THE ART OF MENTAL CALCULATION; WITH DEMONSTRATIONS
By Professor A. C. AITKEN,
M.A., D.Sc., LL.D., F.R.S., F.R.S.E., Hon.F.S.E.
The President extended a hearty welcome to the guests who were present and expressed (he hope that they would have an enjoyable evening.
Professor Aitken, he said, needed little introduction. He was born and educated in New Zealand, but after war-time service with the New Zealand Forces in the 1914-18 War, where he was seriously wounded, he returned to New Zealand and eventually went to Edinburgh in 1923 for post-graduate study in mathematics. In 1925 he was appointed to a lectureship in Statistics and Mathematical Economics in Edinburgh University.
He had written textbooks on algebra and statistical mathematics, was joint author of a textbook on higher algebra, and likewise the author of some seventy memoirs and papers on mathematical subjects.
Notices of the meeting indicated a few of the honours which had been bestowed upon the lecturer, and when he himself visited Edinburgh in May of this year to attend the centenary celebrations of the Society, it was his privilege to hand to Professor Aitken the Diploma of Honorary Fellowship of the Society, which was the greatest honour the Society could bestow.
PROFESSOR AITKEN
I wish first of all to express my sense of the honour conferred on me by the Society of Engineers in electing me an Honorary Fellow in its centenary year, and in inviting me to give a lecture to its members on the present occasion. I was, I must confess, a little doubtful concerning my qualification to do this. It is true that our sub-department of Technical Mathematics in the University of Edinburgh has for half a century taken its part in preparing future graduates in Engineering, who are now to be found in all parts of the world. Yet I am myself not a professional engineer, and my published work, though quite a proportion of it concerns numerical analysis and was conceived with the modern calculating machine in view, is, I fear, rather abstrusely algebraic and does not lend itself to general discourse. I was assured, however, by some Fellows of the Society that if I shook off a certain reluctance and spoke more freely than I have hitherto done on one of my personal proclivities, namely a bent, very unfashionable in the present era, for extensive mental calculation, I should meet a not unfriendly audience. Therefore I decided to do this, strengthened by the reflection that there was an excellent precedent; for in February 1856, nearly 100 years ago, George Parker Bidder, a notable calculator to whom I shall refer later, in his capacity of President of the Institution of Civil Engineers (and doubtless on request), gave an address on Mental Calculation, describing and exemplifying his own methods. (This is to be found in the Proceedings of the Institution of Civil Engineers, vol. xv, 1855-56; and vol. Ivii, 1878-79.) Bidder's address is in fact the locus classicus on the topic, the most detailed and straightforward account in existence, and by a professional man of standing.
I may remark that I think demonstrations of rapid mental calculation should be reserved for educative and instructional occasions. I deplore all tendency to surround the faculty with an aura of the marvellous, as I likewise deplore certain traces of something like charlatanism that I fancy I perceive here and there in the historical records. Before the present audience I shall feel no constraint, since I am sure they will perceive my didactic intention.
First, some brief historical remarks. At rare intervals so-called "lightning calculators" are observed, quite often persons possessing the minimum of general intelligence, almost idiots, but obsessed by menial arithmetic and capable of astonishing feats. You will find these prodigies described and compared in such works as W. W. Rouse Ball's Mathematical Recreations and Essays, in the Napier Tercentenary Celebration Handbook (article by W. G. Smith, M.A., Ph.D., pp. 60-68), or in Mr. Fred Barlow's Mental Prodigies (Hutchinson, 1951); while two in particular, the famous Jacques Inaudi, who died only four years ago, and an Ionian Greek, Pericles Biamandi, were subjected in 1895 to a series of tests by the famous French psychologist Binet. You will lied reference to Jedediah Buxton, a Derbyshire ploughman of the 18th century; to Zerah Colburn, an American youth of the early 19th; to Zacharias Base, of a hundred years ago, whose powers were employed for making tables of factors, prime numbers and logarithms; Henri Mondeux - but I need merely refer you to Rouse Ball or to Mr. Barlow's book. One with much more education and intelligence than most of those was George Parker Bidder (1805-1878), "the elder Bidder," whom I have mentioned, who interests us all here as having been a notable professional engineer, and who especially interests me in that he attended the University of Edinburgh and gained the prize in mathematics there in 1822. His son, bearing the same name, a barrister and Queen's Counsel, was similarly gifted, though not so strongly. Both admit that they practised incessantly. I merely remark that they were fortunate in having leisure to do so.
Of noted mathematicians, not many have been rapid mental calculators: John Wallis, Euler, Gauss; possibly the Indian, Srinivasa Ramanujan, though perhaps he hardly came into the really rapid class. But Gauss was certainly a prodigy, showing it, moreover, as a very young child.
For the most part, then, the faculty is extraordinarily infrequent, the instances are few, and biographically of hardly any general interest whatever. But with perhaps only one exception (and it is a doubtful one), all these calculators, whether undistinguished in other respects, as the very great majority are, or outstanding, like the illustrious Gauss, were child prodigies in this special talent. May I remark at this point that I never was; not until I was at secondary school and approaching fourteen years of age did I look into arithmetic from this point of view, and it was then not arithmetic at all, but algebra, that first stimulated my interest in the processes of calculation. Incidentally, I regard mental algebra as on a much higher plane than mental arithmetic, and incomparably more rewarding.
There is perhaps time for brief digression, for a very remote cast into the past. Recent work in deciphering cuneiform inscriptions seems to prove that the Babylonians of 3 600 years ago were arithmeticians of very high proficiency. There is one inscription which gives a list of triads of integers which can be the three sides of a right-angled triangle. The Babylonians used simultaneously our scale of ten and a scale of sixty, the sexagesimal. Their multiplication table ran to 59 times 59, expressed in sexagesimals, and there has actually been found a table of reciprocals, giving to three places the sexagesimal equivalent of fractions from 1/40 to 1/80. I incline to think (though of course I have no proof) that among these Babylonians there were some who knew by heart their multiplication table.
But now to our subject. What kind of sums are the stock-in-trade of the rapid calculator? I would say, for the most part very ordinary ones, which personally I find rather dull, involving one or two multiplications and perhaps a division or two; with large numbers, of course. For example a problem like the following, which was proposed to the elder Bidder:
"The Equator has 360 degrees of longitude, each degree measuring there 69½ miles; how many coins each of diameter 13/8 inches would span the Equator?"
(Answer: 46 080 by 25 020=1 152 921 600, by noting that mental multiplication by 25 is division by 4, and by 20 is mere doubling; having observed first that 360 by 69½ is 25 020. This was not how Bidder did it.)
Or again - this was asked of another calculator - "What is the cube root of 327 082 769?"
(Answer: 689, since by inspection it must be a little less than 700, for 7 cubed is 343; also it must end in 9, and cannot be 699.) [ Such cube-rooting (and there is a good deal of it in the records) is almost trivial, since the important digits yield by inspection the first I digit, and mere inspection of the last digit of the proposed number gives; the last digit of the answer (for example 7 would give 3, while 3 would i give 7); the rest is commonsense, having no reference at all to the other digits. The real test of ability to do square, cube or any other root is, in my view, to have a number proposed that is not an exact power, and to I be asked to give the answer to several decimals; but this type of question you will hardly find in the published records.
As I have said, I began myself, as a boy of thirteen or more, with algebra. With advancing years I will not now try what sanguine youth then prompted me to try, for instance to square numbers of eight, nine or ten digits. I will be satisfied to leave that to the machine which, whatever anyone may say, is now faster than any mental prodigy. But I will ask for a few numbers of three or four digits to be given to me (Dr. Taylor has very kindly gone to the trouble of preparing questions for me), and I will square them.
[Dr. Taylor here asked for the squares of the three-digit numbers 251, 299, 413, 568, 596, 777 and 983, each of which was correctly given almost instantaneously, 568 and 777 taking a little longer. Dr. Taylor then proposed the four-digit numbers 3 189 and 6 371; in each case the square was given in about five seconds, the lecturer making a momentary error and correcting it in the first case.]
All that I use, perhaps more than once in the course of a calculation, is the algebraic identity a⊃ = (a+b) (a-b) + b², where a + b or a - b is conveniently chosen, one or other of these numbers ending in one or more zeros, and b being relatively small. For example, when Dr. Taylor asked me to square 777, I took b as 23, mentally multiplied 754 by 800, and added the square of 23, namely 529. The same with the rest; some wore especially easy.
You can see that, with some practice, such as nowadays I have not the leisure to afford, it would be possible to square mentally numbers of five or six digits in a very short time; but why do so, when an electrical machine is at one's elbow?
Square Root. The central idea here is Newton's; but, remarkable to state, the Babylonians have it in their cuneiform inscriptions, and so did the wonderful Archimedes, the Syracusan Greek. I will consider the square root of 51. As a first approximation for the square root, 7 will serve. Note then that 51/7 is another. The mean of the two, that is, of divisor and quotient, is 50/7, and this, 7.1428..., is a much better approximation.
But we might have begun with 50/7, and then, dividing 51 by it, we have 7.14. The mean of divisor and quotient is now 7.1414285... , and in fact Barlow's Tables give 7.1414284 to 8 significant digits.
But the resources of mental division are not exhausted even here. An expert would know very well that 7.14141414... is 707/99, and dividing 51 by this we have 5049/707, easily accomplished by dividing by 101 first, yielding 49.990099009900... and then by 7, so that we have 7.141442715700... , and the mean of this and 7.14141414141414... is 7.14142842857 to 12 digits, whereas the true value of the square root of 51 is 7.14142842854 to that degree of accuracy.
I am making the point here that this can be done mentally by an expert calculator. The great thing is to seize on a good and useful first approximation; and here experience, resource and opportunism are required; also a certain steadfastness of purpose, for in mid-calculation it may flash upon one that a better approximation had been available, but one must resolutely ignore that and keep on riding the inferior horse.
I will now ask for some numbers to be proposed for extraction of square root.
[Dr. Taylor here proposed several of the previous numbers, namely 251, 299, 413, 596, 777. In each case the square root was given in two or three seconds to five significant digits, with the remark that for 299 and 596 the last digit might be in excess, as it was. Dr. Taylor then proposed the four-digit numbers 3189 and 8765. In each case the result was quickly given to five digits.]
To revert to 51. How near is 50/7 to the square root? It was got from a divisor 7 and quotient 51/7. These are in ratio 49 : 51. The halfway mark between them is 50; I will say therefore that both 7 and 51/7 "deviate" by 1 in 50. The square of this is 1 in 2500. I double and say, 1 in 5000. Now 50/7 reduced by 1 in 5000 is
4999/700 = 7.1414285
remarkably near the true value 7.1414284..., and in fact identical with the second approximation given above. It is clear by this time that we have several methods to choose from. There is yet another. Looking again at the first pair, 7 and 51/7, we note once more their ratio 49 : 51. Quarter the distance between 49 and 51, and take the first and third quarter, namely 49½, 50½, their ratio being 99 : 101. I say then that
7 by 102/99 = 7.14141414...
is a good approximation to the square root of 51. Alternatively that
51/7 by 99/101 = 7.141442715700...
is equally good; but we have met both of these before, and have seen that their average is spectacularly good. There are even subtler and more powerful approximations still. There is one known to me, quite simple, which I may illustrate by saying that in our example here (rather hard worn by now) we could correct 50/7 by reducing it not by 1 in 5000, but by 1 in 4999½. The result is
7.141428428557..., as against 7.141428428543
and so committing an error of 1 in 500 000 000 000. This is an extreme approximation for square root; and I have never gone beyond it in mental calculation.
Cube Root. I will take for illustration the cube root of 128. You can see that it must be near 5, since the cube of 5 is 125. Trisect the interval from 125 to 128. The "middle third" yields the ratio 126:127. I assert that
5 by 127/126 = 5.0396825...
is very close to the required cube root, which is in fact 5.0396842 eight significant digits. This method of "thirding" is here ever so slightly in defect; it is the business of the algebraist to ascertain the formula for the small error committed. I will not go into these delicate refinements. Here of course a proposed number may prove rather intractable, being remote from any suitable cube of an integer, or of a fraction with small denominator. I may mention also a similar method which I call "sixthing." Here for example we have that 128 is 5 by 5 by 5.12. I divide that former interval, 125 to 128, into six parts, and taking the first and the fifth of these I form the ratio 251:255. We have then
5.12 by 251/255 = 5.0396863,
an approximation almost as good as the other, this time slightly in excess. I am not aware that this approximation has ever received notice.
As for higher roots, fourth, fifth and so on, they can be done by mental calculation using appropriate devices (such as "fifthing" for the fifth root); but they become progressively more difficult.
Decimalization of Fractions. Now I pass to something which I find of great use, as must already have been observed, but which I hardly find at all in the literature of mental calculation; namely the expression of the reciprocals of numbers, or of vulgar fractions generally, as decimals, in particular as recurring decimals.
To begin with a very elementary example, it is not generally known that one can divide by a number like 59, or 79, or 109, or 599, and so on, by short division. Take for example 1/59, which is nearly 1/60. Set out division thus:
6 / 1.016949152...
0.0169491525...
Here we have the decimal for 1/59, obtained by dividing 1 by 60; as we obtain each digit we merely enter it in the dividend, one place later, and continue with the division.
As another example consider 5/23. Write it as 15/69. Then proceed:
7 / 15.21739130...
0.217391304...
In fact 5/23 = 0.2173913043478260869565, a recurring decimal with a period of 22 digits. One could equally well have written it as 65/299, then carrying out division by 3, two digits at a time, and entering in the dividend two places further along.
[Dr. Taylor here proposed 71/83 and 31/67. In each case the first five digits of the decimal were given at once, but the lecturer remarked on a curious and unusual inhibition, preventing him from running on freely with many more digits. On the other hand, 47/91 was given at once as 0.516483, the six digits being said to recur.]
There are other possibilities. For example the mental calculator is, or should be, very familiar with the factorization of numbers; he should know not merely that 23 times 13 is 299, but that 23 times 87 is 2001. For example 5/13 is equal to 435/2001; and if we note that 435 is the same as 434.999999999..., we have another method, in which, as we obtain the digits, we subtract them from the dividend, so many places later. Thus in the present case:
2 / 434 782 608 695 652 ...
217 391 304 347 ...
For example 217 from 999 gives 782, which we then divide by 2, obtaining 391; this, subtracted from 999, gives 608; and so on.
My aim has been to demonstrate, in these various rather simple examples, some part of the repertoire, the armoury of resource upon which (as I hold) a mental calculator may draw, and in regard to the choice of which he must make instantaneous decisions, and keep to them.
[Here the remark was made that memory and calculation were sometimes almost indistinguishable to the calculator. This was illustrated by the recitation of the 96 digits of the recurring period of the decimal for 1/97, checked by Dr. Taylor. Probably because 97 was the largest prime number less than 100, this particular example had been frequently proposed.]
Primes and Factorization. The memory of significant results and serviceable devices plays a predominant part in the equipment of the mental calculator; almost exclusively so in the question of recognizing prime numbers, or of factorizing composite numbers. In my brief introductory remarks I mentioned that Zacharias Dase compiled factor tables. He would doubtless have been told by Gauss such important facts as Fermat's Theorem, namely that every prime of the form 4m+1 is the sum of two squares in one way only. If we find that a number is the sum of two squares in two different ways, then it is not a prime, and there is a simple algebraic technique for its factorization. Let us ask: is 977, the half of 1954, the number of the present year, a prime? One sees it to be 961 plus 16, the sum of the squares of 31 and 4; and a very rapid trial, since we have only to try, for one of the terms, squares above 22 squared, assures us that there is no other such resolution. Thus 977 is a prime.
I will now try to recognize whether proposed numbers are primes, or to factorize them if they should prove to be composite.
[Here Dr. Taylor proposed the numbers 1327, 871, 989, 401, 1193, 1157, 1447, 901 and 1369. The primes 1327, 401, 1193 and 1447 were instantly recognized, except for a moment of hesitation about 1447. The factorizations of the rest, namely 13 by 67, 23 by 43, 13 by 89, 17 by 53, and 37 squared, were given at once.]
Memory. Last of all, before I pass to psychological considerations, before I ask whether the visual, or the auditory, or the rhythmic, plays the principal part in mental calculation, I shall say something about memorizing. My memory is not as good as it was some twenty or twenty-five years ago. At that time I found it easy to remember not only numbers and formulae and mathematical proofs, but music and poetry and indeed most things. The one requisite was that a live interest in the subject should fix an uncdeviating attention. Mnemonics I have never used, and deeply distrust. They merely perturb with alien and irrelevant association a faculty that should be pure and limpid. Our present civilization, not only urban but rural, full of noise and interruption as it is, offers every hindrance to that relaxed meditation upon which the strength of memory thrives best. I grew up in a remote part of the Empire, before the days of radio, when even the telephone, that modern necessity but chief among the interrupters of thought, was a rarity.
This is incidental. Memory in my own case is visual if I desire, though in the main auditory, but resting on a rhythmic foundation. I was interested to find Mr. Fred Barlow, in his book (p. 149), referring to the "appalling waste of time and energy" committed by Dagbert, a French calculator. Dagbert claimed to have memorized it to the 707 places of decimals to which it had been calculated by W. Shanks in 1873. It amused me to think that I had done this myself some years before Dagbert, and had found it no trouble whatever. All that had been necessary was to range the digits in rows of fifty each, each fifty being divided into ten groups of five, and to read these off in a particular rhythm. It would have been a reprehensibly useless feat, had it not been so easy.
[This was illustrated first by reciting p to 250 digits, the five-rhythm being in evidence. Dr. Taylor then proposed that the run of decimals from the 301st place, beginning with 72458, should be given. After 150 had been given, he then asked that the same should be done at the 551st place, beginning with 80943. This was also done, being checked by Dr. Taylor from the value computed to 1000 decimals, mentioned in the next paragraph.].
Some twenty years later I learned that this feat had indeed been a waste of time, in that Shanks, noted and responsible calculator though he was, had gone wrong at the 528th place, and that the last 180 digits of his value were therefore erroneous. Quite recently the famous electronic machine ENIAC has shown up in the strongest light the puerility of the efforts of the human calculator, by evaluating and checking p to 2035 decimals in 70 hours. I amused myself again by learning the correct value as far as 1000 places, and once again found it no trouble, except that I needed to "fix" the join where Shank's error had occurred. The secret, to my mind, is relaxation, the complete antithesis of concentration as usually understood.
Interest is necessary. A random sequence of numbers, of no arithmetical or mathematical significance, would repel me. Were it necessary to memorize them, one might do so, but against the grain.
Finally, the psychological analysis. The fastest mental calculators seem to have been of auditory, not of visual, type. Mr. Wim Klein, whom I recently met and who is notably fast in performing multiplications, finds it necessary to speak rapidly in Dutch while he is calculating. The visualizers, almost without exception, are slower; the Greek, Pericles Diamandi, was found by Binet to take about six times as long in his calculations as Inaudi, who was of auditory-rhythmic type. (This is one of the most interesting comparisons in the literature.) I myself can visualize if I wish, and at intervals in a calculation, and also at the end when all is done, the numbers come into focus; but mostly it is as if they were hidden under some medium, though being moved about with decisive exactness in regard to order and ranging; I am aware in particular that redundant zeros, at the beginning or at the end of numbers, never occur intermediately. But I think that it is neither seeing nor hearing; it is a compound faculty of which I have nowhere seen an adequate description; though for that matter neither musical memorization nor musical composition in the mental sense have been adequately described either. I have noticed also at times that the mind has anticipated the will; I have had an answer before I even wished to do the calculation; I have checked it, and am always surprised that it is correct. This, I suppose (but the terminology may not be right), is the subconscious in action; I think it can be in action at several levels; and I believe that each of these levels has its own velocity, different from that of our ordinary waking time, in which our processes of thought are rather tardy. But here I am conscious of wading into deep waters; I am in danger of vagueness and imprecision. Therefore I cut short such introspective analysis.
The machine, whether desk, hand or electric, or electronic, is bound to have a deleterious effect on mental calculation. When I came from New Zealand thirty years ago and first used an arithmometer, even of the antiquated types then available, I saw at once how useless it was, how gratuitously useless, to carry out for myself any mental multiplication of large numbers. Almost automatically I cut down my faculty in that direction, though I still kept up squaring and reciprocating and square-rooting, which have a more algebraic basis and a statistical use. But I am convinced that my ability deteriorated after that first encounter.
Mental calculators, then, may, like the Tasmanian or the Moriori, be-doomed to extinction. Therefore - if you have borne with me so far - you may be able to feel an almost anthropological interest in surveying a curious specimen, and some of my auditors here may be able to say in the year A.D. 2000, "Yes, I knew one such."
I thank you for your kind attention to these discursive remarks.
VOTE OF THANKS
The President, in proposing a vote of thanks to the lecturer, said he was not really capable of doing justice to the occasion. He was sure many people far better fitted than he had done so on many many occasions, and it would be impertinence on his part to try to pay all the compliments that were due to Professor Aitken.
He had shown ability - if he (Mr. Howard) might say so-quite out of the run of ordinary mortals. Such skill and ability in dealing with calculations established that he was unique, and one could well understand the reasons for the numerous awards and honours that had been granted to him. His method of relaxation was quite different from that of ordinary individuals! !
However, very briefly but most sincerely, on behalf of all present and on behalf of the Council of the Society, he tendered the very best thanks to Professor Aitken for making the journey from Edinburgh to address the meeting that evening.
The vote of thanks was carried by acclamation.
DISCUSSION
Dr. H. G. Taylor said that it was interesting, since this was the Society of Engineers, to link what Professor Aitken had said with engineering. There were, in engineering, a number of problems which could be solved the more easily with the mental ability he had shown. In the structural field, calculations caused a good deal of trouble and took days, if not weeks, to carry out. The practice of his methods would help considerably, if the necessary ability could be developed.
He himself had discovered that Sir Charles Parsons had, in a way, something of the same characteristics; though he was perhaps not a marvellous calculator. In reading a biographical note about him only the other day, he had come across the following. In engineering, although he was Eleventh Wrangler in his year (he was at St. John's. College, Cambridge), he found formal calculation of very little interest. He generally reached his results almost instinctively by some obscure mental process which even he himself did not properly understand.
He rather gathered from what Professor Aitken had said that he did not entirely understand how his wonderful brain worked, and it would seem that Sir Charles Parsons had something of the same sort of brain.
His achievements in engineering were, of course, well known and perhaps this was partly the explanation for them.
To turn to the lecture, he could not refrain from paying tribute to those incredible people who learnt all the sexagesimal scale and who knew multiplication up to 59 x 59. It must have made the child's life far more tiresome than it was now. When one considered the trouble one had to get children to go up to 12 x 12 and remembered all that lay between that and 59 x 59, one realised that it must have been a very foul period. Perhaps not much else was learnt in those days.
Professor Aitken had shown his method of calculating cube roots, and it seemed to be a pretty good method. It worked very well, and some people could make it work with paper and pencil. Everybody knew square roots and most people probably knew there was an arithmetical method of doing cube roots. But when he was faced with the problem of providing questions, he obviously did not know himself what to do! However, one could always take a number and cube it.
Professor Aitken made the very interesting remark that if one started on the wrong track, one should not stop in the middle of a calculation, although one realised that there was a better way of doing it. It seemed obvious that his mind was working at two things at once. When he was asked to produce an answer very quickly, he wasted half the time saying how difficult it was. No doubt, he was only giving himself time.
He had asked Professor Aitken, when he lectured at the branch in Scotland, why if he had such a wonderful memory, he went to the trouble of writing down his lecture. The reply was that he could do so but did not want too much showmanship. Professor Aitken had noticed people in the audience whom he had not seen for many years - Dr. C. E. R. Brace, for instance - and had even remembered their initials.
It was interesting that he had a memory for things other than figures. Perhaps he would give the meeting some information about them. For example, he gave a remarkable display to the psychologists at Edinburgh who wanted to know how his brain worked with music. It was a remarkable achievement in memorising passages of music just as well as he memorised strings of figures.
Professor Aitken said he was not sure how to tackle Dr. Taylor's questions. Which did he regard as the most important?
Dr. Taylor said everyone was interested in cube root.
Professor Aitken thought it would take too long to explain and would convert the room into something like a class room. There was a method - Homer's method - of extracting not merely cube root but the root of any algebraical equation. The ordinary square root one learnt at school was just Homer's method diluted for school children. The cube root was also Homer's method but it was not given in many books. He could think only of Workman's Arithmetic, where it was given in some detail.
This brought him to the question about the ability of the Babylonians and the multiplication table up to 59 x 59, and the trial it must have been to the Babylonian children. It was not; because they did not have to learn it. Education among the Babylonians was not education in the modern sense. There was a consecrated priesthood of Chaldeans - astronomers - and they were set aside. He imagined that what might be called the proletarians had very little to do with calculation at all. It was this high priesthood and their novices and acolytes who, by a special vocation, were dedicated to arithmetic and calculation and astronomy. They were a caste set apart.
The inscriptions to which he had referred were very special indeed. They were marvellously interesting. It was wonderful to see in the very modern books on the history of ancient mathematics these cuneiform inscriptions and to learn to decipher them for oneself, the sexagesimal system, for instance, and to see what they were doing. He had some ideas about that. He might write something of what he saw in these tablets which had, he thought, escaped the commentators.
Mr. Green said he would like to limit himself to one question, if he might. It was clear from what Professor Aitken had said that in some of the calculations there were various stages. There was a certain calculation to be carried out mentally, and there was a different sort of calculation to be worked out. In the end the two had to be collated. The Professor had explained that in some mysterious way he was conscious of two calculations going on at once. But sometimes there must be even more than two. Would he be good enough to explain, if he knew the answer, whether, having arrived at an intermediate result he was able to store it away in his memory so that it was no longer an encumbrance and he could get on with an entirely different sort of calculation? Could he, at will, bring this earlier calculation out of the attic, dust it, and combine it with later ones in order to produce a composite answer?
Professor Aitken replied that he was able to put aside in storage for a future occasion a result that had already been obtained. He knew that he would be able to bring it out correctly. It would have been noticed in his demonstrations that if he was diffident - as he had been once or twice - he was not so good and might make an error.
He thought this ability to put an answer in storage was what distinguished the calculator from what might be called the man in the street. The man in the street forgot the stages between. One of the most interesting things about Bidder was that he did his stages one at a time and each one obliterated the preceding stages. He went from A to B to C to D, and finally he arrived at the answer PQR. He himself did not work in that way: nothing was obliterated. He could pull everything out tit any time in the course of the calculation.
Mr. Lewis said he was expert at going up and down columns of Figures and getting them wrong. The answer was to take the greater number for an expense account and to split the difference and call it a day for an engineering account!
Engineers had to deal with a vast range of science, chemistry, physics, mathematics, and so on. If they could train their memories to store more facts than some of them seemed to be able to do it, it would be of great advantage to them. It was not of much use to an engineer to be able to remember p to one hundred places. But to remember odd bits of chemistry and physics, different formulas, and so on, would be of immense use to him.
He had heard of people who stored numbers. A telephonist had once told him she had an amazing facility for remembering literally hundreds of numbers. He had said to her that with an amazing memory like that she must find it easy to memorise the Saturday morning shopping list. "Oh, no," she had replied, "my memory is like a shelf. If I add one more book at one end, a book falls off at the far end. At this moment it in chock-a-block with telephone numbers."
Professor Aitken said he had never found until recently that if a book was added at one end another fell off at the far end. It might he that he was reaching the age when this was the case.
He had been interested in many things, other than mathematics. Calculation did not interest him nearly so much as might be thought. He was far more interested in algebra; in very high algebra and very high analysis, in modem analysis and probabilities and all kinds of things. His interest was far greater in these than in casual and occasional calculations, which had just happened.
He was also far more interested in literature and music and very many more things. He was interested in history; he was interested in economics. Interest was the thing. Interest focussed the attention. One practised a bit and at first one might have to concentrate, but as soon as possible one should relax. Very few people did that. Unfortunately, it was not taught at school. It was not taught in the Scottish schools where knowledge was acquired by rote, by learning by heart, sometimes against the grain. The thing to do was to learn by heart not because one had to but because one loved the thing and was interested in it.
These were the first stages. It was like Yoga. It was practised in the physical plane but in the mental plane as well. One had to concentrate at first-to put chains on oneself. But there came a time when one in no way felt the chains at all. Then one had moved away from concentration to relaxation.
As to the subconscious, everybody had solved problems in their sleep. He did not suppose there was a single person in the room who had not wakened up with the thing clear in the morning when it was far from clear at night.
He himself made a practice of it, again by confidence. That was to say, at an hour not too late - say 10.30 or 11 p.m. - feeling that a piece of research was becoming rather turgid, one forgot about it and played something on the piano, or the viola, or the violin. In the morning early, perhaps at 2.30 a.m., one put on the light and wrote.
Dr. C. E. R. Bruce said he remembered reading in Hadamard's book how, when this subconscious process was going on, he could look down and watch what was happening. The subconscious kept pushing up little solutions into his mind and his conscious mind could look down and watch the process going on. He supposed that that was not possible in Professor Aitken's calculations. There would be no time.
Professor Aitken said there was no time to split oneself into two and watch what one was doing.
Dr. Bruce, continuing, said he supposed one would get into the position of a centipede asking which leg came before which.
These powers were so exceptional that he wondered whether there was anything of a distribution of them among the populace. Was this something absolutely exceptional, right beyond the pale of distribution? It was an entirely different process from the processes of ordinary mortals. it was really out of all comparison with the normal processes adopted in calculation.
Professor Aitken said he was not sure about this. When he began at school himself it was not so remarkable. He did not win the prize in arithmetic in his first year at secondary school. He thought he was rattier poor at arithmetic. It was only when his interest was fructified by an encounter with algebra that he began to move ahead so quickly.
He understood the question to be whether the ability was different in degree or kind. He thought it was different only in degree. He had heard of calculators and had got to know them. He had not gone out of the way particularly to make their acquaintance. He had met a few who had local or other reputations. They were not so very remarkable. To tell the truth, he thought the ability was very rare, but if one had the germs of it, one could develop it by application, as he himself had done.
He should mention that there was heredity in it. His father was quite good. He had helped his father with his books after school. When they added a column one up and the other down they got the same answer.
His father was an excellent arithmetician and it was probably his example that shielded him (Professor Aitken) from fear about arithmetic. Many persons had fear about arithmetic and that handicapped them. There was no fear in his family at all.
His father's elder brother, whom he himself had never seen because he went to Australia to make his career, the first in that generation, by family tradition, was very remarkable. He did all his calculations, as did Professor Aitken himself, mentally and without paper. He was a farmer's son and it fell to him to work out volumes of timber and barrels, and so on. He had perhaps found in a book on mensuration Simpson's formula or variants of it and did them in his head. His (Professor Aitken's) father used to say that his uncle was the best arithmetician he had ever known by far and by far, and this legend or tradition was preserved in the family. He supposed that was why when he became thirteen or fourteen he wanted to be something like Uncle Tom. His own daughter said quite frankly that one mathematician in a family was enough. She could work out square numbers as quickly as he could without trouble. If she thought it worth while, she could do some of the things he did, but she did not think it worth while. He thought it could be developed by small degrees, but he thought it developed exponentially and that each step was more rewarding, after a while. In the end one got clean away from the ordinary distribution which one would imagine to be among the populace. That was what he thought, but he might be wrong.
Mr. Boyd thanked the Society for allowing him to attend the meeting. It was a long time, he said, since he had so thoroughly enjoyed an evening, and questions simply bubbled up in his mind.
He would like, however, to stick to principles. He wondered whether Professor Aitken had analysed this extraordinary mental machine which he possessed. He obviously had analysed it very thoroughly, and it was to be hoped that he would publish the results for posterity. It had a very utilitarian purpose in connection with the development of calculating machines. No doubt he had already placed it at the disposal of the experts. It must have a more general application, however, and would be of enormous value to students and engineers and all sorts of people. It would be of great advantage to be able to control one's mental processes better.
In his calculating machine did Professor Aitken use a different radix from the one people were accustomed to? It was usual to use the binary system. Did he ever find it useful to do such a thing in his own mind?
Professor Aitken said this was a very interesting question. He was asked whether he used the binary scale, or perhaps the scale of eight, as Mr. E. W. Phillips had once suggested. He had not used this, but he once used the scale of twelve. He used it for amusement to see whether it could be done.
He carried out mental arithmetic in the duodecimal scale with twelve instead of ten and found that he could be very proficient in that too. It was very useful. There was one-third in the duodecimal scale - 0.4 - because 3 x 4 = 12, one-sixth = 0.2; and so on. There were also terminating decimals, and factorials with strings of noughts because there was a high power of twelve. After a while he stopped because he did not see any future in the duodecimal system.
There was an enthusiast about the world at that time - George S. Terry, of Massachusetts - who believed that if mankind would only adopt the duodecimal system, peace in our time would be ensured. The logic might not be clear to those present, and it was not quite clear to him either.
Professor Bickley said it had been enjoyable to watch Professor Aitken at work and see the remarkable things he could do. He would like to know why 67 was so difficult: sometimes one could learn almost as much from the things one could not do as from the things one could.
Professor Aitken said it was really a question of what he might call undercutting or overcutting his drive. 67 ought to have been a gift, but it was not. He just missed it. 67 x 597 was 39,999. This was 40,000 minus 1. Ordinarily he would have made instant use of this.
Mr. G. Coates asked whether Professor Aitken ever worked with much cruder approximations than he had mentioned. He himself was acquainted with the fact that the square root of 10 was 3.162 and that p was 3.142 or thereabouts. He was accustomed to those numbers for about fifteen years before it dawned on him that they were very close together. Whenever he had a square root and it in the same calculation, he played with the square root until it came to something with a 10 and, of course, after that it did not matter. One could still cross out p5 or p7. With that crude approximation one could shorten a number of calculations. If he was lucky it did not take him longer than with a slide rule. If he was unlucky it took him about ten thousand times as long as it would take Professor Aitken.
The point was that this trick which he imagined was known to a lot of people, together with a number of other such tricks, depended on a very crude approximation, about 1 per cent. That was quite good enough for a lot of people who wanted approximate results. Did Professor Aitken use this technique, and if so could be give any examples?
Professor Aitken said he did not use this technique. He might begin with a crude approximation, but it was his method to deduce a better approximation from it, and from that a very much better one. In raising to high powers there would be a fairly considerable error in approximating 3.162 to 3.142. If one started with an error which was 1 in 150 and raised it to 6 in 150, or 4 per cent, this would be outside the limits of tolerance in the engineering sense.
As for the square root of 10 as an approximation to p, Mr. Coates was late in the field. Some of the Greeks had it and the Egyptians had 256/81, which is the square of 16/9 and equals 3.1605.
Mr. J. N. Walker asked whether Professor Aitken made use in recognising prime numbers and factors of the sum of the sum of the digits.
Professor Aitken said he did not. Was it casting out nines that that was meant? No doubt Mr. Walker was thinking of digital root. This had no interest whatever for him. When he was asked a question about, say, 1327, he asked himself whether it was a prime or not. If it was not a prime he asked himself what were its factors and that threw up the thing straight away.
The President, in closing the meeting, said that a formal vote of thanks had already been passed, but he felt sure everyone would wish him to say how much they had appreciated the way in which Professor Aitken had dealt with the discussion and the inimitable manner in which he had replied to the questions that had been put to him.
