Professor Derek del Solla Price in his book Science since Babylon (enlarged edition) says:

"....... Babylonian astronomy, especially in its Seleucid culmination during the last two or three centuries BC, represents a level of mathematical attainment matched only by the Hellenistic Greeks, but vastly different in content and mode of operation. At the kernel of all Babylonian mathematics and astronomy there was a tremendous facility with calculations involving long numbers and arduous operations to that point of tedium which sends any modern scientist scuttling for his slide rule and computing machine." [*]

In other words, the maths is boring and tedious for most Modern man's mind, and would prefer to use a calculator than actually think through the calculation. A difference between Modern Mathematician and Ancient Mathematician way of thinking. It reminds me of Star Trek the original series, Spock is able to do mental calculations that match the Ship's Computer calculations. Ancient Mathematicians pre- Slide rule and calculator, were prepared to duplicate Spock's mental calculations. There are several incidence of 'mental prodigies' able to do amazing calculations in their heads. Today such mathematicians are rare, in the past they must have been more common.

Del Solla continues:

"Indeed some of the clay tablets, presumably intended for educational purposes, contain texts with problems which are the genotypes of those horrors of old-fashioned childhood-the examples about the leaky baths being filled by a multiplicity of variously running taps, and the algebraic perversions (though here expressed more verbally than symbolically) with a series of brackets contained within more brackets ad nauseam." [*]

Illustrating the ancient mathematics is boring as per our modern minds, and hence why many hate maths lessons. De Solla admits it to be boring:

"That is admittedly the dull side of Babylonian mathematics. Its bright side was a feeling for the properties of numbers and the ways in which one could operate with them. One gets the impression that the manner in which Ramanujan worked-in perceiving almost instinctively the properties of numbers far from elementary and in having every positive integer as one of his personal friends-was the normal mode for a Babylonian." [*]

Ramanujan was an extraordinary mathematician, not only was he a 'mental prodigy' able to do long calculations in his head a la Spock, but he was able to do abstract mathematics as well. Usually the rare 'mental prodigies' only do arithmetic, not other things like algebra and calculus.

Consider this account by Mahalanobis who went to visit Ramanujan in hospital, during WWI. Mahalanobis had a maths problem from a magazine, he says:

"I got interested in a problem involving a relation between two numbers. I had forgotten the details; but I remember the type of the problem. Two British officers had been billed in Paris in two different houses in a long street; the door numbers of these houses were related in a special way; the problem was to find out the two numbers. It was not at all difficult. I got the solution in a few minutes by trial and error." [#]

Bear in mind that Mahalanobis was probably a mathematician, so 'easy' for him might not mean 'easy' for everyone else. The talk with Ramanujan went:

"Mahalanobis: Now here is a problem for you.

Ramanujan: What problem, tell me.

Mahalanobis read out the problem from the magazine.

Ramanujan: Please take down the solution. (He dictated a continued fraction.)" [#]

"The first term was the solution which Mahalanobis had obtained. Each successive term represented successive solutions for the same type of relation between two numbers, as the number of houses in the street would increase indefinitely. Mahalanobis was amazed." [#]

What Ramanujan had done was derive an equation for this problem. But that was not the amazing thing:

"Mahalanobis: Did you get the solution in a flash?

Ramanujan: Immediately I heard the problem, it was clear that the solution was obviously a continued fraction; I then thought, "Which continued fraction?" and the answer came to my mind. It was just as simple as this." [#]

The amazing thing was: Ramanujan had to not consciously work out the answer. He merely consciously though 'what is the answer?' and his unconscious immoderately provided the 'answer' without having to consciously work through the derivation.

The way his mind was set up, part of his mind the unconscious part was working like a computer.

If Ancient Mathematicians were like Ramanujan then their own brains acted equivalent to our modern computers. If a person wanted to do a long complicated calculation and had no access to our modern computers, then he would train his brain to be that computer.

De Solla however says:

"I do not wish to exaggerate more than is necessary for effect, or to imply that the ancients all had the genius of Ramanujan. It is plain, though, that their forte was in matters arithmetical, and in this they were supreme." [*]

So, consider another mathematical prodigy:

"Madhava (c. 1340 - 1425) finds pi correct to 11 decimal places. Vieta in 1579 only found 9. Madhava is also credited with what we call the Gregory series for arc tan x, found in 1667. If Madhava was not the discoverer - he may have been given credit by admiring followers - it was still discovered first in Kerala in south India, where it was attributed to Madhava in several texts of the fifteenth century. Madhava, or other Keralese mathematicians, also discovered Leibniz's series for pi, and the power series for sine x and cosine x, usually attributed to Newton." [#]

All very interesting to mathematicians that certain mathematics was discovered in India before it was discovered in Western Europe by such greats as Newton and Leibniz, but probably boring to everyone else? Any way, the important point is:

"Madhava's work is reminiscent of another Indian genius, Ramanujan. Where did Ramanujan grow up? At Kumbakonam, not far from Madhava's birthplace." [#]

Ramanujan was not unique, there were others like him. It seems plausible that there were many other such mathematical prodigies in the past. What get discovered by the few mathematical prodigies in recent history, could have been discovered first by mathematical prodigies of the same extraordinary caliber in ancient history.

De Solla wonders about these ancient mathematicians:

"The origins of this facet of Babylonian civilization are hard to determine. Perhaps it was some peculiar national characteristic; perhaps some facility given by the accident of their writing in clay with little uniform, countable, cuneiform wedges. Possibly there was some urgency in their way of life that required the recording and manipulation of numbers in commerce or religion. Such tenuous speculation seems not only dangerous but unprofitable in view of the fact that the Babylonians were by no means unique in this quirk of mind: the Mayan passion for numbers in their calendrical cycles is not far different, though nothing that we thus far comprehend quite matches the superb Babylonian sophistication." [*]

He is just wondering how Ancient mathematicians were able to handle what was to us tedious calculations that we leave to computers.

De Solla then tells us that the Babylonian mathematical approach to astronomy was very advanced:

"For our purposes, what is significant about the Babylonian attitude toward astronomy is not any accident of its origin, but rather that it existed as a highly developed and penetrating arithmetical way of dealing with the motions of the sun and moon and planets. The Babylonians operated with the vital technique of a place value system for all numbers, integral and fractional. They made use of the very convenient sexagesimal base of sixty, which we still retain from their tradition in our angle measurement of degrees, minutes, and seconds and in our subdivision of the hour. Above all, they were able to make astronomical calculations without recourse (so far as we know) to any sort of geometrical picture or model diagram. Perhaps the nearest thing to their methods in modern mathematics is in Fourier analysis of wave motions, but here the mathematician thinks in terms of concepts of sine waves rather than the mere numerical sequence of the Babylonians." [*]

That was the Babylonians, what of the Greeks that came after them:

"It is inevitable that we should be drawn to compare the high science of the Babylonians with that of the Greeks. For each we can perceive something like a reasonably continuous tradition until the last few centuries BC, when both are concerned with the same very natural problem of the maddeningly near-regular motion of the planets. By that time each is standing ready with a mature and abstruse scheme full of technical refinements and containing, coordinated within the scheme, all the most relevant observations and considerations that had accrued through the centuries." [*]

"It is one of the greatest conjuring tricks of history that these two contemporary items of sophistication are as different from each other as chalk from cheese. Spectacularly, where one has deep knowledge, the other has deeper ignorance, so that they discuss precisely the same basic facts in manners so complementary that there is scarcely a meeting ground between them. For all the Babylonian prowess in computation, one discerns no element of that method of logical argument that characterizes the Greek Euclid. One might go further and accuse the Babylonians of being totally ignorant and incompetent with geometry (or, more generally, with all Gestalt matters), but here one must exercise due caution and allow them the modicum of architectural geometry, mythical cosmology, etc., that any high civilization seems to develop willy-nilly. For example, we know that the "Pythagorean" properties of the sides of the right-angled triangle were known to the Old Babylonians about a thousand years before Pythagoras; but this is precisely the sort of homespun geometry that can readily be acquired, even today, by anyone meditating upon a suitable mosaic floor or in a tiled bathroom." [*]

"What now of the Greeks? Are they not just as lopsided, scientifically speaking, as the Babylonians? We must be careful here to distinguish between the early Hellenic period and the later Hellenistic. In this distinction the history of science provides (as it does so often elsewhere) a perspective refreshingly different from that of ocher histories. For example, the great Renaissance, beloved of the historian of art, seems to diminish a little when viewed by the historian of science and to take much more of the character of a parochial Italian movement whose significance is overshadowed for us by the influential revival of astronomy in Protestant Germany. As for the Greeks, the great centuries of art, philosophy, and literature of the Golden Hellenic age are overshadowed for us by the tremendous scientific vitality of the Hellenistic period." [*]

"Making what we can of the earlier period, we can discern the presence of an aura of logic and of geometry that we know so well from Euclid, but totally lacking is any depth of knowledge of calculation. Again, one must make the exception of the everyday and allow that an inhabitant of classical lands could, when pressed to it, function sufficiently to make out his laundry bill. One also allows the minute amount of arithmetic (in the Babylonian sense) contained in the well-known Pythagorean writings. Although these were concerned with number, and at times more than trivial, they were devoid of any difficult computation or any knowledge of the handling of general numbers far beyond ten. One need only examine the attitudes of each civilization toward the square root of two. The Greeks proved it was irrational; the Babylonians computed it to high accuracy." [*]

Surprise: the Babylonians were better at mathematical calculations than the Greeks that came after them. The Babylonians were great 'mental mathematical prodigies' but did not use the type of Logical thinking that the Greeks were later credited with. There was two different approaches to thinking about mathematics.

De Solla continues:

"Again, we need here have little interest in understanding the series of complex motivations and accidents that had set the Greeks on this particular road of civilization. So far as it concerns science, other civilizations had probably done much this sort of thing before. Modern historians have long lived in consciousness only of the glorious and unique Greek tradition of mathematical argument from which we patently derive so much of our present state of mind; this being so, it is difficult to disabuse ourselves of the tradition and attempt to re-estimate how far Hellenism would have taken us in the absence of the Babylonian intervention so clearly manifest in such later Hellenistic writers as Hero and Hipparchus." [*]

"The true fruition came as a natural but fortuitous consequence of combining the qualitative, pictorial models of Greek astronomical geometry with the quantitative operations and results of the Babylonians." [*]

"From the Greek point of view, the planets appeared to rotate almost, but not quite, uniformly in circles. By the Babylonians, the extent of the lack of uniformity was well measured and accurately predictable. How could the Greeks picture this slight but precise lack of uniformity in planetary motion? They could not conveniently do it by letting the planet move sometimes faster, sometimes slower. Motion that got faster and faster might be allowable, but there was no convenient mathematical machinery for considering a fluctuating speed. The most natural thing to do was to retain the perfect and obvious uniform motion in a circle and to let the Earth stand to one side of that circle, viewing the orbit with variable foreshortening. Such a theory accounts for the most complex actual motion, as we now know it, to an accuracy virtually as great as the eye can perceive without the aid of the telescope. Kepler showed that the planets move in an ellipse with the sun at one focus. The ellipse is, of course, very close to an off-center circle, and the planet appears to move with very nearly uniform angular velocity about the empty focus." [*]

"The Babylonian technique was to use series of sequences composed of numbers that rose and fell steadily or had differences that themselves increased or decreased steadily. All the numerical constants were most cunningly contrived so as to yield the necessary periodicities and provide quantitatively accurate results without the intervention of any geometrical picture or model." [*]

"Thus the Greeks had a fine pictorial concept of the celestial motions, but only a rough-and-ready agreement with anything that might be measured quantitatively rather than noted qualitatively. The Babylonians had all the constants and the means of tying theory to detailed numerical observations, but they had no pictorial concept that would make their system more than a string of numbers." [*]

"This extraordinary mathematical accident of doing the only obvious thing to reconcile Greek and Babylonian, and deriving thereby a theory that was a convincing pictorial concept and also as near true as could be tested quantitatively, was a sort of bonus gift from nature to our civilization. As a result of that gift and its subsequent Hellenistic elaboration by trigonometrical techniques, the great book of the Almagest could stand for the first time as a complete and sufficient mathematical explanation of most complex phenomena. In nearly every detail it worked perfectly, and it exemplified an approach which, if carried to all other branches of science, would make the whole universe completely comprehensible to man. It stood also as a matrix for a great deal of embedded mathematical and scientific technique which was preserved and transmitted in this context up to the seventeenth century.

We must now survey our story and draw what conclusions we may. The fact that our civilization alone has a high scientific content is due basically to the mixture at an advanced level of two quite different scientific techniques-the one logical, geometrical, and pictorial, the other quantitative and numerical. In the combination of both approaches to

astronomy, a perfect and workable theory was evolved, considerably more accurate than any other scientific theory of similar complexity. If one may speak of historical events as improbable, this Ptolemaic theory was improbably strong and improbably early. It was almost as though that branch of science had got an unfair start on all the others, racing ahead long before it should have in the well-tempered growth of any normal civilization, like the Chinese." [*]

"If we are more satisfied and curious about the state of that science that we actually have, rather than what might have been, perhaps it behooves us to analyze further the consequences of our twin origin in the Graeco- Babylonian melting pot. It is more than a curiosity that of two great coeval cultures the one contained arithmetical geniuses who were geometrical dullards and the other had precisely opposite members. Are these perhaps biological extremes, like male and female, with comparatively little likelihood of an hermaphrodite? Possibly there is some special quality of nature or nurture that can make a human being, or even a whole society, excel in one of these extreme ways. Perhaps some men can excel in both, as Ptolemy evidently did. Perhaps the vigor of modern mathematical physics, for example, would demand that it be maintained by men who manage to excel both as Babylonians and as Greeks." [*]

"The example of Ramanujan indicates that perhaps there are Babylonians, almost of pure mathematical breed, abroad among us today. Other mathematicians may surely be classed as of the Greek temperament. Unfortunately there has been very little useful study of the psychology of scientists, but the little that we have accords well with the notion that visual-image worshippers and number-magic prodigies may be surprisingly pure as strains. Certainly we know from experience in the world of education that our population at large consists of those who take to mathematics and those who definitely do not. The problem is evidently fundamental and of too long standing to be attributable solely to any mere bad teaching in the schools." [*]

"Can it be that the Babylonians and Greeks among us do not communicate with one another very well in this sphere where they met only once at a high level? To put it in more Psychological terms, we may have here a problem in which we should do well to distinguish between the visualists and the verbal thinkers (if this is the modern equivalent of the old types) and, if we find them distributed bimodally, we should perhaps arrange for each group to have a teacher and a method of the correct mathematical blood group." [*]

Modern Society is biased towards one type of mathematical - scientific outlook, and does not have any more Ramanujan - Babylonian types.

Reference:

[*] Science since Babylon, enlarged edition, Derek del Solla Price, Yale University Press, USA 1975: blurb Professor Price has enlarged his widely known and influential study of science and the humanities to include much new material, extraordinarily broad in its range: from ancient automata, talisman and symbols, to the differences of modern science and technology. p10 - 21

[#] The Penguin Book of Curious and interesting mathematics, David Wells, Penguin books, UK 1997 p 229 - 230, p 119

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