 is taken to be position of the next instruction.
MULTIP (General Description) . The number in TS 18, 19 is multiplied by the number in TS 20, 21: the result is brought to standard form by shift of decimal point. An error is obtained for the product by using the errors in the given numbers and allowing for rounding off. The result is stored in TS 22, 23.
ADD is analogous to MULTIP.
As an example of a more complicated process, I have chosen the calculation of the value of a polynomial. ”

“ If we like to consider LC as the analogue of a computer then CA must be considered a desk calculating machine. It carries out the four fundamental arithmetical processes (with possible exception of division, see p. 27) , and various others of the nature of copying, substituting, and the like. To a large extent these processes can be reduced to one another by various roundabout means; judgment is therefore required in choosing an appropriate set of fundamental processes. ”

“ Nature loves to present to us these queer paradoxes. It is well known that those wonderful "lightning calculators," who now and again surprise the world by their feats, lose all their mysterious powers directly they are taught the elementary rules of arithmetic. ”

“ The notation of the decimal form is such that the content of DL 10 can be passed out onto a card in the usual way and if the card is then listed the digits of the numbers α,m will then appear on the listing paper in the usual way. Or in other words only the first 10 minor cycles of DL 10 are used, and a decimal digit is represented by the minor cycle in which a pulse occurs, and its significance by the position of it within the minor cycle. ”

“ This latter case is convenient for the purpose of multiplying the partial sum by a small integer without a complicated series of previous transfers; if the multiplicand is taken from the partial sums register then SELF is stimulated. The multiplier may also be taken from a variety of sources. It may be taken from TS 2 or from CI or from TS 11, and we accordingly stimulate NOR, GIV or ERR. The multiplier consists of four consecutive digits from whichever source is chosen. The choice of the digits is made by means of a choice of one of the pulses P1 to P32 to enter on a certain line (DIGIT) . ”

“ The rule is to divide the principal by 4, and to multiply the quotient by one-third of the product of the rate by the time in months. Six times 5 1/2 = 33, and one-third of 33 is 11. If the time be expressed in years, multiply one-fifth of the principal by one-half the product of the rate by the number of years, and remove the decimal point one place to the left. ”

“ Good professional calculators could work out on paper and in about the same time all that Maudi did mentally; in fact, he simply used his extraordinary memory for figures where the common calculators used pen and paper. ”

“ Normally it is intended that after an operation of type A the next instruction to be followed will be that with the next number, and it might be supposed therefore that the formula CD = CD’ + P17 would apply. ”

“ Rather it means that normally the amount of mathematical argument required to get a reliable prophecy of the behaviour of a circuit is out of proportion to the small trouble required to try it out, at any rate if one is in an electrical laboratory. In practice one compromises with a rough mathematical argument and then follows up with experiment. ”

“ The popular data have also another important function, which only appears when we consider that the same instructions will be used on quite different jobs. If we were just to number the instructions serially throughout all the instructions ever used on any job, then, in the set of instructions actually used in any particular job there would be large gaps in the numbering. ”

“ Those which are rather down at heel may then be removed, and when the conditions are restored to normal there should be a good margin of safety. We cannot of course rely on this 100%. We need a second string. This will be provided by a variety of checks of the types normally employed in computing, i.e. wherever we can find a simple identity which should be satisfied by the results of our calculations we shall verify it. ”

“ Instruction tables will have to be made up by mathematicians with computing experience and perhaps a certain puzzle-solving ability. ”

“ Take the number in 741; enter it as multiplier; read the next coding line The coding is:
741, ——, 7
We do not punch anything in the middle field: the machine is “educated” and “knows” that it has started a multiplication and needs a [Pg 102] multiplier, and it expects this multiplier in the third coding line. To have no code for the multiplier counter is, of course, a departure from the general rule, but it saves some punching and permits the use of this space for certain other codes, thus saving some operating time. ”