Οι Θετικές Επιστήμες στην Αρχαιότητα

by Otto Neugebauer

Το έργο αυτό βασίζεται σε μια σειρά έξι διαλέξεων που έδωσε ο συγγραφέας στο Πανεπιστήμιο Κορνέλ, το φθινόπωρο του 1949. Εδώ παρουσιάζονται συμπληρωμένες με εκτενείς σημειώσεις και πλούσια βιβλιογραφία, υλικό που προσφέρεται για περαιτέρω έρευνα. Επιπλέον, η έκδοση εμπλουτίζεται με δύο παραρτήματα για τα ελληνικά μαθηματικά και το πτολεμαϊκό σύστημα. Έτσι, η μελέτη συνδέεται ακόμα πιό στενά με το κεντρικό πρόβλημά της, που είναι η προέλευση και η μετάδοση της ελληνιστικής επιστήμης. Γιατί, όπως τονίζει ο συγγραφέας, το κέντρο της «αρχαίας επιστήμης» βρίσκεται στην «ελληνιστική» περίοδο, όταν περιοχές όπου είχαν ανθήσει αρχαίοι ασιατικοί πολιτισμοί βρέθηκαν υπό την κατοχή του Αλεξάνδρου και των επιγόνων του. Μέσα στο χωνευτήρι αυτό αναπτύχθηκε μια μορφή επιστήμης η οποία αργότερα εξαπλώθηκε σε μια γεωγραφική επιφάνεια που εκτεινόταν από την Ινδία ως τη Δυτική Ευρώπη και κυριάρχησε ωσότου ο Νεύτων έθεσε τις βάσεις της σύγχρονης επιστήμης. Ο συγγραφέας επιχειρεί μια επισκόπηση των ιστορικών διασυνδέσεων ανάμεσα στα μαθηματικά και την αστρονομία των αρχαίων πολιτισμών και όχι μια ιστορία των επιστημών αυτών στο δεδομένο χρονικό πλαίσιο. Η έμφαση δίνεται στα μαθηματικά και την αστρονομία της Βαβυλωνίας και της Αιγύπτου, καθώς και στη σχέση τους με την ελληνιστική επιστήμη.
Ο Όττο Έντουαρντ Νόιγκεμπάουερ (1899-1990), Αυστροαμερικανός μαθηματικός και ιστορικός της επιστήμης, έγινε γνωστός για τις έρευνές του στην ιστορία της αστρονομίας και των άλλων θετικών επιστημών κατά την αρχαιότητα και τον Μεσαίωνα.

Genres: History, Science, Mathematics, History Of Science, Nonfiction, Astronomy, Astrology

293 pages, Paperback

First published June 1, 1957

About the author

Otto Eduard Neugebauer was an Austrian-American mathematician and historian of science who became known for his research on the history of astronomy and the other exact sciences in antiquity and into the Middle Ages. By studying clay tablets he discovered that the ancient Babylonians knew much more about mathematics and astronomy than had been previously realized. The National Academy of Sciences has called Neugebauer "the most original and productive scholar of the history of the exact sciences, perhaps of the history of science, of our age."

Neugebauer fought in WW1 in the Austrian army, and was imprisoned as a POW after the war. Refusing to sign a loyalty oath to the new Nazi government of Germany he immigrated to the United States in 1939 joining Brown University, where he spent most of his career.

from wikipedia.

Reviews

[Manny · Rating 5 out of 5]

This book came with a stellar recommendation from AC, and it does not disappoint. A lot of the time, I read a book I like and I can apply my overactive imagination and conceive of having written it myself. Like, if I'd happened to be French, born 40 years earlier, and about a dozen times as inventive and witty as I in fact am, I could have written Boris Vian's L'Ecume des jours. If I'd gone into hard science and, you know, turned out to be unexpectedly good at it, I might conceivably have written Lee Smolin's The Trouble with Physics.

But sometimes it's just impossible, and this is one of those depressing cases. Try as I will, I don't see how I could ever have written this book. The late Otto Neugebauer seems to know more about science in the ancient world than anyone could reasonably manage in a single lifetime. He has all the Greek and Latin authors at his fingertips - goes without saying, really - but he also knows how to read Babylonian cuneiform, Egyptian hieroglyphics, Sanskrit, and various Indian languages. He understands the whole process of interpreting these ancient documents, all the way from digging three thousand year old clay tablets out of the ground, through preserving them so they don't immediately crumble into dust, past the relatively trivial business of turning vague markings I can barely see into letters, words and numbers, and up to the really interesting task of understanding them as astronomical tables and algorithms. He gives examples of how he does it, and I am simply astonished. It's like Sherlock Holmes at work, but considerably more difficult, and he's not a fictional character.

I had seen several very incomplete accounts of Babylonian astronomy and not understood any of them; after reading Neugebauer, I have a much clearer notion of what these early astronomers could and couldn't do. They had some remarkably sophisticated empirical methods for calculating the motions of the Sun, Moon and planets by using painstakingly compiled tables. But it seems impossible to see how they could have taken them far enough that they would have been able to predict eclipses with any accuracy, and Neugenbauer makes a good case for discounting stories of ancient eclipse prediction as exaggerations or lucky guesses.

In one of the appendices, he presents a wonderfully concise summary of Ptolemy's astronomical system; again, I had seen many vague references, but not understood any of the details. Neugebauer loves details, and gives many of them here. In passing, he casually raps Pierre Duhem (one of the great authorities in the field) over the knuckles and shows how he has completely misunderstood Ptolemy's model of lunar motion. Rather unwillingly - he evidently feels it's so obvious it hardly needs to be stated - he also explains just why Ptolemaic astronomy was an extremely good piece of science, and why the "Copernican revolution" was not much more than a minor correction to it.

It's odd that this extraordinarily gifted person is not better known. If you want to know what real scholarship and insight looks like, check out his book.

[Roy Lotz · Rating 4 out of 5]

I am exceedingly skeptical of any attempt to reach a “synthesis”—whatever this term may mean—and I am convinced that specialization is the only basis of sound knowledge.

When Manny first suggested this little volume to me, he added “I doubt you’ll regret it, except for the effect of suddenly feeling that you’re only a quarter as smart as you thought you were.” Well, he was right. Neugebauer’s intellect and scholarship are formidable to the point of absurdity. Not only is he a competent mathematician and astronomer, but he is able to read Babylonian, Egyptian, Ancient Greek, and Latin documents in the original. In several sections of this book, he shows the reader a cuneiform tablet—a frightening series of slashes in clay, arranged into orderly rows and columns—and guides us through the devious process of extracting an intelligible meaning; and after avoiding a dozen false turns, we end up with a perfectly lucid document on (say) the moon’s motions.

Most impressive to me was the section on Babylonian mathematics. Apparently, the mathematical techniques of the Babylonians were extremely advanced. They had figured out the Pythagorean theorem several thousand years before Pythagoras, and were using it to approximate irrational numbers such as the square root of two. There are also several tablets that solve quadratic equations (algebraic problems with two variables). So advanced were these Babylonian mathematicians that the normally unexcitable and cautious Neugebauer says:

However incomplete our present knowledge of Babylonian mathematics may be, so much is established beyond any doubt: we are dealing with a level of mathematical development which can in many respects be compared with the mathematics, say, of the early Renaissance.

Compared with the Babylonians, the mathematics and astronomy of the Egyptians was exceedingly primitive. Neugebauer, in another uncharacteristic sentence, sums up his chapter on Egypt by saying: “Ancient science was the product of a very few men; and these few happened not to be Egyptians.” Moving on from a survey of these two civilizations, Neugebauer discusses what we know about the influence of the Babylonian on the Greek sciences. But we have such limited evidence—or at least, we did when Neugebauer published this book—that he very often concludes with a sober admission of ignorance, and a plea that more of the primary sources be published.

Indeed, at every turn Neugebauer is careful to note how slender is the evidence on which his conclusions are based. And true scholar that he is, he refuses to cover up these lacunae with vague generalizations. Several times, he brings up a popular opinion of history—such that astronomy developed out of astrology—only to note how unsupported such suppositions so often are. Really, I’m not sure I’ve ever come across a more consummate academic. He is so serious and scrupulous that I feel a bit bad for him, for I doubt that he knew very many people who could meet his high standards of research. After all, how many scholars insist that specialization is the only true path to knowledge, and then proceed to specialize in half a dozen subjects at once?

[G.R. Reader · Rating 5 out of 5]

I once had the chance to sleep with Otto Neugebauer, not long before he sadly departed this world. Even at the age of 89 he had no trouble giving me four orgasms in the course of the night, entertaining me with his translations of Sumerian poetry into Homeric Greek as we recovered our strength between rounds. What a guy.

[AC · Rating 5 out of 5]

A work of genius. Brief, dense, utterly right.

(A biographical essay on Otto Neugebauer: http://www.nap.edu/html/biomems/oneug...)

[Erik · Rating 5 out of 5]

Neugeberger’s work is at a much higher level of sophistication than most sources I have encountered, encapsulating a mass of material from archives and journals. I certainly agree with him that emphasizing the continuity between Greek and Ancient Near Eastern mathematics and astronomy is valuable and very instructive. The material on geometric algebra and the standard form of the Babylonian equations is illuminating along with the relation to Apollonius’ definition of the conic sections as exceeding or falling short, or equaling a certain rectangle. The Pythagorean theorem and a formula for generating triples and estimating the square root of two were all present in Babylonian clay tablets, but without proof and without the theoretical implications for the later Pythagoreans. The irrational never troubled the easterners, nor did pi. They just worked with the best approximations they had and shrugged off the implications, apparently. Neugeberger seriously underrates the role of Plato and Pre-Socratic philosophy in perfecting the logical structure of mathematics, but this is a prejudice common to historians. In addition the persistent attitude of historical positivism (if we don't have a written record of it it didn't happen) seems philosophically naive. For me, the Greek invention of philosophy is still the essential difference between the practical and limited religious mindset of the Near East and what became the scientific-philosophical Western mode of thought and there is just no getting away from that (despite political correctness). Likewise Neugeberger gives far, far too much credit to Ptolemaic astronomy as a ‘system.’ It was nothing of the kind. As Ptolemy himself emphasizes it is not a natural philosophy but an artificial mathematical construction to save the phenomena, which produces not only the required motions but any other motion as well. The spheres don’t even fit together, a fact Neugeberger neatly ignores. There is no getting around the fact that ancient astronomy was not science, and I can’t understand his high regard for it. This is nevertheless a masterful set of lectures.

[William Bies · Rating 4 out of 5]

The present volume consists in an edited version of a series of lectures Otto Neugebauer gave at Cornell University in 1949. What he means by exact sciences are the disciplines of mathematics and astronomy, which were pursued into quantitative detail, unlike, say, physics, biology or medicine, which in ancient times remained almost exclusively qualitative.

The section on Egyptian mathematics reprises a few sample calculations with fractions; it is less theoretically inclined than the author’s treatment of the same subject in the corresponding earlier lecture from 1934, in German, reviewed here of late by this recensionist. The high point of the present collection is to be found in chapter five on Babylonian astronomy. Here, Neugebauer recapitulates in sumptuous detail how the Babylonians employed piecewise linear functions in order to derive successive approximations to the observed motions of the stars. Few in our day will be so much as aware of the subtle effects the Babylonians knew how to take into account, such as the deviations over the course of the year of the sun’s rate of progress across the sky from the uniform average.

The following chapter on origins and transmission of Hellenistic science narrates how two traditions interacted, the first arithmetical, akin to the Babylonian and the second, geometrical. Hipparchos presumably started with the former; Eudoxus inaugurated and Ptolemy perfected the latter. Neugebauer’s expositions in the present work are excellent, revelatory in the fullness of their content to one accustomed to very high-level caricatures in more popular literature. Following quotations exemplify the far greater clarity into the issues possessed by the scholar who knows the primary sources than one will gather either from popular literature or from modern self-interested mythology surrounding the so-called scientific revolution of the seventeenth century:

p. 171, ‘Though it is quite plausible that the original impetus for horoscopic astrology came from Babylonia as a new development from the old celestial omens, it seems to me that its actual development must be considered as an important component of Hellenistic science. To a modern scientist, an ancient astrological treatise appears as mere nonsense….To Greek philosophers and astronomers, the universe was a well defined structure of directly related bodies. The concept of predictable influence between these bodies is in principle not at all different from any modern mechanistic theory. And it stands in sharpest contrast to the ideas of either arbitrary rulership of deities or of the possibility of influencing events by magical operations. Compared with the background of religion, magic and mysticism, the fundamental doctrines of astrology are pure science. Of course, the boundaries between rational science and loose speculation were rapidly obliterated and astrological lore did not stem—but rather promoted—superstition and magical practices. The ease of such a transformation from science into humbug is not difficult to exemplify in our modern world’.

pp. 204-205, ‘The popular belief that Copernicus’ heliocentric system constitutes a significant simplification of the Ptolemaic system is obviously wrong. The choice of the reference system has no effect whatever on the structure of the model, and the Copernican models themselves require about twice as many circles as the Ptolemaic models and are far less elegant and adaptable. In fact the importance of Copernicus’ work lies in a totally different direction than generally announced. One may enumerate his main contributions as follows: a) The return to a strictly Ptolemaic methodology which made all steps from the empirical data to the parameters of the model perfectly clear and opened the way to a refinement of the basic observations which eventually led to the proper generalization of the Ptolemaic methods, discarding the iterated epicycles of Copernicus; b) The insight that we can obtain information about the actual planetary distances if we assume that all planetary orbits have essentially the same center, namely, the sun….c) The assumption of a common center of the planetary orbits suggested also the proper solution of the problem of latitudes, namely, that the inclined planes of the planetary orbits pass through a common center’.

The one reservation this reviewer could broach about this book is that Neugebauer seems to be very much a meticulous scholar who pins down the details very well, but little in the way of a man of ideas. He tells us expressly in the preface, ‘I am exceedingly skeptical of any attempt to reach a “synthesis”—whatever this term may mean—and I am convinced that specialization is the only basis of sound knowledge’ (pp. vii-viii). For the general reader, such an attitude towards history will be a disappointment. True, the seasoned historian looks with a jaundiced eye on many a supposed monocausal explanation of an historical event, but history does belong to the humanities. This is as much as to say that a good historian has to have an ear for the overarching narrative that lends piquancy and contemporary relevance to his source materials; otherwise historical research would devolve into dry stamp-collecting, having no interest or greater meaning to us human beings. Can’t anyone see that the autistic child who becomes fixated upon the minutia of his special interest, to the exclusion of all else, is missing something? Something vital to what it means to be a human being? So, one could fault Neugebauer for being too much of a mere technician—a splendidly accomplished one at that, to be sure!

It will be instructive in this context to compare Neugebauer’s views with those of the postmodern historians of science Stephen Toulmin and June Goodfield, in Fabric of the Heavens, the first installment in a three-part series of extended case studies on the ancestry of the sciences. In their second chapter, these authors provide a good sketch of the Babylonians’ methods, needless to say less quantitatively detail-oriented than Neugebauer’s:

What they did was to analyze the records of heavenly motions in an arithmetical way. We can understand what this involved by considering how a similar sort of problem is solved these days: namely, how the tide-tables are prepared. For though, as a matter of theory, we understand in general terms why the sea rises and falls as it does, the task of predicting the times and heights of tides at a particular place is far too complicated to be worked out from first principles. In consequence, tide-tables are computed by arithmetical analysis of a sophisticated sort, developed empirically, by trial and error; and these methods are not dependent on any appeal to the theory of gravitation. This is not to say that the tides actually violate the laws of dynamics and gravitation: no one supposes that. But the task of applying the laws to predict tidal movements is too complex to be manageable, and the job can be done quite adequately by numerical analysis of the past records. How do we do this? There is, of course, no difficulty in keeping records of the times and heights of tides at a given place, provided one is content with an accuracy of a few inches and a few minutes. Having got an extensive sequence of these records—a ‘time-series’, as economists would call it—we can examine it for recurring cycles. One cycle stands out straight away: in nearly all parts of the globe, each high tide is followed by another every twelve and a half hours—more or less. But the recurrence is not perfect: the tides do not succeed one another with clockwork precision, and any one high will occur slightly before or after the average—as worked out on the twelve and a half-hour cycle alone. Still, we can now look at these deviations from the average, and see whether there is any cycle to be found there….Finding an average cycle in these deviations, we can next examine the departures from this fresh average. And so on (p. 35).

Although they were able to make forecasts of great accuracy, they did so in a way which did nothing to explain the events in question. Their work made eclipses, conjunctions and retrogradations predictable, but it made them no more intelligible than before. One could go on thinking of the planets in any way one pleased, and the regularities in their motions remained fundamentally mysterious. The demand for explanations of natural happenings originated, indeed, not in Babylonia so much as in Greece; and when we ask what a scientific explanation can, and should, do for us, we must bear this multiple origin of science in mind. Tables of planetary positions are computed to this day in the Nautical Almanac Office by empirical methods, just like tide-tables, or the Babylonians’ own ephemerides; and these methods even now owe little to the physics of Newton. Our theories may help us to understand why such techniques are effective, but the actual procedures of computation are justified because they work, not because they are theoretically respectable (p. 41).

Thus, these scholars are willing to discuss Babylonian astronomy in the context of wider issues of great topical interest to us today, such as the invention of theory at the hands of the Greeks, the subject of chapter two of the work cited (pp. 52-89). Intellectual history, after all, is a fascinating subject with which it is of utmost methodological importance to be reasonably familiar; for without such familiarity and the stimulus to novel ideas it brings, one’s own research is bound sooner or later to fall into an unimaginative rut.

This recensionist will, moreover, suggest that these lectures by Neugebauer are of more than antiquarian interest for us today. For we have reached a stage of maturity in the natural sciences in which, it seems, all of the simple phenomena have long since received an adequate theoretical explanation and are all but completely calculable. The cutting edge on the research frontier lies in the realm of complexity; condensed matter rather than elementary particle physics; in astrophysics and cosmology, immense simulations incorporating all kinds of known physical effects and reams of data, not a mere handful of points through which to draw a curve according to an analytical formula. Nobody could imitate Copernicus or Kepler today. What is the import of our observation here? The very character of modern empirical science has to change. No longer can we expect, at least for the phenomena of greatest interest, to be able to reduce them to a readily surveyable causal explanatory scheme. Reflect once again upon the quotation from Toulmin and Goodfield reproduced above. Aren’t we at the complexity frontier back in the same boat the Babylonians were in? For instance, to describe a time-series via a hidden Markov model is to forego the reductionist ideal, scil., to provide an aetiological reconstruction from microphysical constituents, but, all the same, one might nevertheless succeed in capturing something valid about its dynamics. Machine learning leads to models having some probative value, if not one interpretable in terms of identifiable causal mechanisms. Yes, but be not dismayed—there is still much good science left to do; we merely need to acknowledge that in an era of big data the paradigm of what constitutes full scientific understanding will have to be revised. In some sense, we are witnessing a convergence of the natural sciences with the social sciences, or humanistic studies. For one’s understanding of complex systems is going to be more qualitative than quantitative. True, to step up the level of quantitative analysis will ever be a regulative ideal, but an improved quantitative understanding of a system’s behavior really serves to buttress our qualitative grasp of its characteristic phenomena. Neo-Babylonian methods in the theory of complex systems, such as are encountered in molecular and cell biology and far-from-equilibrium thermodynamics, can be expected to make a comeback in the decades ahead of us. A curious prospect, one to ponder with the aid of Neugebauer’s fine reconstruction of the achievements of the past in the exact sciences.

Scarcely necessary to a full comprehension of the text, but entertaining all the same to the geek who loves to chase down footnotes are Neugebauer’s chapter-end notes, which provide extensive references to the then-recent literature and digressions on what would appear to be minor points (as of the date of publication in 1957); besides, in places he gives us an inside look into some of the squabbles that take place among scholars. For instance, he tells of how in 1931 he managed to trace down a long-lost and controversial text at the closely guarded collection at the University of Jena, whereupon the university officials informed him that only by mistake had he been admitted to the collection and forbade him to publish his finding; apropos of which Neugebauer tartly remarks, ‘Nevertheless I reserved for myself the privilege of remembering my newly acquired knowledge, and since then my copy of the text has been used by other scholars’. What the author modestly omits to mention is that he was Jewish and within three years would be forced to emigrate from Nazi Germany to America. Another such tale concerns the pioneering work on Babylonian lunar theory by the Peruvians Epping and Strassmaier that appeared from 1881 onwards. Apparently, Strassmaier had been granted access to the collection at the British Museum and, while there, made copies for his own use of a good number of not otherwise archived astronomical texts; he was, in fact, the first to decipher any such texts and to recognize them as astronomical. At Strassmaier’s suggestion, Epping set to work on the former’s copies and, in due course, single-handedly laid the foundation of our understanding of the origins of scientific astronomy, as it arose in Babylon. He was the first to perceive the role of arithmetical progressions in the prediction of lunar phenomena, to an accuracy of a few minutes of arc, what is a most remarkable technical achievement for naked-eye observers that rivals even Tycho Brahe’s. Nevertheless, the British Museum evinced little interest in organizing its collection, to which thousands of tablets were being added every year, in order to retrieve a more complete documentation of Babylonian astronomy. Indeed, it refused ever to publish any of Strassmaier’s texts or to release any information about potentially related texts it may have acquired afterwards. Is it supererogatory to point out at this juncture that Epping and Strassmaier were Catholic priests, the latter a Jesuit? To extrapolate from what one gathers about the American scene, it was not perhaps until the 1960’s that the anti-Catholic prejudice long nursed at elite Ivy League universities began to fall by the wayside, to the extent that it ever has.

[Stephan Frank · Rating 4 out of 5]

(not exactly easy beach reading, but that is in fact where I read it -- New Jersey beach)

One has to keep in mind that this is from a lecture series.
Amazing breadth of information, clearly beyond the scope of what I was ready to study in depth (at the beach), hence I could fly over some of the details without truly digesting them all. Thus, it was a fun, quick read that allowed me to catch glimpses of how research into the mathematical aspects of Astronomy in the Antiquity basically started with Neugebauer et al., and was put apparently on solid footing. Also made me realise that - while somewhat interested in such endeavours - the actual task of an in-depth-study is in the end too stale of a subject for me personally, as a lot of time has to be spent on deciphering methods of essentially tedious book-keeping.

Neugebauer is best, however, when he summarises his lengthy chapters in one or two very poignant sentences towards the end : dry humour at its best, coupled with extreme insightfulness. Sometimes, one wishes he had not spent thirty pages beforehand to lay down what can be said in such condensed versions.

[Michal Paszkiewicz · Rating 5 out of 5]

Excellent book on ancient science, particularly mathematics & astronomy.

Particularly great was the fact it went into lots of detail! And there was a bonus detailed explanation of the Ptolemaic model at the end. Awesome!

[Morgan Lupton · Rating 4 out of 5]

Probably most useful as a reference for understanding ancient mathematics, but there’s a good bit of history in there too.

[Kyle · Rating 5 out of 5]

Neugebauer explains how ancient math and astronomy worked. This is done very well, although I'd argue that he doesn't always make it very clear what he's talking about if you don't already know (his treatment of the Almagest and Ptolemy's map projections take a lot of definitions for granted). In order to understand the explanations of many of the steps, you will have to reread them many times and stare at the figures (just like you have to for any math/physics books with derivations that aren't trivial). Still, the explanations are there, and Neugebauer in the main of the text gives good information about the history of science, his opinions, and I thought his Babylonian math explanations were much clearer than the geometric ones.

Overall, I think it is definitely worth reading just to get a sense of the achievements of ancient cultures, and to know what ancient math and astronomy consisted of.

My one big qualm is that this edition didn't have the plates referred to in the book. I was able to look at some of them online, which was helpful. I wouldn't say the plates are necessary, but they are helpful.

[Bernard English · Rating 3 out of 5]

The chapter entitled "Origins and Transmission of Hellenistic Sciences" was far easier to digest than the chapters on most others. It just felt far less technical and more about history proper rather than deciphering ancient systems and symbols. I thought it a bit too much. Some Youtube videos on celestial mechanics or basic astronomy will be necessary for lots of readers, I suppose. There is an important chapter on the sources which really shows the sad state excavation was in when he wrote the book over half a century ago as well as the challenge of keeping up with all the unearthed artifacts. He points out that after museums spend money for the "removal of many tons of sand and debris," the archeologist would leave exposed rocks making it easy for locals to dig out much else and sell them at high prices--sometimes to the same museums which paid for the dig in the first place. There is a 15-page appendix explaining the Ptolomaic system in detail, which is not easily found in other books.

[Ron Me]

Another mistitle: It should be "Math and Astronomy in European Antiquity". No physics or chemistry. There's a little bit about India from Greek sources, but only derivative, and no mention of China whatsoever. That said, I enjoyed parts of the book, especially when he shows in considerable detail how he went about figuring out the meaning of a cuneiform tablet. In general, though, the book is way too detailed for anyone but a real specialist.

[Karen · Rating 4 out of 5]

This book on Babylonian, Egyptian, and Hellenistic mathematics and astronomy is excellent and very detailed. I'm wondering if newer information has come to light because the text is based on lectures Neugebauer gave in 1949 and updated in 1957. I would hope that more research and decipherment of the ancient texts has been made.

[Kevin Bjorke · Rating 4 out of 5]

Amazing and I would give it give stars EXCEPT.. there's a trio of major holes: Asia, Africa, and America(s). All have legacies of math and astronomy, yet they are 300% ignored.

[O. Thurmond · Rating 4 out of 5]

Ancient Western intersection of math, astronomy, asrology, history, numbers, etc. There was some pretty amazing scientific accomplishments, thought, traditions and explanations which have surprisingly widespread effect even today. (Base 10, base 6, and binary systems, time keeping, etc.) Very cool you can follow it, even showing Cuniform, old babylonian, ptolemy, byzantine, Selucid....If someone matched this up with Ancient Chinese,Indonesian, Africn Incan, Myan, and or Australian Aboriginal math it would be an accomlishment for all the ages. Yes, that was a challenge! Add inmodern computers and WOW!

[Ricardo Vega · Rating 5 out of 5]

A wonderful tease to a monumental work by the author that has been out of print for quite some time. I am still looking for a decent used copy that does not cost and arm and a leg.