1.1. Review by: W E H Berwick.
The Mathematical Gazette 14 (195) (1928), 198.

This volume contains an interesting account of the methods of manipulating rational fractions known to the ancient Egyptians. Several books on this subject have been published in the last twenty years, and the field is now fairly well explored. Mathematical advancement in Egyptian times was hampered by the notation existing then. In arithmetic, the distinction between prime and composite numbers was appreciated by the Greeks only at a later period. The solution of an arithmetical problem by Egyptian methods was rather like that of a present-day puzzle in which the methods permitted to be used are severely restricted by conditions. Until the ordinary decimal notation for numbers came into being about the same time as the invention of printing, progress in arithmetic was necessarily slow. Dr Neugebauer's book contains a useful bibliography of literature on the subject, with several tables requisite for dealing with fractions by Egyptian methods.

2.1. Review by: B L van der Waerden.
Mathematical Reviews MR0016320 (8,1d).

Edition of all mathematical cuneiform texts discovered (chiefly in American collections) since the publication of Neugebauer's Mathematische Keilschrifttexte [Springer, Berlin, 1935]. For all texts a transcription, translation and commentary are given; a number of photographs and copies are added. Among the table texts are two new types: a table of approximate reciprocals of "irregular" numbers, correct to three or four sexagesimals, and a table of exponents of powers of two. Among the problem-texts the most remarkable is one containing a list of seventeen right-angled "Pythagorean triangles" with whole numbers $l, b, d$ as sides, ordered according to decreasing values of $b/l$. The first column gives the square $b^{2}/l^{2}$ or possibly, as the authors suppose, $d^{2}/l^{2}$, the next two columns $b$ and $d$. All numbers are calculated according to the classical formulas $l = 2pq, b = p^{2} - q^{2}, d = p^{2} + q^{2}$, where $p$ and $q$ are simple numbers consisting of factors 2, 3, 5 only. Another text gives a very good approximation to √2, namely, 1; 24, 51, 10. Other texts, dealing with bricks, digging of canals, etc., contain valuable information about metrological questions, wages, dimensions of bricks, etc. Two texts contain lists of coefficients used in diverse applications. Goetze has contributed a chapter on the dialects of the Old Babylonian texts.

2.2. Review by: Albrecht Goetze.
Journal of Cuneiform Studies 2 (1) (1948), 33-37.

Mathematical cuneiform texts have been known for a long time, but their study was sorely neglected. Not without reason. On the one hand, the texts, by their style and conciseness, were not attractive to Assyriologists; on the other hand philologists lacked the mathematical knowledge which is necessary for their intelligent interpretation. Just as the understanding of Babylonian law required jurists with Assyriological training, the recovery of Babylonian mathematics was dependent either on Assyriologists with a flare for mathematics, or on mathematicians patient enough to acquire sufficient Assyriological knowledge. Such a combination is rare indeed. So far it has been achieved only twice: by the late F Thureau-Dangin and by O Neugebauer. It is to these two scholars that we owe the fundamental publications on Babylonian mathematics.

The Mathematical Cuneiform Texts here to be reviewed are a kind of supplement to Neugebauer's Mathematische Keilschrifttexte. The volume presents the material that has come to the authors' attention since the publication of the earlier work. In itself it is extensive enough to be representative of this kind of material; beyond that it also offers some new and even unique types. In the production of the new book Neugebauer was able to avail himself of the patient and competent cooperation and the Assyriological advice of Abraham Sachs. His hand is visible particularly in the treatment of lexicographical questions throughout the volume. The swift development of the new branch of Assyriological studies is evidenced by the series of supplementary studies which Sachs has begun to publish in this Journal.

After a brief introduction comes a chapter on Table-texts. It consists chiefly of a list of the tables of reciprocals, multiplications, squares and square roots, and cube roots that have come to light in various museums, particularly in the University Museum in Philadelphia; the tablet MLC 2078 with logarithms (something new) deserves special mention. The bulk of the volume is devoted to the presentation and interpretation of no less than 39 problem texts, in the main from the collections of Yale University. Most of the texts were written down in the Old-Babylonian period; the reviewer has tried to distribute them locally and to point out the dialectical differences they exhibit. The volume closes with indices among which the Vocabulary is of particular importance to those interested in language. Twenty-three plates of copies are added at the end of the book and a further series of 26 plates presents photographs of most of the tests.

It can hardly be the duty of a philologist to evaluate the significance of the publication for the history of mathematics. Undoubtedly it is quite considerable. The historian of civilisation marvels at the achievements of the ancients. He wonders where the origin of their mathematical insight and technique lies. It is obvious that there must have been a long development before the stream of tradition we possess comes to the surface, in about the second quarter of the second millennium B.C. The terminology the Babylonian mathematicians use suggests that here as elsewhere they were dependent on the Sumerian. But before we can be sure of that, we should like to possess some older texts from the Pre-Akkadian period. So far none has been found.
...
The remarks I have made must not detract from the fact that the book is in every respect of the highest quality. Not only historians of the sciences but also Assyriologists must be grateful to the American Oriental Society and the American Schools of Oriental Research, which have published the volume, and to the American Council of Learned Societies and the Mathematical Association of America which have made subventions towards its printing.

2.3. Review by: A Leo Oppenheim.
Journal of Near Eastern Studies 6 (2) (1947), 126-128.

Since about 1934, after long decennia of neglect and careless treatment, the mathematical texts in cuneiform writing have become the subject of much systematic research work done by a series of scholars, mathematicians as well as philologists. The present book, which contains an impressive amount of tablets from American museums (together with two from the British Museum and one tablet from Berlin), represents a new and important step forward with its new material and its equally relevant new interpretations of already published texts.

Written by a mathematician (O Neugebauer) and by an Assyriologist (A Sachs), the book should, of course, be reviewed only by a similar team. Since this is not practicable, the Assyriologist's use of this impressive publication is necessarily restricted to the somewhat melancholy task of turning over leaf after leaf full of strange formulas, forbidding-sounding mathematical terms, and argumentations which patently bespeak the high standing of Mesopotamian mathematics and the difficulties which are involved in the understanding of the manifold "problem" texts. This task is melancholy because the reviewer fully realises that he thus misses an important aspect of the Sumero-Akkadian civilisation - an aspect which is at least as revealing as, for instance, that offered by its artistic activities.

The philologist, and especially one who is interested in lexicography, cannot help being fascinated by the highly technical and difficult vocabulary of these texts. Praise and admiration are due to Dr Sachs for his work; it shows his fine philological training as well as the exactness and sobriety of his methods of interpretation.

2.4. Review by: L G Simons.
The American Mathematical Monthly 53 (7) (1946), 391-392.

Many of us who have had the privilege of developing courses in the history of mathematics live over again, with the appearance of this volume, the thrilling experiences that were ours through the revelations of Babylonian achievements in mathematics made by Professor Neugebauer and set forth in his Mathematische Keilschrift-Texte, and several articles. What delight we had in the solution of the quadratic equation with the positive sign before the radical as taken from a tablet of about 2000 B.C., and from other seemingly incredible items! We have here, then, the continuation of that work in one of inestimable value to the history of mathematics.

The preface tells us that "this volume is devoted to the edition of hitherto unpublished mathematical texts, chiefly from American collections. Not only does this new group of documents supplement the previously published material in many respects, but is itself sufficiently extensive to offer a fair impression of the main types of Babylonian mathematical texts. New frontiers, the existence of which had been no more than suspected are actually reached by a tablet (Plimpton 322) involving 'Pythagorean' Number Theory." What more exciting adventure could an explorer embark upon than this one! And the authors made discoveries which have greatly enriched the world in a scholarly and intellectual sense.

From this same preface comes the gratifying knowledge concerning "The Yale Babylonian Collection the source of the largest part of the material presented here" that: "More mathematical problem-texts have now been published from this collection than from any other museum in the world." All of this shows to what extent this country has come to the front in the matter of cuneiform texts.

An additional feature of great value is the contribution to this volume by Dr A Goetze "of his important chapter on the dialects of the Old-Babylonian Akkadian mathematical texts." The preface states that: "In addition to numerous table-texts, most of the problem-texts published in this book were discovered by Dr A Goetze."

Without making any hollow pretence to reading the whole book, much may be gleaned by reading all that is set forth so well in general form. The chapters are five in number. Chapter I Introduction gives The Texts; The Sexagesimal Number System; Translation and Transcription; Metrology; Examples of Metrological Calculations. Chapter II Table-Texts gives Introduction (which tells us that: "The Mathematical tables by means of which all numerical calculations were carried out, played a basic role in the high development of Old-Babylonian Mathematics and of Babylonian astronomy in the Seleucid period"). Reciprocals; Multiplication Tables; Squares, Cubes and Varia; (under the last, surprisingly enough, occurs "Logarithms" since Tablets which contain tables of exponents $a^{n}$ where $n$ is an integer between 2 and 10 and a is one of the numbers 9, 16, 1,40, 3, 3,45 are known.") Chapter III Problem-Texts: This chapter contains a wealth of material for study and reference under the headings: Introduction; Pythagorean Numbers; Cube Root; Geometrical Problems. The second of these (Plimpton 322) is of outstanding interest. It tabulates the answers to a problem containing Pythagorean numbers (or Pythagorean triangles). "It is the oldest preserved document in ancient number theory" (italics mine). We learn that this tablet falls between 1900 and 1600 B.C. A review is no place for a complete treatment of a text but the elucidation of this one makes a fascinating study. Under "Historical Consequences," it is pointed out that: "We now have a text of purely number theoretical character treating a problem organically developed from other problems already well known..." Chapter IV The Akkadian Dialects of the Old-Babylonian Mathematical Texts, by A Goetze. Chapter V Indices covers Bibliography and Abbreviations; Concordance of Museum Numbers; Vocabulary; Subject Index; Map showing Ancient Sites which are mentioned. The utter completeness and meticulous care of the entire work must be noted. As a climax and crowning feature, there are forty-nine Plates which present all the texts whose descriptions are so perfectly presented.

We learn that the: "Greatest amount of new information about practical questions is yielded by texts dealing with bricks. Here we learn for the first time the dimensions of several standard types of bricks and the metrological system used in counting bricks. It is clear that such information is not only of use for the better understanding of similar mathematical texts, but will also influence our interpretation of economic documents, and archaeological data." And most interesting is the fact that: "For the first time we have texts which have the character of pages from a general handbook."

"It lifts the heart," to use a phrase coined by J B Priestley, to live with a piece of scholarship like this one even though one brings to the living very little ability to follow its details.

2.5. Review by: E J Dijksterhuis.
Isis 37 (1/2) (1947), 96-97.

The important publications of O. Neugebauer (especially the Mathematische Keilschrift Texte, 1935) on which our actual knowledge of Babylonian mathematics is chiefly based, are supplemented in a most welcome way by the volume mentioned above, which contains a number of hitherto unpublished mathematical texts, mainly from American collections. The new volume presents all the features of philological exactitude and penetrating mathematical insight which are so highly characteristic of all Dr Neugebauer's publications. As to the contents, it not only furnishes many important corroborations to conclusions drawn already from the previously published material, but it is in itself sufficiently extensive to give a fair impression of the whole trend of Babylonian mathematics. Moreover one of the documents reveals a type of arithmetical reasoning the existence of which had hitherto been no more than presumed.

As in MKT, the texts now published can be grouped under two main headings: table-texts and problem-texts, the former of which furnish the indispensable auxiliaries for practical computation, while the latter are somewhat of the type of the collections of problems in our mathematical school books; evidently they were used in mathematical teaching.

For the dates of the great majority of the table-texts no narrower time-limits can be given than 1800 and 300 B.C. Only in some cases did a more precise determination of date prove possible. The problem-texts are mainly Old-Babylonian, i.e., they date from the centuries around 1700 B.C., but none is older than 1800 B.C. As to the provenance a rather large number of texts are from Nippur or Kiš, while the authors confess to be almost completely in the dark about the origin of the problem-texts. It was only possible to distinguish on dialectic differences between northern and southern Old-Babylonian Akkadian texts; a very valuable aid in this investigation was provided by the chapter on the dialects of the Old-Babylonian Akkadian mathematical texts which was contributed to the volume by Dr Goetze. Two of the younger texts (to be dated in their present form to the end of the eighth or the beginning of the seventh century B.C.) were found at Niniveh.
...
On reading the translations given by the authors and the valuable commentaries accompanying them, one is deeply impressed by the brilliant way in which the numerous difficulties in the reading and interpretation of the tablets have been overcome. These difficulties are of several different kinds: first there is the bad state of preservation of many tablets, then the extreme minuteness of the writing, the absence of the oral interpretations which were no doubt given to the scribes by their teachers, the enigmatical character of several words which are evidently used in a special technical sense. Sometimes the interpretation was only made possible by previous experience of the authors on analogous problems; in other cases they were compelled to postpone the translation in expectation of new material.

Everyone who is interested in the fascinating subject of Babylonian mathematics should be thankful to the authors for the excellent fulfilment of their arduous task.

3.1. Review by: A W Gomme.
The Classical Review 63 (3/4) (1949), 120-122.

An adequate review of this important book would require not only a know ledge of ancient astronomy equal to Dr Neugebauer's and of the evidence for the nature of the Attic calendar equal to Dr Pritchett's (neither of which is to be found in more than half a dozen scholars all told), but a re-examination of much of that evidence on the stones themselves. I will only attempt here to state their main conclusions and the methods by which they have reached them.

The two immediately most important conclusions are, I think, these first, that Aristotle's statement that the first four prytanies of the year consisted each of 36 days and the last six of 35 (with the same principle, we must suppose, applied, mutatis mutandis, to intercalary years) cannot be ignored, as recent studies have ignored it, and that the evidence for his day, 341/0 to 307/6 B.C., is consistent with it; and that the evidence for the periods of the twelve phylae and the thirteen phylae is also consistent with that principle, and that it may well have applied as well to those years of the fifth century when the boule and its prytanies held office for a solar year. Secondly, the authors maintain, it is not necessary to suppose that the Athenians ever carried their love of a complicated calendar so far as to use both backward count and forward count of days in the last third of the month, at the same time and in the same language; and that the evidence is consistent with the view that the backward count only was used. If they have proved their case, the result is obviously of great value, for it takes contemporary evidence into account, and provides a simpler solution than the old; and they are well aware of the difficulties of their task. The first and most obvious of these is the paucity of the evidence: the lacunae in the inscriptions are many, and even when (as often) the number of letter-spaces in the missing part is known or can be reasonably conjectured, we are faced with several possibilities of restoration; there were variations in spelling and still more in writing, two letters in one letter-space and spaces left vacant, and so forth; and occasionally an error by the stone-cutter (or by the writer of his 'copy') must be assumed in order to restore.

Examples enough of all these things can be found for certain on inscriptions; but they make it disturbingly easy to restore according to a given theory. These difficulties (or should we call them facilities?) are to be found in the way of any theory; but Pritchett and Neugebauer have a new one as well, or rather, they lay more stress on an old one than their immediate predecessors have done; and this is a third important novelty in this book. It is widely known that on certain inscriptions, of the second century B.C., we have, besides the prytany date, two monthly dates. The authors note that the former is normally in agreement with the proper prytany date (which shows that prytany periods did not vary) and that it is always in advance of the latter; that is, that some days have been intercalated by the archon, though the civil year was not thereby lengthened, days being dropped, it must be supposed, in Scirophorion, to ensure that the new year began with a new moon on Hecatombaion ist. They go on from this to the natural assumption that on all our other inscriptions the monthly date is a date, and that intercalation of days was common and not determined by any system. This gives great flexibility to their argument; for an equation of a monthly date with a prytany date, which at first sight appears either to support a different theory (e.g. the forward count, or that Aristotle is wrong, or that a particular year is intercalary), or to be a mistake, may now, by the assumption of a previous intercalation of days in a civil month, be made consistent with their own. There is nothing improper in this: intercalation of days is testified for certain, and we know from Aristophanes that the civil calendar was liable to get out of line with the moon. The probability that this happened fairly often, and would be reflected in the calendar-equations of the inscriptions, is rightly taken into account; still, it adds to the 'facilities'. The only gap in the authors' argument that I find is some explanation of the reason for these intercalations of a few days here and there; for a priori one would have expected them to be made in order to correct anomalies due to faulty astronomical observation rather than to create them; and that it would have mattered less if the prytanies (in the period of the twelve phylae) rather than the civil and religious months had got out of line with astronomy. The juggling with the calendar in order to please Demetrius Poliorcetes is no sort of parallel.

There is much else of interest and importance in this book ...

3.2. Review by: A G Woodhead.
American Journal of Archaeology 53 (3) (1949), 322-323.

The complex nature of the study of Greek calendar problems is all too familiar to scholars: such are the complexities, indeed, that, after any break in continuous study of them, the student will usually find it necessary to review all the ground previously won before he can presume to go further. This the authors of The Calendars of Athens have recognised, and their method of presenting and developing their thesis is gratifyingly lucid. They state their main problems and solutions at the outset, and thus provide the reader with a torch to light him through the necessarily involved paths of proof. They give clear definitions of their terminology and one is grateful that they have not disdained a table of the Attic months, usually taken for granted, but so easily forgotten.
...
Calendars of Athens is a stimulating and well-reasoned study. Its appeal is naturally limited, but among the small band of epigraphists and students of the Greek calendar it will at once assume and will long maintain, a deserved position as a work indispensable for study and reference. And both in the clarity of its method and in the validity of its general conclusions it has much from which any classical scholar would be well advised to profit.

3.3. Review by: William Bell Dinsmoor.
The American Historical Review 54 (2) (1949), 333-337.

The history of the great days of ancient Athens was recorded in the pages of the great historians, Herodotus, Thucydides, and Xenophon; but the contemporary administrative, financial, and commemorative documents inscribed on marble, while fairly numerous (1,086 included in Vol. I of the Corpus of Attic inscriptions), are few in proportion to those of later times. The story of the gradual decline of Athens after the defeat by Sparta was written by numerous historians of lesser repute, most of whose works, neglected by the medieval copyists, have either been lost or survive only in fragmentary quotations; but, in partial compensation, the proportion of contemporary documents vastly increases (13,247 included in Vols. II-III of the Corpus, completed in 1940). The present American excavations in the Agora alone have added 6,100 items, very few of which were absorbed in the Corpus. It is upon the interpretation of these broken marble documents, therefore, that we must chiefly rely for the details of Athenian history after 404 B.C.

One of the essential processes of interpreting such documents is the establishment of their dates and sequence; and for this, apart from internal (and often elusive) characteristics of handwriting and changes of formula, our chief reliance must be placed, at least in the case of administrative documents, upon the names of the successive archons and secretaries, the officers who sanctioned them. If lists of these officers had survived from antiquity our task would have been easier; but the list of archons by Demetrius of Phaleron down to his own archonship in 308 B.C., and the list in the annals of Philochorus down to about 260 B.C., are both entirely lost, while the annalistic list by Diodorus of Sicily, which evidently once descended to 54 B.C., now breaks off with 301 B.C. A compilation of all our written evidence, however, made it possible to replace part of our loss with an almost unbroken list from 511 to 293 B.C. From this point onward everything was chaotic until W. S. Ferguson's discovery of the law of succession of the secretaries, which bears his name, enabled him and his followers, since 1898, to carry the list through the later centuries. But this research is still far from complete; and a chance discovery of a new inscription often results in considerable readjustment.
...
The work now before us is the joint product of a collaborative investigation between a student trained by Meritt in Greek epigraphy (Kendrick Pritchett) and an expert in chronological problems of the ancient Near East (Otto Neugebauer). For a study of this sort, such collaboration would seem to offer great promise. As the cornerstone for their investigation, however, they have employed an entirely revolutionary principle: the prytany half of the equation is the fixed standard upon which we must build. They base this upon a somewhat general statement of Aristotle - possibly a stenographer's interpretation ...
...
The book is full of interesting material and will certainly inspire renewed examination of the documentary evidence; though many of its conclusions, to the reviewer's mind at least, should be employed with caution. But we must be grateful to the authors for their insistence that, when backward count of the days "after twenty" had been definitely established (in the reviewer's opinion not until 306 B.C.), it was consistently observed without the intermittent vacillation to forward count which has been permitted in all other recent studies.

Review by: Malcolm F McGregor.
The American Journal of Philology 70 (4) (1949), 422-425.

The last generation has witnessed a series of thoughtful studies which have carried our knowledge of the Athenian calendar far beyond what earlier scholars might have dreamed was possible. For the fruitfulness of the results, which aim at providing for Greek history a sound chronological basis in general and a sharper perspective in its particulars, we are indebted primarily to the inscriptions, especially to the rich yield of the Athenian agora, and to the persistent and ingenious work of such men as Meritt, West, Ferguson, Dinsmoor, Pritchett, and Dow. Now, at just the right time, comes an important book by Pritchett and Neugebauer which takes stock of progress, evaluates method, and issues a few needed words of caution to the epigraphists, whose dexterity sometimes carries them beyond the credible limits of the evidence.

Although the authors admit the double datings as their point of departure, they were soon led to survey all the epigraphic material bearing on the calendar from the fifth to the second centuries B. C. and thus to examine the now familiar puzzles, e.g., intercalation, length of prytanies, calendar cycles, backward or forward count. What has obviously impressed them most is the correctness of Aristotle's statement concerning the rigid prytany calendar.

A major virtue of this book is that the problems are clearly put and the terminology is precisely defined. As the authors recognise, all do not belong to the small circle of chronological experts; yet most Greek historians are interested in the difficulties and the methods. The first chapter (Problems of the Athenian Calendar) will be instructive to the experts as well as to others, for here the leading arguments are anticipated without the usual assumption that the reader is intimate with the evidence and with the previous literature. So, for example, the nature of a lunar (astronomical) calendar is described as are the character and purpose of cycles constructed by a system of intercalation. The same principle is adhered to throughout the volume. An illustration is the table of Athenian monetary symbols presented in note 10, which, elementary as it may be to epigraphists, will earn praise from non-specialists; cf. the useful list of the days of the Athenian month, with the illuminating Notes.
...
This study is carried out in an expert manner and with scrupulous honesty. The epigraphic responsibility is Pritchett's, the astronomical contributions are Neugebauer's. Both are scholars of reputation and this impressive performance will enhance both reputations.

3.4. Review by: G E M.
Archaeology 1 (4) (1948), 228.

The problems presented by the Athenian calendar of the fifth, fourth, third and second centuries have attracted the attention of our leading scholars and became instrumental for the appearance of the monumental publications of Ferguson, Dinsmoor and Meritt on the subject. To those studies is now added a worthy addition. Our volume is addressed to the specialist, and the average reader who has a general interest in archaeology will find it hard to follow and understand the close-knit arguments, based upon a re-examination of the available evidence including the inscriptions brought to light by American scholars in the Agora of Athens.

Our authors challenge the accepted theories advanced to explain the peculiarities of the Athenian calendar and maintain that Aristotle's statement in the Constitution of Athens provides "the only authoritative basis for the calculation of the sequence of days in any given year at the time of the composition of that work."

The evidence is presented in an exemplary fashion and the whole study does credit to the problem. Whether we agree with the authors or not, we have to admit that their work will prove of the utmost importance to the archaeologist and the historian.

4.1. Review by: E J Dijksterhuis.
Mathematical Reviews MR0046956 (13,809a).

This book originated from six lectures delivered by the author at Cornell University in 1949 and consequently does not aim at an exhaustive discussion of the vast subject. The main emphasis is laid on mathematics and astronomy in Babylonia and Egypt in their relationship to Hellenistic science. Chapter I deals with the writing of numbers in antiquity and with some elementary arithmetic, Chapter II with Babylonian mathematics. In Chapter III the technical treatment is interrupted by a discussion of the way in which the source material for the investigation of ancient science is brought to light and made available to scholars, and of the dangers it is exposed to. Chapter IV resumes the thread of the exposition with a concise sketch of Egyptian mathematics and astronomy. In Chapter V the author enters more closely into a discussion of Babylonian astronomy. In Chapter VI the puzzling question of the origin and transmission of Hellenistic science is tackled. To each chapter a bibliography and a most valuable collection of notes and references is annexed.

4.2. Review by: Raymond J Seeger.
Science, New Series 117 (3036) (1953), 257-258.

Despite all the lip service given nowadays to general education, rarely is science assigned more than a minor technical role of "information, please." Any integration of science with culture is supposedly the responsibility of self-styled humanists, who rely primarily upon the scholarship of other humanists. It is not surprising, therefore, that the historical interrelationships between science and civilisation are somewhat distorted. What is needed as a basis for any generalisations are researches by scientifically trained historians and/or by historically trained scientists. Otto Neugebauer belongs to this class. As he grate fully remarks in the preface, with respect to its dedication to Richard Courant: "I owe him the experience of being introduced to modern mathematics and physics as a part of intellectual endeavour, never isolated from each other nor from any other field of civilisation."

The present book, a modified form of the author's 1949 Cornell University "Messenger Lectures on the Evolution of Civilization," is a semi-popular, scholarly account of mathematics and astronomy in Babylonia and Egypt in their relationship to Hellenistic science. It is based upon the author's belief that "The investigation of the transmission of mathematics and astronomy is one of the most powerful tools for the establishment of relations between different civilisations." The author modestly concludes his account with the remark: "Perhaps it is vain to hope for anything more than a picture which is pleasing to the constructive mind when we try to restore the past."

After a review of the early history of number symbols, the author discusses the characteristic features of mathematics in the Old Babylonian period of the Hammurabi dynasty. To an amateur, such as myself, nurtured upon classical tradition, it is startling to learn of the highly developed numerical skills utilised at this time. Tables still exist containing squares and square roots, cubes and cube roots, and sums of squares and cubes. Special types of cubic equations were solved; particular exponential functions (for the computation of compound interest) were used; arithmetical progression was known. From a Seleucid text one finds "the correct application of the 'quadratic' formula for the solution of quadratic equations." Their computed value of 1.414213 (actually 1.414214) for the square root of 2 was still used by Ptolemy. In connection with such numerical work, "The determination of the diagonal of the square from its side is sufficient proof that the Pythagorean theorem was known more than a thousand years before Pythagoras." Even the "fundamental formulas for the construction of triples of Pythagorean numbers were known. Geometrical concepts play a very secondary part in Babylonian algebra."

After this fascinating revelation of "a level of mathematical development which can in many aspects be compared with the mathematics, say, of the early Renaissance," it is somewhat of a let-down to read about the status of early Egyptian mathematics and astronomy. For example, "Egyptian mathematics did not contribute positively to the development of mathematics." One of the major results, however, was a "deeper insight into the development of computation with fractions." The whole process was entirely additive. In the case of astronomy there is apparently only one very beneficial influence - namely, a calendar with a fixed time scale and no intercalations, which became the standard astronomical system of reference through the Middle Ages. "This calendar, indeed, is the only intelligent calendar which ever existed in human history." Incidentally, one "Egyptian contribution to astronomy is the twelve divisions of daytime and of night." Noteworthy by its omission in the text proper is any reference to the astronomical or mathematical significance of the Pyramids. The author concludes this lecture with the interesting judgment that "Ancient science was the product of a very few men; and these few happened not to be Egyptians."

After this interlude we find ourselves searching for clues to ferret out the mysteries of Babylonian astronomy. Right at the start we are emphatically warned that "mathematical theory played the major role in Babylonian astronomy as compared with the very modest role of observations, whose legendary accuracy also appeared more and more to be a myth." The Babylonians, of course, were primarily interested in lunar, solar, and planetary phenomena close to the horizon. We are reminded that sandstorms frequently obscure the desert horizon so that "the almost proverbial brilliance of the Babylonian sky is more a literary cliché than an actual fact." Eclipses and occultations, on the other hand, are usually observable under more favourable conditions. Hence, "Ptolemy states that practically complete lists of eclipses are available since the reign of Nabonassar (747 B.C.), while he complains about the lack of reliable planetary observations. Not a single text is known which could be called a wholly observational record.

We know so little about the underlying empirical material which was so skilfully applied to provide the basic parameters of a real mathematical theory." Incidentally, the zodiac (first mentioned in a Babylonian text of 419 B.C.) was invented to assist in the description of celestial motions. "Arithmetical progressions were skilfully utilized for the prediction of lunar phenomena with an accuracy of a few minutes." Babylonian astronomy was fully developed at about 300 B.C.

The last chapter, on the "Origin and Transmission of Hellenistic Sciences," is a natural climax for this challenging story. By this time we are conditioned to expect something like the following:

If modern scholars had devoted as much attention to Galen or Ptolemy, they would have come to quite different results about the remarkable quality of the so-called Greek mind to develop scientific theories without resorting to experimental or empirical tests... Plato's role has been widely exaggerated. His advice to astronomers to replace observations by speculation would have destroyed one of the most important contributions of the Greeks to the exact sciences. ...

On the other hand, "the traditional stories of discoveries made by Thales or Pythagoras must be discarded as totally unhistorical."

Professor Neugebauer cites evidence for his conclusion that the mathematics of the Hellenistic period is part of an unbroken tradition from earliest ancient history to modern times. On the other hand, "The Elements of Euclid concern, with very few exceptions, a purely Greek development in a sharply defined direction." The axiomatic style of Eudoxios is to be sharply differentiated from that of Ionia and of southern Italy. Nor can credence be given to anyone claiming "repeated land measurements responsible for geometry;" it is "completely impossible to test any such hypothesis." In Hero's later degenerate geometry, indeed, one finds a reflection of the arithmetical or algebraic tradition of Mesopotamia.

The history of Greek astronomy presents a more involved problem than the history of mathematics, with its unique contribution over a relatively short period. For example, "there existed linear methods' of far wider extent than one could possibly have deduced from the silence of Ptolemy and his commentators." Furthermore, "essential parameters ascribed by Ptolemy to Hipparchus are identical with the corresponding parameters of the Babylonian theory." Hipparchus, indeed, used both geometric and arithmetic (linear) methods. The latter were particularly used also by astrological authors for horoscopes. Hence one finds "astrology an exceedingly helpful tool for the transmission of Hellenistic thought." The Hindu and Babylonian contact, moreover, has been made primarily through the Greeks. Accordingly, "we stand today at the beginning of a systematic investigation of the relations between Hindu and Babylonian astronomy, an investigation which is bound to give us greatly deepened insight into the origin of both fields."

One of the small pleasures I personally derived from this stimulating book was the explanation of the arrangement of the Greek planetary week, which we still use today. It is "totally misleading when this order is called Chaldean in modern literature." Something new about something old! I strongly recommend this important summary to every scientist, particularly mathematical and physical scientists, and to every so-called humanist, particularly historians and philosophers. The excellent bibliography, notes, and references are instructive for mature specialists.

There are, of course, minor blots on this excellent record - for example, the spelling of Greek names. I felt somewhat unhappy, too, about the chronological table at the end. To be sure, "dates are only approximate." But why 1670 for Newton? What is the basis of the approximation? My major critical remark concerns the title itself. What are "the exact sciences in antiquity" or "the modern exact sciences" mentioned in the text? Are mathematics and astronomy to be regarded as a special single category of the sciences? As a physicist I would merely note the following predominant features: logic for mathematics, observations for astronomy, and experiments for physics. Webster's dictionary cites the phrase the "exact sciences" as an example of a usage of the word "exact" for denoting "capable of great nicety, especially in measurements." Perhaps this meaning might be applicable to some branches of physics, but not to most of astronomy, and certainly not at all to mathematics.

In the last instance one might substitute an alternate dictionary meaning, namely, rigorous - a fighting word among modern mathematicians. I would personally prefer to give up this outmoded terminology.

4.3. Review by: R W Sloley.
The Journal of Egyptian Archaeology 39 (1953), 126-127.

In this book, amplifying six lectures delivered at Cornell University in 1949, Professor Neugebauer gives a valuable survey of the historical interrelationship between mathematics and astronomy in ancient civilisations. The main emphasis is on mathematics and astronomy in Babylonia and Egypt (of which excellent summaries are given) in their relationship to the science of the Hellenistic period - the period following the Alexandrian conquests of the ancient sites of oriental civilisations. During this period a form of science developed which later spread over an area reaching from India to Western Europe and was dominant until the creation of modern science in the time of Newton. In the development of this science astronomy played a very important part.

Most valuable for the student are the chapters on 'The Sources and their Evaluation' and 'The Origin and Transmission of Hellenistic Science'. We are shown the oriental background of the mathematics and science of the Greeks and the influence of Babylonia and links with India are emphasised. Two widely separated types of Greek mathematics must be distinguished - one, represented by the strictly logical approach of Euclid, Archimedes, and others: the second type is part of general Hellenistic mathematics, the roots of which lie in the Babylonian and Egyptian procedures.

There are two clearly marked periods in Babylonia - the 'Old Babylonian' (c. 1800 to 1600 B.C.) during which mathematics reached the highest level ever attained in Babylonia, and the 'Seleucid' datable to the last three centuries B.C., when the only essential progress made was the introduction of a sign for zero. Early Babylonian astronomy was crude and merely qualitative on a par with contemporary Egyptian astronomy - but texts from the 'Seleucid' period are based on a consistent mathematical theory of lunar and planetary motion.

A direct survival of Babylonian method is seen in a problem of mathematical geography expressing the latitude of a locality by means of the ratio of the longest to the shortest daylight for the region in question. In the theory of lunar motion a Greek papyrus of purely mathematical character is based on a Babylonian method but adjusted to the Egyptian calendar.

The author points out that the relatively primitive level of mathematical knowledge in ancient Egypt makes it possible to investigate a state of development which is no longer available in so simple a form except in Egyptian documents. The whole procedure was essentially 'additive', based on simple counting. Multiplication was performed by breaking up one factor into a series of duplications - the same principle is employed in modern computing machines. Some original and interesting comments on the methods of handling the 'unit-fractions' are given. Such fractions influenced the Roman administrative offices and thence spread through the Roman empire. In Ptolemy's Almagest final results are often expressed in these fractions. They are occasionally used to this day in stock exchange quotations in Cairo, e.g., for.

The 365-day Egyptian calendar - 'the only intelligible calendar which ever existed in human history' - became the standard astronomical system of reference. It was kept alive throughout the Middle Ages and used in the time of Copernicus. Another Egyptian contribution to astronomy is the 12-division of daytime and night which we still use. An astronomical concept of real Egyptian origin is that of the 'decans' and it is suggested that the decans did not form a closed ring on the heavens, but that a decan may represent any constellation rising heliacally during an interval of ten days (cf. the paranatellonta of the Greeks). On this assumption, however, it is not easy to explain the diagrams of the Cenotaph of Sethos I; but it should be noted that the author in a private communication to the writer claims to have successfully resolved the difficulty. The diagrams in the tomb of Senenmut show two stages of design. Faint traces in blue of an earlier arrangement are visible and indicate that artistic principles largely governed the arrangement of the scenes. Thus it seems a hopeless task to attempt to identify the star groups depicted with the modern system of constellations.

A major incentive for the study of astronomy was the attempt to achieve some regularity in the intercalations of the lunar calendar. Astronomy did not originate in astrology as has so often been stated, but the widespread belief in astrology, as the one science which gave insight into the causes of events on earth, influenced the transmission of astronomical knowledge from one nation to another. Astrological documents in Mesopotamia belong to the Seleucid period and their number is insignificant compared with that of the astronomical texts. In Egypt the earliest horoscopes, Demotic and Greek, are from the time of Augustus.

The author illustrates the difficulties which beset the investigator in the field to which he has devoted himself for many years with remarkable success. Many editions of the classical authorities are untrustworthy or incomplete and an enormous amount of material in the form of cuneiform tablets is still unpublished and even unexamined. There is no reliable edition of Ptolemy's Geography one of the most influential books of antiquity and as yet we know practically nothing of the history of the zodiacal and planetary symbols. A timely warning is given to those attracted by pan-Babylonian theories which still exercise a baneful influence in the literature.

Professor Neugebauer pays a deservedly high tribute to Sir Harold Bell's 'Egypt from Alexander the Great to the Arab Conquest' (Oxford, 1948) not only as a summary of the history and methods of papyrology, but as a brilliant study of the diffusion and decay of Hellenism, the general problem, of which one facet is the subject of this book.

It is satisfactory to learn that complete editions of all available cuneiform and Egyptian astronomical texts are in course of preparation and will shortly be available for students.

4.4. Review by: Francis J Carmody.
Isis 43 (1) (1952), 73

Dr Neugebauer has revised his series of lectures given at Cornell in 1949, adding technical notes and demonstrations. Presented thus in lecture form, the learned aspects of the book are unobtrusive and the dominant tone is one of vulgarisation; the endless hours of labour devoted to deciphering Babylonian texts have yielded important yet simple results. Thus two goals are served, a useful introduction to the subject matter, one which will be of value to less experienced historians of science, and a document constructed soundly about authoritative research.

The presentation is strictly of the kind needed in the history of science: the linguistic problem is explained and satisfied, and the mathematical detail clearly exposed for what it was. We can follow a chronological progression of technical events without worries about fundamental truths or precursors; in short, the thought is wholly objective within full consciousness of the subsequent developments. In this sense, Dr Neugebauer's book is more authentic than those histories of mathematics that admire ancient quaintness and look for the roots of modern developments. Dr Neugebauer can speak for instance of non-eccentric systems of epicycles and of ancient astrology for their place in the history of human thinking: with geometric devices at man's command, he may prefer simple arithmetic and through the latter locate exact answers. The immense labour of computation done by Ptolemy was wasteful in time, but none better can be imagined.

Dr Neugebauer explores the mysteries of the manipulation of numbers, underlining the importance of arithmetic, for example in problems of lunar motion and their expression in symbols, ciphers, fractions and their reciprocals, the latter strongly algebraic. The sexagesimal system is presented at the start as the basic practice of Babylonian calculations. Chapter 3 discusses the difficulties encountered if one would interrelate Hellenistic and other systems. An important note on this topic seems to me practically hidden. The Egyptian contribution was simple and static yet far-reaching, for it fixed multiple year-cycles for calendar systems and innovated in the presentation of fractions. The Babylonians developed the various lunar cycles in great detail. Dr Neugebauer could have done further service along these lines by presenting the pertinent formulas for date - era chronologies, an elucidation of the material presented by the Arabs, al-Farghânî, al-Battânî and az-Zarqâlî; his remarks on mediaeval astronomical tables are important, someone must explore them in detail and study the methods used in setting the given time and angle values.

The author deals with Hindu influence as one aspect of the transmission of Hellenistic science. The end result of Hindu influence is found in the works of the Arab astronomers and astrologers from about 840 on. Thus Abû Mashar, using Hindu material, was translated into Greek and Latin and thus perpetuated many ancient doctrines and practices. This topic has been treated by a number of scholars (Steinschneider, Boll, etc.) in studies known to Dr Neugebauer; new and conclusive precisions are here made; but, given the importance of the topic, more might well have been said, even by way of mere repetition, especially since the author feels so rightly that astrological texts have been unjustly neglected. Dr Neugebauer mentions the problems and names the outstanding modern books which have shown their rich possibilities, but he says too little about these books. Of Cumont's L'Egypte des astrologues he could easily have said (the detail bears repetition) that it reveals an immense store of ancient usages and customs. Of Honigmann's Die sieben Klimata he could have added that the great astrologers, Abû Mashar, al-Qabi'şî and Alî ibn abi'r-Rijal record long lists of place names arranged by climates and which reveal geographical knowledge of determinable date, end results of certain stages in Arabic explorations, expansion and map making. My regrets indicate that I agree completely with Dr Neugebauer's methods and always hope for more than space allows a man.

4.5. Review by: William H Stahl.
The Classical Weekly 46 (3) (1952), 44.

This volume is not, as the reader might assume from the title, a systematic treatment of the exact sciences throughout antiquity. What little attention is given to Greek scientists is largely for the purpose of demonstrating Mesopotamian influence upon them. Nevertheless it is an important book. The author, a leading authority on Babylonian and Egyptian science, presents a survey of the development of our knowledge of those fields, material previously available in English only in scattered periodicals and reference works.

Of particular interest are Dr Neugebauer's remarks about the desperate shortage of trained Assyriologists and the consequent inadequacies in handling the vast stores of clay tablets that have come into museums from excavations during the past century. The bulk of the texts remain untranslated; many of them have never been catalogued. Provenience is of primary importance in establishing the worth of an archaeological document, but excavation and accession records are so faulty or completely lacking that Dr Neugebauer does not know of a single mathematical or astronomical tablet whose provenience was ascertained therefrom.

The illustrations are excellent and the analytical bibliography is very helpful.

4.6. Review by: Marshall Clagett.
The American Historical Review 58 (1) (1952), 80-81.

In this day of half-digested syntheses and ill-conceived sociological investigations into the history of science, it is a relief to hear our most productive historian of antique astronomy and mathematics assert that he is "exceedingly sceptical of any attempt to reach a 'synthesis' whatever this term may mean." For Neugebauer "specialisation is the only basis of sound knowledge."

This volume bears out his philosophy of history. It does not pretend to be an overall history of the exact sciences in antiquity - in spite of its title. Rather it is a carefully selected and prepared group of essays on Egyptian and Babylonian mathematics and astronomy, with a short added discussion of some problems of Greek mathematics and astronomy. Insight is gained into the subject by means of the "problem" method. This method consists in centring the discussion on some given document or fragment of a document and analysing it in great detail. Neugebauer by this method of analysis is not only able to introduce the reader to the kinds of sources but he is able to draw general conclusions on the nature of the topic under discussion.

The first chapter describes the highly general Babylonian system of numeration - sexagesimal in nature, with its all-important place-value notation. The next essay deals with the more important features of Babylonian algebra, as well as with the widely used tables of reciprocals, and other similar topics. This discussion of Babylonian mathematics is rounded out in the third chapter with a lesson on the decipherment of a single mathematical tablet, with the philological and mathematical conclusions to be derived from such a decipherment.

From Babylonian mathematics Neugebauer turns in the next chapter to Egyptian mathematics and astronomy. The additive nature of Egyptian mathematics is illustrated by examples of multiplication and division. The crude concepts and procedures of Egyptian fractions are discussed; and the celebrated "table of twos" is analysed. Egyptian astronomy is characterised as being on a very elementary level so far as mathematical procedures are concerned, although the Egyptian official calendar is described as "the only intelligent calendar which ever existed in human history."

Chapter VI is the master essay in this volume. It is, in my opinion, the best forty pages on the difficult subject of Babylonian astronomy in any language. It makes a fitting introduction to Neugebauer's papers on this subject and to the classical works of Kugler and his predecessors. It deals mainly with Babylonian lunar theory as illustrated by an analysis of the various columns of a lunar ephemeris. From this analysis we get an excellent view of the rather remarkable mathematical procedures developed to reduce complex functions to a series of linear approximations.

A final chapter on the origins and transmission of Hellenistic science gives fresh insights into some very old problems. The similarity between the geometric algebra involved in problems of "application of areas" and the algebraic procedures of the Babylonians is pointed up. Equally interesting are Neugebauer's suggestions regarding the persistence of Babylonian linear methods in Greek astronomy on a "far wider extent than one could possibly have deduced from the silence of Ptolemy and his commentators."

The volume includes abundant critical bibliographical references. Prepared from Neugebauer's "Messenger Lectures," these chapters are worthy successors to those of James Henry Breasted, the initiator of that series.

4.7. Review by: Arnold Dresden.
Journal of the American Oriental Society 73 (1) (1953), 53-55.

This volume follows closely the "Messenger lectures on the evolution of civilisation," which the author delivered at Cornell University in 1949. Obviously, the limitations imposed by a volume of less than 200 pages, necessitate careful selection of the topics to be treated, from the vast field indicated by the title. On account of his own interests and his own important contributions, and because he believes that "the investigation of the transmission of mathematics and astronomy is one of the most powerful tools for the establishment of relations between different civilisations," the author has devoted this volume to a survey of the historical interrelationship between mathematics and astronomy in ancient civilisations.
...
Each chapter is followed by a bibliography, and by extended notes and references. In those sections, a large number of interesting details receive further attention. As a single example, let us say, in closing, a few words about the myth of the "Babylonian Saros" (i.e. the relation which equates 223 synodic months to 242 draconitic months). Starting with Edmund Halley's deduction, based on an incorrect reading of Pliny, published in the Philosophical Transactions for 1691, presented as a fact by Montucla in his Histoire des Mathématiques, "it has been accepted doctrine that the Babylonians used the 'saros' for the prediction of eclipses." The origin and survival of such historical myths remain of course a subject of great interest to the historian. The present volume makes an important contribution to methods of dealing with them decisively.

4.8. Review by: B Farrington.
The Classical Review, New Series 3 (3/4) (1953), 207-208.

By the exact sciences Neugebauer means mathematics and astronomy. His purpose is to survey the relationship between mathematics and astronomy in ancient civilisations. In particular he wishes to discover the effects on Hellenistic science of developments in Babylonia and Egypt. He has written a very readable and a very important book. His name is familiar for his researches, so that it is with reluctance that he turns aside from them to address a wider public. But in so doing he has rendered a signal service to scholarship, and his masterly book explains its somewhat inaccessible subject to the layman with greater clarity than any other study known to me.

In form the book consists of six chapters, originally lectures, on Numbers; Babylonian Mathematics; The Sources, their Decipherment and Evaluation; Egyptian Mathematics and Astronomy; Babylonian Astronomy; Origin and Transmission of Hellenistic Science. The treatment of each subject is selective rather than comprehensive. What is said is designed mainly to explain the origin of Hellenistic science. But each chapter is completed by a bibliography and by notes and references, which are not less interesting than the text itself. Finally the fourteen plates are more than a pictorial supplement: they are an integral part of the exposition.

In his chapter on the sources Neugebauer sets forth the limitations of the Babylonian material available. There are perhaps 500,000 tablets in various museums and these no doubt represent but a small fraction of what may yet be uncovered by the spade. But that is not the most serious obstacle to research. This consists in the fact that at the present rate of decipherment - so few are the competent workers in the field - it would take many centuries to publish the material. The recovery of a knowledge of Babylonian science rests on the work of a handful of men. Father Strassmaier towards the end of last century spent many years copying tablets in the British Museum. With the help of Father Epping of Quito, it was realized that they contained arithmetical progressions skilfully utilised for the prediction of lunar phenomena with an accuracy of within a few minutes. In a little paper of ten pages in a Roman Catholic theological magazine was laid in 1881 the foundation of this new and most important branch of science. But Budge in his Rise and Progress of Assyriology (1925) records that Strassmaier thought it a waste of time to try to compile an Assyrian dictionary while so many thousands of tablets in the British Museum and elsewhere remained unpublished. 'Today', writes Neugebauer, 'one may repeat this statement, only replacing "tens of thousands" by "hundreds of thousands".'

What emerges from the scanty material as yet interpreted may be suggested in a few sentences. Already before 1500 B.C. Babylonian arithmetical procedures had been fully developed. These procedures formed the basis of Babylonian mathematical astronomy, which, however, is not attested by the existing material before the time of the Seleucids. Contrary to the general opinion, Babylonian astronomy is mainly mathematical rather than observational. The results and procedures of Babylonian astronomy were available to the Greeks from the time of Hipparchus and form part of the heritage of Ptolemy. The Egyptian contribution to this heritage was slight, the chief element in it being the adoption by the Hellenistic astronomers of the Egyptian calendar of 12 months of 30 days each with 5 additional days at the end of each year. This Neugebauer calls 'the only intelligent calendar which ever existed in human history'. Again, contrary to general opinion, what is called Chaldaean astrology. was a Hellenistic creation. Neugebauer quotes with approval the judgement of Cumont: 'Hipparque, dont le nom doit être placé en tête des astrologues comme des astronomes grecs'. Astrology, however, was not the only original creation of Greek students of the heavens. Babylonian mathematical astronomy rested on arithmetical procedures. Geometrical astronomy was the achievement of the Greeks. 'By and large, one has to distinguish two widely separate types of "Greek" mathematics. One is represented by the strictly logical approach of Euclid, Archimedes, Apollonius, etc.; the other group is only part of general Hellenistic mathematics, the roots of which lie in the Babylonian and Egyptian procedures. The writings of Heron and Diophantus form part of this oriental tradition.' ...

4.9. Review by: William D Stahlman.
Journal of the History of Medicine and Allied Sciences 8 (1) (1953), 97-101.

Professor Neugebauer's latest book represents the printed version of his Cornell University "Messenger Lectures on the Evolution of Civilization," delivered by him at Cornell in the autumn of 1949. Since these were designed for non-specialists as well as specialists in other fields, the book will appeal to a wide audience anxious to know something of the results of recent research on ancient exact sciences. By "exact sciences" in this context the author means mathematics and astronomy, and inasmuch as no discussion of the latter could hope to present a balanced picture without including classical astrology, there are numerous references to it. The author further restricts himself to non-Greek accomplishments in these fields since Greek science is comparatively well covered in numerous volumes. Yet in spite of these restrictions, the reader will be pleased with this "slight synthesis" at the hands of one who has spent many patient years at the difficult task of being an ultra-specialist.

In the Introduction he makes it plain that he will not deal with the history of ancient medicine or natural sciences but at the same time emphasises the close connections between medicine and astronomy in the Greek medical schools, for example, and the effect on medieval medicine of Hellenistic astrology. But even more important to anyone interested in the history of the ancient world is the fact that astronomy offers one of the most potent tools for the establishment not only of pure chronology but also of the history of intercultural contact, and this because the transmission of technical astronomical knowledge requires what might be termed a "penetrating" contact. Ptolemy's epicyclic planetary theory cannot be transmitted in the manner of an old wives' tale, passing between traders over a cup of beer. Further, since it is most improbable that parallel astronomical systems, complete to many details, should develop independently in diverse cultures and because of the many inherent checks available to the investigator, it follows that the study of these schemes often throws light on non-astronomical history.

The author disarms us at the beginning by disclaiming any effort at historical "synthesis"; and yet, precisely because it is carefully qualified, this book is a synthesis in the best sense of that term. The pitfalls involved in synthesis could not be more apparent than to a specialist like Professor Neugebauer, and accordingly when he puts parts of his work together he knows whereof he speaks. We are thus repeatedly warned that only part of a history has been laid before us, and in fact, his primary purpose has been "to convey to the reader some of the fascination which lies in active work on historical problems."
...

4.10. Review by: William H Stahl.
Archaeology 6 (1) (1953), 59-60.

Dr Neugebauer has probably done more than anyone else in the present generation to acquaint scholars who are not professional orientalists with the extent of mathematical and astronomical knowledge attained by the Babylonians and the ancient Egyptians. A volume by him summarising the results of recent studies and explaining the problems involved in this difficult field of research is an important addition to the literature on the ancient Near East.

The book exhibits a lack of homogeneity, a defect which appears to stem from the very nature of its composition. It originated in a series of lectures on the evolution of civilisation, and it is obvious that Dr Neugebauer has striven to make a technical subject attractive to a lay audience. Despite his efforts, the mathematical and astronomical portions will not be clear to untrained readers.

As a concession to his audience, Dr Neugebauer has interspersed some fascinating obiter dicta about the grievous conditions existing in the field of Babylonian studies. Ever since Layard's memorable find at Nineveh in 1849, clay tablets have been coming into museums in a volume far exceeding the capacities to classify and publish them. Most of the efforts and funds of archaeologists have been expended in field work, which has attained a high level of efficiency and care, but there the care has ceased. The fascination ends with the initial excavation reports. Tablets extricated so painstakingly from the earth have been stored away uncatalogued in museum basements. The texts of one lot that had been lying thus for over fifty years could be plausibly assigned to an excavation only from the date of the newspapers in which they had been wrapped. Tablets have disintegrated even in exhibition cases because they have not been properly treated to withstand their new environment. Much of the difficulty can be traced to the shortage of trained scholars. If all the available Assyriologists were to work at their present pace, several centuries would be required to publish the backlog of texts, and still they continue to come in faster than they can be handled.

The title of the book is misleading. Readers would naturally assume that the major portion deals with Greek science, but in apology for the disproportionate emphasis upon oriental science the author observes that Greek mathematics has already been adequately covered in the works of Sir Thomas Heath and that the highly technical character of Greek astronomy makes it impossible to deal with in this book. The latter assertion is a rather startling one in consideration of the technical nature of his discussion of Babylonian mathematics and astronomy. The six lectures that comprise this volume are on ancient number systems and symbols, Babylonian mathematics, deciphering ancient texts, Egyptian mathematics and astronomy, Babylonian astronomy, and the origin and transmission of Hellenistic science.

It is also confusing when Dr Neugebauer speaks of the beginnings of Greek astronomy about 200 B.C. Heath's survey of Greek astronomy in Aristarchus of Samos virtually ends before that date.

The plates are excellent - eight of them being published here for the first time - and are frequently keyed to the text so that the reader feels that he is being initiated into the mysteries of handling and deciphering clay tablets.

Occasional faults in grammar, idiom, spelling, punctuation and typography indicate that the usual editorial care was not given to this volume, which was printed abroad.

4.11. Review by: John L Myres.
Man 52 (1952), 149.

This is a handy and valuable book on a difficult subject. It only deals with mathematics and astronomy, and only comes down to the end of the Hellenistic Age, but it begins with a historical retrospect, and account of numbers and the simplest mathematical processes. The 'September' plate from the Duc de Berry's Book of Hours (1416) illustrates the continuity between ancient astronomy and late mediaeval calendars, and the great break which comes with the introduction of mechanics into astronomical matters by Newton and his contemporaries; even Copernicus and Kepler constantly refer to Ptolemy, and modify Hellenistic tradition. Early numerals are explained in ascending order, with an explanation of the sexagesimal system in Babylonian computations, and the use of special signs for commodities such as silver. Each chapter has a convenient bibliography, and footnotes for special points.

Babylonian mathematical texts, divided into 'table texts' and 'problem texts' have their own peculiarities and rules for their use, of which samples are discussed. 'Pythagorean numbers' illustrate problems concerning relations between numbers, never sharply separated from 'algebraic' methods. Interest in procedures is shown by collections of formulae. Geometrical components are less significant. Tables of 'coefficients' were compiled for many substances and relations such as 'diagonal' and 'inheritance.' Fresh discoveries are still being made, as the very numerous collections of tables are deciphered, and fresh advances in theory from tablets from Susa. But 'Babylonian mathematics never transgressed the threshold of pre-scientific thought.' Further details must be sought in the author's Vorgriechische Mathematik (Berlin, 1934).
...

4.12. Review by: W van der Wielen.
Mnemosyne, Fourth Series 8 (2) (1955), 153-154.

The well-known historian of science O Neugebauer (author of "Vorgriechische Mathematik", 1934) has written a remarkable book on ancient mathematics and astronomy. For a classical student the title is in so far misleading that only one, though the most important, chapter deals with Greek science, its origin and its influence on Western and later Oriental science. This is, however, the only disappointing feature of the book, and the classical scholar, too, will be amply rewarded by the reading of the preceding chapters, which deal with oriental science. The beautiful illustrations are an essential part of the book; thus the reproduction of a page from the "Très belles heures du Duc de Berry" forms the starting-point for the first chapter, called "Numbers". This and the next four chapters (II "Babylonian Mathematics", III "The Sources; their Decipherment and Evaluation", IV "Egyptian Mathematics and Astronomy", V "Babylonian Astronomy") lead up to the sixth chapter, "Origin and Transmission of Hellenistic Science".

The third chapter contains much that is of the utmost importance for the historian as well as for the archaeologist. The author often expresses himself in a paradoxical manner, as e.g. the opening sentence of the chapter: "There are many forces which cooperate in the destruction of source material, none more powerful than continuous peaceful life". Prof Neugebauer, a specialist in Babylonian and Egyptian science, rightly points out the former's great influence on Greek science. Egyptian science was less important ("Ancient science was the product of a very few men; and these few happened not to be Egyptians)." So he is inclined to question the enormous originality of the Greeks in the field of mathematics and astronomy. I think he is right in discarding some "textbook facts", e.g. the much-exaggerated role of Thales and "Pythagoras" in the early stages of the development of Greek science, but I cannot agree with his contention, "that Plato's role has been widely exaggerated." It is evident from the whole book that Prof Neugebauer does not believe in "un miracle grec" and is unwilling to attach any meaning to the expression "the Greek genius". The last paragraph of the notes to Chapter VI is so typical of this attitude that I venture to quote it at length: "In our discussion we have frequently used the word "Greek" with no further qualification. It may be useful to remark that we use this term only as a convenient geographical or linguistic notation. A concept like "Greek mathematics", however, seems to me more misleading than helpful. We are fairly well acquainted with three mathematicians Euclid, Archimedes, and Apollonius who represent one consistent tradition. We know only one astronomer, Ptolemy. And we are familiar with about equally many minor figures who more or less follow their great masters. Thus what is usually called "Greek" mathematics consists of the fragments of writings of about 10 or 20 persons scattered over a period of 600 years. It seems to me a dangerous generalisation to abstract from this material a common type and then to establish some mysterious deeper principle which supposedly connects a mathematical document with some other work of art". It would be easy to give an impressive series of facts from the history of Greek mathematics that belong to a rather short period (say from 400 to 212 B.C.) and prove just the contrary!

Nevertheless, this is a brilliant and inspiring book, written by one of the greatest experts in this field. Brilliant because of the lucid style and the penetrating manner in which the main problems are discussed. Inspiring because of the great stimulation to further study, which may be best summed up in the author's own words: "The history of the ancient mathematical sciences is a field in which one need not go far to find fertile soil ready to be cultivated."

4.13. Review by: Leland R Phelps.
The Classical Journal 48 (5) (1953), 184-185.

Professor Neugebauer has performed a definite service to the history of science with the publication of this book. Its chief value lies in the approach to the problem of the two sciences with which the book is concerned, viz., mathematics and astronomy. Emphasis is placed not on chronological development but on what the author has called "the historical between interrelationship mathematics and astronomy in ancient civilisations." Those civilisations with which the book is primarily concerned are the Babylonian, Egyptian, Hindu and Hellenistic, and the major problem in which the author is interested is that of the origin and transmission of Hellenistic science (specifically mathematics and astronomy).

This presentation is based on a series of six lectures which were delivered at Cornell University in 1949 on the following topics: 1. Numbers, 2. Babylonian Mathematics, 3. The Sources; their Decipherment and Evaluation, 4. Egyptian Mathematics and Astronomy, 5. Babylonian Astronomy, and 6. Origin and Transmission of Hellenistic Science. To each of these the author has appended a bibliography and a series of detailed notes and references which will satisfy the most demanding scholar.

In the light of evidence obtained from archaeological sources Professor Neugebauer has found a number of the most widely accepted beliefs concerning the sciences of antiquity to be invalid. Deciphered cuneiform texts have cast a great deal of light on the development of geometry. Although the Babylonians did not treat geometry as a special mathematical discipline they made extensive use of geometrical relationships. It has been revealed, for example, that they were familiar with the principle of the Pythagorean Theorem. A large amount of the empirical data assembled astronomical by the Babylonians for their calculations was later utilised by the Greeks in the development of their geometrical theories and astronomy. The assumption that the Babylonians had an astronomy which was essentially of a magico-astrological nature has been proved untenable, for deciphered texts have revealed that they possessed a rather extensive tradition of mathematical astronomy. Actually a great deal more is known about Hellenistic astrology than Babylonian.

The author refers to the Hellenistic period as the centre of ancient science. In investigating the roots of this science, for it was not an independent organism but one which was dependent on the developments which took place in the civilisations which preceded it, critical texts from these previous civilisations are examined and their relevance for Hellenistic science is discussed. Only through continued patient investigation by specialists such as Professor Neugebauer who are trained not only in the ancient languages but also in the technical fields they are investigating will the development of the sciences finally appear in true perspective.

A title more applicable to the contents of the book could perhaps have been selected. The all-inclusive nature of The Exact Sciences in Antiquity is somewhat misleading. One expects a range of subject matter beyond that of mathematics and astronomy. The author was perfectly justified in limiting himself to those fields which he felt qualified to discuss. The only criticism offered is against the misconception created by the title of the book.

4.14. Review by: A P Treweek.
The Journal of Hellenic Studies 73 (1953), 179.

This work is a presentation in book form of the six lectures delivered by Neugebauer at Cornell University during the autumn of 1941, hence the original form has forced upon him a much more free and summary treatment than would have been given in a purely written work. He has, however, taken the. opportunity to add some detailed notes and a critical bibliography to each chapter.

As Neugebauer points out in his preface - which is worthy of careful reading for its judicious common sense he has tried to survey the historical interrelationship between the mathematics and astronomy of various ancient civilisations. Abandoning to Heath the field of Greek mathematics, and passing over the complicated technicalities of Greek astronomy, he deals mainly with mathematics and astronomy in Babylon and Egypt, their relationship to Hellenistic science, and the transmission of the latter to the Hindus and the Arabs. To put a title to such a series of lectures is a difficult one, and the words 'The Exact Sciences' are somewhat misleading; they apply quite well to the arithmetic discussed, but the astronomers considered were rather people travelling hopefully towards exactitude than actually arriving there.

After an introduction which emphasises the fact that the transmission of mathematical and astronomical methods and material gives us very accurate information about the time and circumstances of the contact of civilisations, N. surveys the fields of the exact sciences in six chapters, dealing with Numbers, Babylonian Mathematics, the Decipherment and Evaluation of the Sources, Egyptian Mathematics and Astronomy, Babylonian Astronomy, and the Origin and Transmission of Hellenistic Science.

Chapter I discusses various systems of numbering, of both integers and fractions, as seen in late medieval France, Greece, Rome, and the two periods of Babylonian mathematics.

Chapter II deals with the mathematical methods of the Babylonians, in both the old Babylonian and the Seleucid periods, these being illustrated by quotations from texts. The use of multiplication tables is illustrated, including the use of reciprocal sexagesimal tables for division; there is also a fine example from a tablet of the relations between the sides and diagonals of a series of rational right-angled triangles - a tablet which raises quite a number of still unsolved problems about method. Quadratic equations are also treated. This, however, is all arithmetic, there is no real geometry or algebra; as Neugebauer observes, 'Babylonian mathematics never transgressed the threshold of pre-scientific thought.'

In Chapter III he considers the sources from which we draw our knowledge of ancient science. After showing how misleading modern editions of an ancient mathematical text can be - and those who have worked in this field well know how cavalierly such editions can treat abbreviations and diagrams - and lamenting the lack of modern editions of various minor and even major works, he passes from manuscripts to papyri, which have their own peculiar set of frustrations; but all this trouble is, as Neugebauer observes, 'child's play compared with ancient Mesopotamia'. The difficulties of excavation, preservation and publication of tablets, and the urgent need of 'excavating the source material in museums', are also mentioned, before he turns to a sample decipherment of a text.

Having disposed, in Chapter IV, of the crudity of Egyptian mathematics and astronomy, Neugebauer deals in Chapter V with Babylonian astronomy. After exploding the still commonly held myth of its great age and excellence, he deals with the work of Epping and Strassmaier and their successors on lunar theory, and then with procedure texts and ephemerides, with the two systems of interpolation for solar, lunar, and planetary tables.

Chapter VI considers the oriental as distinct from the purely Greek background of Hellenistic mathematics. The differences between the Greek (geometrical) and the oriental (arithmetical) approaches to astronomy are mentioned, as is the survival of primitive arithmetical devices in the Demotic and Greek astrological papyri. The antecedents of Hindu and Arabic mathematics form the final topic.

The book has a useful chronological table at the beginning, diagrams throughout the text, an index and an excellent set of plates, including a delightful reproduction in colour of 'September', from the Book of Hours of the Duke of Berry, which Neugebauer uses as a starting point for his discussion of numbers.

This work, like the lectures, is less for the expert in any particular part of this field than for the intelligent layman or the beginner; but it should encourage such readers to turn their attention to the more detailed and specialised works in various domains of this field, including the more solid works of Neugebauer himself.

4.15. Review by: I Bernard Cohen.
Scientific American 186 (5) (1952), 80-81.

Neugebauer's book deals mainly with Babylonian mathematics and astronomy. Our textbooks tell us that only two "facts" are certainly known about Babylonian exact science. One is that the Babylonians discovered the precession of the equinoxes and the other is that they determined that eclipses of the sun or of the moon occur at intervals of 223 months. Neugebauer shows that both of these ideas are wrong. It was the Greek Hipparchus who first discovered the precession of the equinoxes, and the discovery of the 223-month eclipse cycle was credited to the Babylonians only through a misinterpretation by Edmund Halley, the 17th-century British astronomer.

On the other hand, the Babylonians did make a number of important contributions to mathematics and exact science which anticipated the Greeks. They possessed enormous skill in dealing with numbers and worked out tables of squares, square roots, cubes, cube roots, sums of squares and cubes, reciprocals, and fractional ratios which enabled them to solve such complex problems as cubic equations of certain types, exponential functions used in the computation of compound interest, and so on. They knew and applied the "Pythagorean" theorem "more than a thousand years before Pythagoras." Recognition of the important mathematical achievements of the Babylonians does not mean that we should cease to admire the achievements of Euclid and other Greek mathematicians. But Neugebauer suggests that we must re-evaluate the Greek contributions. Plato, for example, has been credited with many Original ideas although, Neugebauer insists, his "contributions to mathematical knowledge were obviously nil." Furthermore, "Plato's doctrines undoubtedly have had great influence upon the modern interpretation of Greek sciences. But if modern scholars had devoted as much attention to Galen or Ptolemy as they did to Plato and his followers, they would not have invented the myth about the remarkable quality of the so-called Greek mind to develop scientific theories without resorting to experiments or empirical tests."

One of the most interesting chapters of Neugebauer's book is a thrilling record of the steps by which our knowledge of the earliest exact science has been uncovered by a small band of devoted scholars, of whom the author is the outstanding living representative. Neugebauer writes in a vivid and trenchant style. Although some technical portions of the book are difficult to read, the important points are everywhere made clear. The reader should be warned that the title is somewhat misleading; this book is not a comprehensive history of "the exact sciences in antiquity" (Greek science has been well covered elsewhere) but "a survey of the historical interrelationships between mathematics and astronomy in ancient civilisations." As a history of the ancient mathematics and astronomy that was not produced by the classical Greeks, it is a magnificent creation.

5.1. Review by: Editors.
Mathematical Reviews MR0090506 (19,825a).

In the present edition many additions referring to recently obtained results have been made and there are two new appendices, one on Greek mathematics, the other on the Ptolemaic system and its Copernican modification.

5.2. Review by: Oystein Ore.
Philosophy of Science 26 (2) (1959), 155.

The first edition of Neugebauer's valuable book on ancient mathematics and particularly astronomy appeared in 1951. The main content of the present edition is approximately the same, with new notes bringing it up to date on the many new contributions made in the last few years by the author and other scholars.

To this has been added two new appendices, one on the Ptolemaic System, the other on Greek Mathematics. The latter deals largely with Greek trigonometry and is interesting for several reasons. As the author explains:

"The conventional picture of Greek mathematics a very sophisticated branch of geometry followed by some not quite successful attempts at algebra and number theory during the later periods of decline is wrong for two reasons. First, as we have seen on many occasions in the preceding chapters, we are not dealing with a decline from scientific geometry to less exact methods of algebraic tendency, but with two contemporary phenomena: a comparatively rapid development of rigid mathematical reasoning on the highest level in contrast to a much older and little changing background of ancient oriental and Hellenistic mathematics with a strongly algebraic character inherited from its Babylonian origin. Secondly, even the strictly Greek development is not adequately described by the Euclidean-Archimedean development, which is most familiar to the modern reader, but we must add many methods which concern numerical and graphical problems which originated in mathematical astronomy."

The introduction to this section contains a credo which might be worth pondering for writers on the history of science:

"I do not consider it as the goal of historical writing to condense the complexity of historical processes into some kind of 'digest' or 'synthesis.' On the contrary. I see the main purpose of historical studies in the unfolding of the stupendous wealth of phenomena which are connected with any phase of human history and thus to counteract the natural tendency toward oversimplification and philosophical constructions which are the faithful companions of ignorance."

5.3. Review by: William H Stahl.
Archaeology 12 (4) (1959), 293.

This book, first published in 1952, quickly established itself as a key work in its field. The second edition is a thoroughgoing revision, incorporating corrections and an enormous amount of new material. Dr Neugebauer is an indefatigable and brilliant worker. It would be hard to name a scholar who is contributing more to our knowledge of ancient science at the present time.

The author's own rigorous outlook upon scholarship goads him to frequent acid comments upon the defects of scholarship in the history of science. He observes that specialization is the only sound basis of knowledge, that syntheses and digests contribute nothing, that competently edited texts of Greek, Latin and Arabic treatises are most urgently needed, that textual evidence extant treatises, papyrus and cuneiform documents is the basis of sound conclusions, and that constructs not based upon texts lead to mythical theories. He points to Duhem's misconception of Ptolemy's lunar theory (1914) as "a good example of the rapid decline of the history of science." He has equally melancholy things to say about the frustrations of scholars working with cuneiform texts. The excavation of clay tablets turns out to be their destruction and not recovery, as they are allowed to be unsystematically deposited around the world and to deteriorate in museum storerooms without any prospect of being studied.

The author's obsession with the importance of texts leads him to some extreme positions. He places the beginnings of Greek astronomy about 400 В.С. (200 В.С. in the first edition), discounts the traditional stories of Pythagoras' discoveries as "totally un-historical" and his connection with the elementary theory of numbers as "purely legendary," and remarks that Euclid's Elements and Ptolemy's Almagest reduced all their predecessors in mathematics and astronomy to "objects of mere historical interest."

One suspects that the contemptible synthesisers have had a salutary influence upon Dr Neugebauer. He has added two appendices ("The Ptolemaic System" and "On Greek Mathematics") to make the new addition more nearly conform to the title. He has regard for the interests of a lay audience in original form this book was a set of lectures delivered at Cornell University. He has allowed himself to be drawn into making pointed observations about the implications of his research and to make statements of appraisal about the wealth of biblio-graphical and source material discussed in his text. If this volume were merely a set of detached technical lectures, it would not be one of the most significant books on the history of ancient science to appear in this generation.

5.4. Review by: René Taton.
Revue d'histoire des sciences et de leurs applications 14 (2) (1961), 186.

The first edition of this important work having been previously presented to the readers of this journal, we will simply note that this new edition has been carefully revised by the author, who has added various supplements to take into account recent publications, as well as two appendices, one on Greek mathematics, the other on Ptolemy's system and its modification by Copernicus. Furthermore, the chapters on Egyptian astronomy and on the Babylonian planetary theory have been partially rewritten for this second edition. In this new form, the work of this eminent specialist in ancient astronomy and mathematics covers more comprehensively the vast field announced by its title. Consequently, the widespread success it has enjoyed thus far can only be confirmed.

5.5. Review by: E J Dijksterhuis.
Isis 49 (4) (1958), 455-456.

Ever since this work first appeared it has occupied a place of honour in the literature on the history of science. No wonder a new edition has proved necessary. The structure of the book has remained unchanged. The Notes and References, added to the separate chapters, have been revised in the light of new knowledge gathered since 1952. In some chapters new paragraphs have been inserted.

A most welcome addition is composed of two appendices, entitled "The Ptolemaic System" and "On Greek Mathematics." The first contains an exposition of the kinematical models on which the calculation of the numerous tables in the Almagest is based. The second deals with certain aspects of Greek mathematics, which fall outside the framework constituted by the writings of Euclid, Archimedes and Apollonius, but which are indispensable to a complete understanding of the Greek attainments. The reader is brought into contact with the Analemma (a sort of descriptive geometry), with the instrument which was to become known as the astrolabe, and with cartography.

Throughout the work the writer makes several critical remarks on current misunderstandings concerning the historical development of the exact sciences. He points out in how many respects our knowledge is still distressingly incomplete, and shows the way toward remedying these defects. One can glean a whole sheaf of aphorisms of profit to the historian of science in his daily reading. Here are four, by way of example.

"The common belief that we gain 'historical perspective' with increasing distance seems to me to utterly misrepresent the actual situation."

"What we really need is not bibliographies and summaries, but competent publication of Islamic, Greek, and Latin treatises."

"Perhaps it is vain to hope for anything more than a picture which is pleasing to the constructive mind when we try to restore the past."

"I do not consider it as the goal of historical writing to condense the complexity of historical processes into some kind of 'digest' or 'synthesis'."

Everywhere the writer's deep and innate distrust of rash generalisations and superficial syntheses shines forth. Again and again it is good to listen to his earnest warning voice.

5.6. Review by: J F S.
Science Progress (1933-) 47 (185) (1959), 196.

Few scholars have contributed as much to our understanding of the achievements of the ancient civilisations as the author of this work. Our knowledge of the mathematics and astronomy of the races of antiquity, hitherto somewhat fragmentary, has been vastly increased by his painstaking and exhaustive research, the fruits of which are to be found in the present volume. Not the least impressive part of the work is the light which is thrown upon the mathematics of the Babylonians; this, the author clearly shows, was far more highly developed than earlier writers have led us to believe. The popular view, that Babylonian science was little more than an offshoot of magic, myth and astrology, can no longer be maintained, and according to the author "we are dealing with a level of mathematical development comparable in many respects with the mathematics of the Renaissance" (p. 48). In consequence of all this, the conventional picture of the achievements of the early Greeks is hardly a true one. Even the discoveries credited to Thales and Pythagoras have no historical justification.

Not everyone will agree with Dr Neugebauer's estimate of the importance of Egyptian mathematics. "The role of Egyptian mathematics is probably best described as a retarding force upon mathematical procedures. Egyptian astronomy remained throughout all its history on an exceedingly crude level." This hardly gives credit to the truly remarkable results which were obtained by that race in geometry and astronomy.

The book deals in succession with Number, Babylonian mathematics and astronomy, Egyptian mathematics and astronomy, and the origin and transmission of Hellenistic science. There is an Appendix on the Ptolemaic system, which is not easy reading, but which amply repays the effort, and another on Greek mathematics.

The work is abundantly illustrated, and a most valuable feature is the large number of plates which are reproduced. The bibliographies at the end of each chapter will prove of great use to the reader. This is a book which will command the attention of scholars and laymen alike.

5.7. Review by: Martin Levey.
The Jewish Quarterly Review 49 (3) (1959), 214-216.

We are much indebted to Prof Neugebauer for having brought an earlier work up to date. New publications, particularly in ancient astronomy in the past 6-7 years containing much new material have led to a better perspective in Babylonian and Egyptian astronomy. In this expanded edition (about 20% longer than the previous), the author offers the reader a deeper appreciation of astronomy and mathematics in ancient Egypt and Mesopotamia as well as of their transmission to the Greeks.

Neugebauer has retained the early form of the book six chapters based on his Messenger Lectures at Cornell University in the fall of 1949. As a result, he has not found it expedient to document fully all that he has written, even though the chapters have bibliographical notes and references of value. The work is not intended as a text; it is, on the other hand, very readable as an outline of ancient knowledge in this field.

One of the most admirable attributes of this work which should serve as an example to scholars in this and in other fields is a result of Neugebauer's consideration of his subject as a dynamic one. The history of astronomy and mathematics has been invaded over the past 2000 years by many false ideas which must be brought to light and eradicated. Neugebauer does not hesitate to expose some of them.

In the account of Thales' alleged prediction of a solar eclipse in -584, it is shown that this would hardly have been possible using the myth of the Saros. Kugler, in 1900, pointed out that eclipses were computed during the Seleucid period by using the latitude of the moon in relation to the syzygies. However, Neugebauer does state that there is a possibility that the periodic recurrence of lunar eclipses as a crude 18 year cycle was also made the basis for matters concerning other lunar phenomena. In the case of Egypt, contrary to the usual opinion, the only texts which deal with numerical astronomy belong to the Hellenistic or Roman period. The earlier documents are of an observational or religious nature, partly practical in purpose.

On the other hand, many weaknesses in the history of physical science are bared and the necessary direction of research is indicated by the author. Our knowledge of Greek science comes from the works of only about twenty Greek writers. It would, however, be of great value to enlarge this basis of our knowledge. For example, Books V-VII of Apollonius' Conic Sections have never been edited; they are now only in Arabic. E S Kennedy has pointed out in a recent article ("A Survey of Islamic Astronomical Tables" (1956)) the great wealth of still unpublished material. There is also the "Alfonsine Tables" of ca. 1270 which is preserved in about 75 MSS and in a number of rough early editions (cf. Alfred Wegener, "Die astronomischen Werke Alfons X," (1905)). There are a number of medieval Hebrew MSS on astronomy still unpublished in spite of the important contributions in this field by medieval Hebrew scientists and translators. Other material deserving of scholarly study includes Byzantine astronomical handbooks, the papyri which are deteriorating in northern climates and also the relevant cuneiform tablets many of which are now crumbling in museums because of climatic conditions.

The main changes in the second edition are in the much enlarged sections on Egyptian astronomy and on Babylonian planetary theory - both entirely rewritten. Moreover, two appendices have been added. In order to explain some of the basic procedures in the Almagest of Ptolemy, Neugebauer has written a description of the cinematic models by means of which the computations for the tables were made, allowing one to find for any given moment the longitudes of the sun, the moon, and the five planets. The second appendix is concerned with the stimulus given to Greek mathematics by mathematical astronomy. Neugebauer refutes the idea of the Euclidean-Archimedean corpus as an accurate description of Greek mathematics. In the Greek period, many methods were developed which concern numerical and graphical problems originating in mathematical astronomy. These are described by Neugebauer to give the reader a fresh insight regarding the value of studying the history of ancient mathematics and astronomy.

5.8. Review by: Serge Chermayeff.
Scientific American 206 (6) (1962), 196.

The second edition, in inexpensive form, of a distinguished work in its field, a group of essays based on lectures given at Cornell in 1949 by a foremost student of ancient science. The topics include Babylonian and Egyptian mathematics and astronomy, the origin and transmission of Hellenistic science, certain problems of Greek mathematics - in trigonometry and geometry - that were stimulated by astronomy. Diagrams and plates.

6.1. Review by: Jean Trouillard.
Revue de l'histoire des religions 158 (2) (1960), 239.

This substantial study focuses on a section of Hellenistic astrology, horoscopes, from the 1st century BCE to the rise of Islam. The authors rely not only on astrological treatises but also on inscriptions and papyri that provide the necessary elements for the subject. For the majority of papyri, the Greek text is given, insofar as it can be reconstructed, along with an English translation. For the remaining texts, only the English translation is provided. A reasoned method of classification and interpretation, and a dictionary of technical terms, are presented at the beginning of the book. At the end, there is an index of Greek words, a bibliography, and an index of concepts and proper names. The work concludes with plates reproducing horoscope dials and photographs of inscriptions and manuscripts.

This book is a meticulous presentation of documents. Its aim is to gather the essential texts before the reader and to help them read and interpret them. It employs only the element of systematisation inherent in all scholarly work, particularly in such a subtle subject. One will not find a synthesis here. If one wishes to find a judgment, one might consider the formula of Father Festugière, which the authors have placed as an epigraph to the entire work:

"Hellenistic astrology is the amalgam of a seductive philosophical doctrine, an absurd mythology, and learned methods employed out of step with the times."

6.2. Review by: Gérard Deledalle.
Les Études philosophiques, Nouvelle Série 15 (4) (1960), 555.

The work by Neugebauer and Van Hoesen is a treatise on Greek astronomical techniques. It includes the edition of all known unpublished documents: papyrus, ostraca, graffiti, with the exception of the "literary horoscopes", the text of which has already been published, the English translation of all the documents, including the literary horoscopes, a study on horoscopes in general, another on the authors of the literary horoscopes: Vettius Valens, Critodemus, Antigonus of Nicaea. .., a Greek, Coptic and English glossary, concordance tables, diagrams and photographs of documents.

6.3. Review by: E Flintoff.
The Journal of Hellenic Studies 82 (1962), 188-190.

One must feel grateful to the authors of this work for the incredible industry with which they have assembled together within this book all the authentic Greek horoscopes which survive to the present day, and even some which do not. From now on we shall be on rather more certain ground when we come to discuss the progress and decline of astrology in antiquity, its techniques, the social levels which indulged in it. In view of the multitude of these possible ways of treating the subject the authors must have found it singularly difficult to know where the focus of the book should be, what points it should include or exclude. In the event, they decided that the primary aim should be to enable us to study some of the techniques of Greek astronomy. This meant that, as they themselves tell us, a number of questions of a papyrological, legal, political, medical and even astrological nature have had to be left untouched. Furthermore, the aim of the book is narrower than it might suggest, since the authors have tended to assume in the reader a knowledge of mathematics and astronomy not necessarily possessed by everyone who might have occasion to study the book; whereas next to no assumptions are made about the reader's knowledge of astrology. It is assumed, for example, that the reader's memory will constantly have to be jogged about which planet rules which sign of the zodiac, whereas he will be able to take in his stride diagrams of the complexity of fig. 17, and fig. 18. It is thus primarily a book designed for people who already know more than a little about ancient science, and its aim is to teach them more on the same subject. It is not, however, as I have indicated above, without interest for other studies, and as my own interests tend to lie rather in these fields than in the purely scientific, I feel that it is only fair to warn the reader of my own preoccupations, the filter so to speak through which the photograph is taken.

6.4. Review by: Gerald Holton.
Scientific American 203 (2) (1960), 182.

Other than the information to be found in Ptolemy's Almagest and closely related works, little is known about the techniques of computing the positions of sun, moon and planets during any period of antiquity. This scholarly book examines the extant Greek horoscopes from the first century B.C. to the beginning of Islam, drawing on all available sources - monuments, papyri, ostraca and astrological treatises - to determine what they reflect of the contemporary methods of Greek astronomy. The horoscopes are translated, a commentary is given on each and the authors evaluate the astronomical and historical implications of the whole material. Glossaries, bibliography, illustrations.

7.1. Review by: Owen Gingerich.
Science, New Series 193 (4252) (1976), 476-477.

Was Ptolemy a fraud? Are the observations reported by this famed Alexandrian astronomer a hoax?

Allegations that Claudius Ptolemy did not actually observe the celestial positions dating around A.D. 135 and described in his Almagest were made originally by the French astronomer Delambre at the beginning of the last century. They have recently been repeated with increasing insistence by R R Newton of Johns Hopkins University, who has concluded that "the science of astronomy would be further ahead if Ptolemy had never written the Almagest."

A totally different appreciation of Ptolemy is afforded by O Neugebauer's new three-volume work on early astronomy. The inclusion of the word "mathematical" in the title is deliberate, for Neugebauer eschews the vague, speculative cosmologies of pre-Socratic philosophers. This is not the place to learn about Philolaos or even the Aristotelian spheres. But for Ptolemy, it is the source par excellence.

Divided into six "books," this compendium distils much of a lifetime of scientific research into its three volumes, and it is surely one of the landmark publications of this century in the history of astronomy. Book 1 opens with the Almagest, the standard against which both pre- and post-Ptolemaic astronomy of antiquity must be compared. Book 2 summarises in massive detail Babylonian astronomy, a field that Neugebauer and a few close associates have made their own.

"Egypt has no place in a work on the history of mathematical astronomy," Neugebauer writes in introducing book 3; "Nevertheless, I devote a separate 'Book' on this subject in order to draw the reader's attention to its insignificance." Ten pages later he proceeds to book 4, on early Greek astronomy (but not before recounting a magnificently funny anecdote about the Jet Propulsion Laboratory and Egyptian astrology), and 212 pages later he takes up the Roman imperial period and late antiquity to the 7th century A.D. At least half of book 5 is devoted to Ptolemy's minor works, ranging from the Geography and Tetrabiblos to his Planetary Hypotheses and Handy Tables. The final volume contains the book of appendices (on chronological, astronomical, and mathematical concepts), a bibliography with at least a thousand entries, over 600 figures, and nine plates.

loom Ptolemy's accomplishments large both at the beginning and at the end of this treatise. Although little is known of the man himself, we can easily imagine Ptolemy surrounded by assistants and graduate students at the famed Alexandrian library. Clearly he had at his command both computational assistance and a considerable library of earlier astronomical observations.

The magnitude of Ptolemy's astronomical accomplishment emerges in the rich fabric of Neugebauer's analysis. Among Ptolemy's greatest achievements were the introduction of the equant and his discovery that the tropical year was constant. The role of the equant in planetary theory (including Copernicus's abhorrence of it) is well known to students of early astronomy; suffice it to say that it is an elegantly simple device that permitted a notable increase in the accuracy of predicted longitudes.

The constancy of the tropical year - the time required for the sun to return to its same position with respect to the equator is a more lasting discovery, ranking in subtlety with Hipparchus's discovery of precession (around 135 B.C.). The difficulty of discovering such an apparently elementary fact is revealed by Neugebauer's examination of the Hipparchian eclipse observations, which were sufficiently faulty to obscure this fundamental property of the sun's motion.

Ironically, after Ptolemy had established this constancy, he adopted Hipparchus's numbers for the length of the year and for the seasons. He claims to have observed the time of the equinox in Alexandria, but apparently he merely confirmed that the equinox came at least a day earlier than a strict 3654-day year would require. Had he gone out two days earlier, he could hardly have missed the fact that the year was even shorter than Hipparchus had guessed.

Both Delambre and R R Newton (among others) have claimed that Ptolemy's equinox "observations" are simply extrapolations from Hipparchus. In Newton's estimation, this makes Ptolemy a fraud. Neugebauer, in contrast, passes over this circumstance in silence when discussing the Almagest. However, he examines the related problem of precession in some detail in the section on Hipparchus.

Delambre, who questioned whether Ptolemy made any observations at all, argued that the great catalogue of over 1000 stars in the Almagest had been taken over from Hipparchus, but with the longitudes increased by an erroneous value for precession. (Precession is the slow change in the stellar coordinate system discovered by Hipparchus; Ptolemy set it at 1 degree per century compared to the correct value of 1 degree per 72 years.) Hence Ptolemy's stars have a systematic error that makes their longitudes about 1 degree too small.

Neugebauer brings together convincing evidence to show that Ptolemy's star catalogue was quite independent of the earlier, smaller one of Hipparchus, and he further reports that (apart from the systematic error) Ptolemy was the more accurate observer. How was the value for precession bungled? In order to get the zero point of the stellar coordinate system, it is necessary to relate the day-time position of the sun to the night-time position of the stars, no mean task. According to Neugebauer, the fault must lie largely in the observational technique, which involved measuring the star Spica with respect to the moon during a lunar eclipse. Clearly, the theories of the motions of the sun and moon are tightly bound up in this procedure. Since Ptolemy remarks that Hipparchus's eclipse data were seriously marred, Neugebauer points to this as the crux of the problem.

Neugebauer writes

In all ancient astronomy direct measurements and theoretical considerations are so inextricably intertwined that every correction at any one point affects in the most complex fashion countless other data, not to mention the ever present numerical inaccuracies and arbitrary roundings which repeatedly have the same order of magnitude as the effects under consideration. In the history of the most causal of all empirical sciences, in astronomy, the search for causes is as fruitless as in all other historical disciplines.

The difference in attitude between Neugebauer, a mathematician who has immersed himself in the languages and techniques of ancient science, and R. R. Newton, a physicist who is eager to extract specific results on the deceleration of the earth's rotation, is shown perhaps most, clearly in their respective re-examinations of the lunar eclipse of A.D. 135.

According to Ptolemy's epicyclic model, the moon approached twice as close to the earth at quarter phase as when new or full, a situation clearly in conflict with the simplest observations. Ptolemy not only chose to ignore this untenable discrepancy, but in determining the lunar distance he picked the time of closest approach. The result was badly wrong, 40 earth radii instead of 60; nevertheless this apparently confirmed a model that had quite satisfactory distances for eclipses. The erroneous answer at quadrature, which fit so well with all the rest of the theory, was achieved by reporting a lunar position off by 2/3 of a degree.

Did Ptolemy forge this observation, as R R Newton would have us believe? Or do we just have here evidence of "uncontrollable" observational and "quite unnecessary" trigonometric inaccuracies yielding "one of the most unsatisfactory topics in the whole Almagest"?

In my own opinion, Newton's attack on Ptolemy is at the very least anachronistic. Ptolemy, using clumsy mathematics invented only a generation earlier, made possible for the first time calculations of the local circumstances of solar eclipses. His planetary theory allowed tolerably accurate predictions to be made for over a millennium. It is hard to imagine that such success rested on fabricated observations. It is nevertheless possible that, in those days before error theory was understood, selected observations were adjusted for pedagogic purposes and thus recorded in the Almagest in close agreement with the theory - a theory undoubtedly resting on far more data than Ptolemy specifically reports.

In summarizing the section on lunar parallaxes Neugebauer gives both an evaluation of Ptolemy and an effective appreciation of his own History:

No ancient astronomer had any possibility of analysing sources of errors in observations made long before his time or at far distant localities. It makes no sense to praise or to condemn the ancients for the accuracy or for the errors in their numerical results. What is really admirable in ancient astronomy is its theoretical structure, erected in spite of the enormous difficulties that beset the attempts to obtain reliable empirical data.

He goes on to say, about Ptolemy,

Without the cinematic theories of the Almagest it would have been impossible to introduce, on the basis of better observational techniques, those improvements which found their explanation in Newton's celestial mechanics.

7.2. Review by: A G Molland.
The British Journal for the History of Science 11 (1) (1978), 74-75.

For many years the figure of Otto Neugebauer has dominated the historiography of early astronomy. And now we have this massive new work, whose title may suggest to the unwary that we should welcome the appearance of a synthesis. But this would doubtless be to raise the author's ire, for his scepticism of such endeavours is well known although, as Sarton wickedly pointed out, and as Neugebauer himself came near to admitting, his beautiful book The exact sciences in Antiquity can merit the name. No such charge can be laid against the present volumes, but as a perhaps unintended corollary it is not clear that the work is for reading. This must not be misunderstood: the volumes will have to be the constant companions of any scholar (even one of radically different orientation) working on themes that Neugebauer treats, but as an aid to be used in conjunction with other primary and secondary sources, rather than as a history that can for much of the time stand on its own. The work makes little attempt to tell a story, and despite Neugebauer's delightful epigraph from A A Milne ('The opposite of an Introduction is a Contradiction'), it could provide sore bewilderment for the tyro. This is not just because it is difficult - there is no more a royal road through this field than into geometry - but arises from features of the author's approach to the subject. One of these is the problematic assumption that mathematical astronomy can be almost completely lifted from its context, and certainly be abstracted from any philosophical 'irrelevancies'. Another is the presentation of much of the material in modern mathematical notation, which robs the reader of the flavour of the original. This may be almost inevitable for the Babylonians, but is less desirable in the case of Ptolemy.

The first volume comprises Books I and II. Book I analyses the mathematical astronomy of the Almagest and treats Apollonius and Hipparchus, while Book II is devoted to the Babylonians. In the second volume, Book III is a dismissive account of Egyptian astronomy, Book IV considers early Greek mathematical astronomy, and Book V continues with 'Astronomy during the Roman imperial period and late Antiquity' (including other of Ptolemy's works). Book VI occupies the whole of the third volume: in it we have a synoptic treatment of relevant chronological, astronomical and mathematical concepts, subject index, bibliography, 619 figures, and nine plates.

To emphasise at length the massiveness and rigour of Neugebauer's scholarship would be both otiose and temerarious. The modesty of his reference in the preface to the ignorantia auctoris in comparison with the scholarship of preceding centuries makes so much the more terrifying his corresponding reference to the ignorantia multorum among his prospective readers. Oracle, or no oracle, I dare not bark; the achievement is far too daunting. One must reflect that it may never have occurred if the author had not so dogmatically followed those 'tastes and prejudices' which can, perhaps cogently, be disputed by those who nevertheless will not produce a rival monument of scholarship. Extraordinary, magnificent, invaluable, infuriating, these volumes demand a permanent place on the shelves of the misnamed 'self-renewing library', even if for years at a time they should merely gather dust.

7.3. Review by: Asger Aaboe.
Isis 69 (3) (1978), 441-445.

To find a fitting antecedent of Neugebauer's A History of Ancient Mathematical Astronomy we must, I believe, go all the way back to Delambre's Histoire de l'astronomie ancienne, which appeared in 1817. Not until then do we meet a similar comprehensiveness, thoroughness, and meticulous attention to technical detail, and a like display of analytical powers combined with a nearly unbelievable capacity for hard work. This is perhaps not too surprising, for Neugebauer follows in his approach, as he states, the tradition of the late eighteenth century, that of Montucla, Bailly, Lalande, and Delambre, who paid the ancient astronomers the compliment of treating them, not as philosophers with one or two ideas each, but as serious colleagues whose work deserved detailed and critical study. In its content, however, A History of Ancient Mathematical Astronomy stands apart from its predecessors in its choice of subjects and in the novelty of much of the material it presents. Neugebauer has set himself rather strict limits even stricter than those suggested by the title outside which lie such topics as observational techniques, relations between theory and observation except when discussed in the ancient texts themselves, the quality of the various astronomical schemes, and philosophical cosmology. Some topics are deliberately excluded while others, Chinese astronomy for one, are not dealt with because of Neugebauer's professed want of competence.

The new material in the History falls in several categories: there are many results that have not been published before; much has hitherto been available only scattered in various journals; Neugebauer has also succeeded in gathering much material from texts long published but deemed arid ground by historians of astronomy; and he has managed to see a great deal that is new in texts that were long thought well understood and about which much has been written. Indeed, though he is well aware of the secondary literature, he has never let this knowledge prevent him from taking a fresh and unbiased look at the old texts themselves, often with startling results, and this is entirely in accord with his view of the proper role of secondary literature: he considers it, not as a substitute for reading the ancient texts themselves, but as a guide to them.

Neugebauer emphasises repeatedly how far we are and likely shall remain - from being able to write a real history of ancient astronomy, and his book is indeed no conventional history. It is rather a penetrating analysis and interpretation of the astronomical texts that have happened to survive the erosive forces of centuries and millennia to become known and available through the investigations of modern scholars, prominent among them Neugebauer himself. The texts he has examined are of many kinds: cuneiform clay tablets, demotic and Greek papyri, inscriptions, Sanskrit, Arabic, and Persian manuscripts, Byzantine copies of ancient Greek treatises and tables, and Latin translations from the Greek and Arabic.

The History appears in Springer's new twin series Sources and Studies in the History of Mathematics and the Physical Sciences as Volume 1 of the Studies. It is in three parts, separately bound but continuously paginated, amounting in all to nearly 1,500 pages. The first two parts contain the text proper, while the third is given to three useful appendixes (on chronology, astronomical concepts, and computational techniques), to various indexes, and to all the figures and plates.
...

8.1. Review by: Anne Tihon.
Mathematical Reviews MR0709937 (85d:01031).

This volume contains a sample of some 43 papers published by the author between 1932 and 1980, covering five topics: (1) generalities; (2) Egyptian astronomy; (3) Babylonian astronomy; (4) Greco-Roman astronomy; (5) Medieval and Renaissance astronomy.

The first section (generalities) regroups papers and lectures published between 1932 and 1963. Even though with time the state of the questions has obviously evolved (for example, many more treatises and texts are now edited and translated) and some bibliographies ought to be completed, this section provides historical and methodological syntheses which remain of fundamental importance and can be useful for non-specialists. For years the author repeatedly insisted on the importance of studying the texts and the documents duly placed in their own historical and philological contexts. He never ceased to urge philologists to make astronomical texts available to scientists. This is a good illustration of this concern.

The subsequent sections contain more specialised studies. It would be impossible to describe them here at length: the reviewer wishes only to stress some of their points of interest.

Section (2) (Egyptian) contains, among others, papers devoted to the Egyptian calendar and sothiac period. When they were first published, these studies brought significant progress to the field of Egyptian chronology by doing away with false problems and speculations about the sothiac cycle.

Section (3) (Babylonian astronomy) gives some initiating papers on the methodology for approaching Babylonian astronomical texts, and some others dealing with, e.g., lunar eclipses or the alleged Babylonian discovery of the precession of the equinoxes.

Section (4) (Greco-Roman) gives some basic papers. One of them is The early history of the astrolabe. Although this study could be re-examined on certain points of details, it has so far never been replaced. The study On the Hippopede of Eudoxus remains, after Schiaparelli's study of 1874, the only one to be used today for the problem of homocentric spheres' mechanism: every handbook quotes it and reproduces the author's diagrams. Of equal importance are the papers on Apollonius. One wishes also that the conclusions of the short note by the author on Heraclides Ponticus (1972), which were confirmed at about the same time by G Evans (1970), were accepted at last, and that consequently people would cease to attribute to Heraclides a heliocentric theory of Venus.

Section (5) (Medieval and Renaissance) deals with Jewish, Hindu, and Ethiopic astronomy, with a few pages on astronomy in the Renaissance or western Middle Ages. These essays do not replace the author's other books [The exact sciences in antiquity, second edition, 1957; A history of ancient mathematical astronomy, Part 3, 1975], nor his extensive studies on, e.g., Egyptian and Babylonian astronomical texts, but they provide a useful complement to them, and this publication gives easy access to many papers which are otherwise only to be found with great difficulty in a variety of orientalistic or mathematical periodicals.

8.2. Review by: Sabetai Unguru.
SIAM Review 27 (2) (1985), 305-309.

Under five headings (General, Egyptian, Babylonian, Greco-Roman and Medieval-Renaissance) this book collects selectively somewhat less technical papers on astronomical and mathematical issues from the impressive and rich output of the dean of American historians of astronomy and mathematics. At eighty-five, after sixty years of ongoing publication (an incomplete bibliography published in Centaurus in 1979 by J Sachs and G J Toomer runs to 294 items), Neugebauer's drive and energy have yet to be exhausted. His is indeed a remarkable career that started in Göttingen, took him to Scandinavia and, finally, to the United States where he has been associated with the Department of the History of Mathematics at Brown University and with the Institute for Advanced Study in Princeton. It is no exaggeration to state that Neugebauer's scholarship and personality have to a very large extent defined the limits and the content of the history of ancient and medieval mathematical astronomy.

The selection criterion for inclusion in the book was that the chosen essays be of a level of technical sophistication similar to that of The Exact Sciences in Antiquity, thus enabling the moderately informed reader who is willing to invest some effort (plainly "the layman" does not fit the bill) to grasp intelligently their content in possible preparation for the much more demanding plunge into the author's three volume History of Ancient Mathematical Astronomy. Generally speaking, and keeping in mind that the inclusion standard itself (The Exact Sciences in Antiquity) is technically not homogeneous, the forty-three essays of varying length and difficulty incorporated in the book reflect more or less faithfully the heterogeneous technical level of The Exact Sciences. They supply the diligent and studious reader with a solid foundation of knowledge about pre-Hellenic, Seleucid and Greek astronomy and mathematics, as well as with basic astronomico-mathematical elements drawn from the Greco-Roman and Medieval-Renaissance periods. In addition to a mathematical analysis of the sources, they also deal with methodological issues and touch on matters having to do with the cultural and social background of ancient and medieval mathematical astronomy, illustrating the role played by astronomy in the development of science by means of examples ranging from ancient Babylon to Copernicus, Kepler and beyond. They display convincingly the fundamental part that astronomy and astrology can play in the study of historical chronology. They tackle issues of transmission of astronomical and mathematical knowledge between different cultures (the guide being always structural) and they set the historiographic record straight by means of close attention to the sources, puncturing long-standing myths and glib oversimplifications.
...
My overall impression about this venture is that bringing together in this book these originally widely scattered articles is a valuable effort that will benefit the reader curious to learn something basic about the history of astronomy and mathematics from the pen of one of the towering figures in the community of historians of science. The essays are lucid, trenchantly written, precise, exceedingly learned and represent some of the best historical writing gleaned from the author's rich and impressive output.

8.3. Review by: G Aujac.
Revue d'histoire des sciences 39 (4) (1986), 371.

Every historian of science owes a debt of gratitude to Springer-Verlag for compiling into a single, substantial volume some forty articles by Otto Neugebauer, the renowned specialist in ancient astronomy, published in various journals, often difficult to access. This selection, initiated by Noel Swerdlow of the University of Chicago, encompasses a wide range of topics, all related to ancient and medieval astronomy. Astrology is included, as its study is particularly revealing of the methods employed to gain a better understanding of the heavens. Some articles present overarching questions in a relatively straightforward manner; these are gathered in the first section. Others deal with points of detail, in light of certain discoveries, such as that of a world map of Greek origin in Byzantine manuscripts, or a table for determining the time based on the length of shadows in a Ptolemaic papyrus. Still others attempt, in light of authentic texts, to refute certain received ideas that O Neugebauer seeks to demonstrate are often unfounded. Hipparchus, for example, despite his immense reputation, appears less innovative in theoretical astronomy than Apollonius of Perga or Ptolemy. O Neugebauer devoted important articles to the study of Apollonius's planetary theories and to the equivalence he recognised between the hypothesis of eccentrics and that of epicycles for explaining the movement of the planets. Greek science is abundantly represented in this volume; but it also contains reflections on Egyptian and Babylonian astronomy, which so greatly influenced Greek scholars. A dozen articles deal with medieval astronomy, among them a very detailed analysis of the astronomical data contained in the Très Riches Heures du Duc de Berry. This speaks volumes about the richness and variety of the topics covered, all of which are treated with the aim of remaining as close as possible to the available texts and documents. As O Neugebauer rightly points out in his preface, this selection provides excellent preparation for reading his more technical work, the monumental "History of Ancient Mathematical Astronomy."

8.4. Review by: George Saliba.
Isis 4 (1984), 735-736.

It is hard to imagine the existence of a historian of science who has not used the works of O Neugebauer at one time or another. It is harder still, maybe even impossible, to find a historian of astronomy who has not studied these works with great diligence. But I assume that it is also hard to find any one library other than that of Neugebauer himself - which would include all of these works. Therefore, the need for making Neugebauer's articles available in print again has always been felt, but could never be realised due to the immensity of the volume involved.

It was Noel Swerdlow who finally hit on the brilliant idea of making a selection of those essays that Neugebauer published over the years in a variety of places, hoping that this selection could serve as an introduction to Neugebauer's monumental work A History of Ancient Mathematical Astronomy, and the forthcoming joint work with Swerdlow on Copernicus's De revolutionibus. But once one selects, one has to develop a reasonable criterion for exclusion which has to keep the final product economically feasible. It was this criterion that I presume caused the greatest pain, for I can imagine a hundred arguments and one why certain articles should not have been excluded. This reviewer feels that the working criterion that was finally adopted must have come as a result of the continuous irritation that one feels when one still sees in secondary literature "myths" that were supposed to have been put to rest years ago by Neugebauer; such myths as the "simplification of the Ptolemaic system in Copernicus's De Revolutionibus" or "the discovery of precession by the Babylonians" do not die easily. The volume under review was therefore put together as an introduction to the more technical works of Neugebauer mentioned above, and as another attempt to attack these myths again and maybe this time put at least some more of them to a final rest.

The forty-three essays selected range over a variety of subjects. Eight are of a general nature - this reviewer's favourites among them are the one-page note on "The Study of Wretched Subjects" and "The Transmission of Planetary Theories in Ancient and Medieval Astronomy." Four cover Egyptian astronomy. The six on Babylonian astronomy include the essay on "The Alleged Discovery of the Precession of the Equinoxes." The fifteen on Greco-Roman subjects include such celebrated essays as "The Early History of the Astrolabe," "The Astronomical Origin of the Theory of Conic Sections," and "Apollonius' Planetary Theory." Finally, ten cover medieval and Renaissance subjects, including the essay "On the Planetary Theory of Copernicus," required reading for anyone interested, even if only tangentially, in Copernican astronomy.

The geographical area covered by these essays also ranges over all of the ancient world, with the exception of China. The selection includes cultures and civilisations little known from other sources, such as "Hindu Astronomy," "Tamil Astronomy," and "Ethiopic Easter Computus."

Besides being an excellent selection of essays on ancient astronomy that should be on the shelf of any historian of science, this reasonably priced paper volume will also put invaluable source material within the student's reach in all courses on ancient and medieval astronomy. For the greater general audience, anyone who ever expects to understand the more technical works of Neugebauer, this volume will be an excellent place to begin.

9.1. Review by: Kristian Peder Moesgaard.
Mathematical Reviews MR0752698 (86f:01013).

For several generations to come these two companion volumes will form an indispensable work of reference for any student of Copernicus and of astronomy in the Renaissance. Cosmology, philosophy, life, work and historical basis of Copernicus' achievement get their proper 13% share of the book in its introductory chapter. In the remaining 87% of the pages the authors follow, step by step, Copernicus' technical and numerical foundation of his new doctrine of astronomy. This includes not only explanations of Copernicus' actual routes of deriving parameters, but also possible reconstructions of intermediate calculations in cases where Copernicus omitted the details. Copernicus' results are evaluated, his possible sources sorted out, and the results of modern scholarship commented upon. As Swerdlow's preface runs: "the work ... could well end up being more on the subject of Copernicus's mathematical astronomy than anyone cares to know." With all the tedious computational work done once and for all, the result in published form is a gift to the history of astronomy, a platform from which future scholarship may take off and shape its course. The division into two volumes with tables and figures bound separately facilitates the use of the work. Sample inquiry reveals a very high reliability of the numerical material.

9.2. Review by: Victor E Thoren.
Science, New Series 227 (4688) (1985), 744-745.

The great work from which historians date the beginning of modern science, Copernicus's De revolutionibus, has been republished six times in its original Latin and translated more times than that into at least five modern languages. Yet for all this attention, nothing has been produced that can be termed a critical edition, or even a satisfactory translation, because none of the enterprises has been based upon the detailed technical understanding and extensive re-computation necessarily involved in comparing De revolutionibus with its great predecessor, Ptolemy's Almagest. Years ago, long before the appearance of the spate of new editions and translations in connection with the 500th anniversary of the birth of Copernicus in 1973, Otto Neugebauer envisioned an extensive commentary on Copernicus, as the final phase of a study of the entire tradition of Greek astronomy. The project proved too ambitious even for the venerable Neugebauer, however, and it has only now been completed by one of his disciples. The result is some 400 pages of analysis, accompanied by well over 200 diagrams and some 20 graphs, covering every aspect of Copernicus's mathematical astronomy.

This book is not for tyros. It is not that either the mathematics or the astronomy is severe; in fact, the closest I can come to criticising Swerdlow's study is to express the opinion that many readers will wish for analytical expressions of some of the interminable geometry. But the historical discussion has to assume some background and sophistication in order to be useful to the professionals to whom it is directed, and most non-professionals will, accordingly, find a fair number of bewildering allusions and even apparent gaps in the presentation. For those who persevere, however, Swerdlow has a number of interesting general points to make, amid the mass of detail of individual calculations. What all of these points convey, in various ways, is that science was no easier to do in the 16th century than it is now. If the rules for doing science were so much looser as to allow Copernicus to indulge in shortcuts that will be viewed variously as tragic or comical, the methods available for coping with the task "legitimately" were correspondingly feebler. Four passages from Swerdlow's introductory summary should suffice to illustrate the problems:

This [procedure] would be difficult enough if the parameters were correct, but mostly they were not, resulting as they did from very sensitive derivations from less-than-accurate observations.

Nevertheless, it is one of the most confusing sections of De revolutionibus, containing many errors and internal contradictions, due to an inconsistent revision of an originally flawed exposition.

The agreement between the observation and computation comes out perfectly, but only because Copernicus first altered the time of the observation by 40 m and the longitude of the star by 10'.

the analysis required to discover these consequences was very difficult and required observations of a sort that it never even occurred to Copernicus to make.

Inevitably, a commentary such as this is Monday morning quarterbacking. Yet Swerdlow never allows the reader to forget that Copernicus was not always even trying to do as much as modern readers might presume he was and that even what he was trying to do was very difficult. And, though Swerdlow's analysis is probably more pointed than what most historians of science indulge in, it is probably gentler than most people without historical training will find comfortable. Most controversial for scientists will probably be the fundamental chain of assumption underlying the analysis: (i) Copernicus truly seems to have assumed (more or less correctly) that Ptolemy's models were mathematically correct representations of the phenomena, but physically impossible ones; so (ii) all he (Copernicus) had to do was find physically reasonable models (without equants) that were mathematically equivalent to Ptolemy's; therefore (iii) Copernicus "can hardly be blamed for not undertaking to investigate [the phenomena themselves, which] Ptolemy and Regiomontanus both considered to be settled;" and so (iv) "comparison with modern computed positions (of the planets) is not really of interest and gives a positively misleading idea of the contemporary judgment that is of real historical interest." Thus, though Swerdlow asserts that Copernicus's theories give better results than Ptolemy's did because the initial conditions were more accurate, some readers will be disappointed by his refusal to provide the comparisons with modern theory that have been traditional for this enterprise.

Although many readers will agree that this book contains "more [detail] than anyone wants to know," there can be little doubt that it will join Neugebauer's History of Ancient Mathematical Astronomy as one of the classics of the field.

9.3. Review by: George Saliba.
Renaissance Quarterly 40 (1) (1987), 109-112.

There are very few historical scientific figures who have been burdened in the secondary literature on history of science with more myths and misconceptions than Nicolaus Copernicus. This, despite the fact that some very fine scholarship has already been devoted to the works of Copernicus. The authors of the present volume have themselves contributed extensively within the last two to three decades to the Copernican scholarship - see, e.g. Neugebauer's major article in Vistas in Astronomy, 1968, "On the Planetary Theory of Copernicus," and Swerdlow's definitive critical work in The Proceedings of the American Philosophical Society, 1973, on Copernicus' De Commentariolus yet the myths persist and the secondary literature continues to churn out one article after another and one book after another about Copernicus' contribution to mathematical astronomy as being "the simplification of the Ptolemaic system," or the desire "to get rid of the epicycles," or some naive assertion that Copernicus wished "to eliminate the equant" from the planetary models.

One would think that since Copernicus was so much misunderstood, and because he was such a famous figure to whom at least one revolution has been attributed, many a scholar would devote a life-time to Copernican studies. And yet, no such thing has taken place. Besides the works of the two authors under review, no one else has seriously undertaken the task of investigating the works of Copernicus in a systematic fashion with the hope of understanding them within the context of the general history of astronomy. Instead, those who have studied Copernicus did so either to buttress specific ideas they had about the nature of the scientific revolution that they named after him, or else to add glory to fame and to praise him as the innovator, the revolutionary, or even the champion who single-handedly took on the most stupendous power of the Renaissance, the Catholic Church.

At last, this badly needed book has seen the light of day. Originally conceived as a continuation of the monumental work of Neugebauer - A History of Ancient Mathematical Astronomy, published by the same house in 1975 - it was supposed to bring the discussion of the problems already treated in the literature of Antiquity up to pre-modern times. But once Swerdlow had published the most comprehensive study of Copernicus' earlier work on planetary astronomy, the Commentariolus, it was thought that efforts should be combined to produce a similarly definitive study of the later Copernican work the De Revolutionibus, which combines the kind of scholarship that was exhibited in HAMA, as Neugebauer's work is now known, and on the Commentariolus. This is indeed achieved in the present volume.

Beyond being a definitive study of the mathematical astronomy of Copernicus, as it is known from the De Revolutionibus, this volume should put the above mentioned myths to a final rest. It demonstrates beyond any doubt to anyone who can read sophisticated mathematical astronomy that Copernicus did not "simplify" the Ptolemaic system - indeed he made it more "complicated," he did not abolish the epicycles - he added some of his own, and he did not dispose of the equant; instead he went to great pains to preserve it.

But most important, the astronomy of Copernicus, whether in the Commentariolus or in the De Revolutionibus, will now be seen in a totally different light, for this volume also brings together the results of the research done in Islamic astronomy for the last three decades or so, and also demonstrates beyond a shadow of a doubt that the work of Copernicus must now be seen as a continuation of the same astronomical research done by a group of Muslim astronomers who lived in the thirteenth and fourteenth centuries, and who are now known by their collective name as the "Maragha School" astronomers. In fact, the researches of Victor Roberts (1957, 1966), Edward Kennedy (1959, 1966, 1976), Fuad Abbud (1962), and the present reviewer (1979, 1980) - all cited in the bibliography - have already explored most of the planetary models that may be conceived of as background to Copernican astronomy. What this volume does is to add, in the form of a comprehensive book, all the details that complement that line of research. In the words of Swerdlow and Neugebauer, for example, when discussing Copernicus' lunar model: "Since the recent discovery that Copernicus' lunar model was anticipated some two hundred years earlier by Ibn al-Shațir of Damascus (1306-1375) [the last member of the "Maragha School"], nothing of much originality is left to Copernicus' lunar theory." In another place in the same volume, when discussing the background to Copernicus' astronomy under the heading "Arabic Astronomy and the Maragha School," the authors conclude with the following statement: "The question therefore is not whether, but when, where and in what form he [meaning Copernicus] learned of Maragha theory." Again, when discussing the general planetary theory, the authors conclude: "In a very real sense, Copernicus can be looked upon as, if not the last, surely the most noted follower of the 'Maragha School."

This book then establishes one of the most important shifts in our understanding of Renaissance astronomy. We now have to perceive this most fundamental work of Renaissance astronomy as a continuation of a long tradition already begun in the Muslim East some three hundred years earlier, and thus we have to rethink such concepts as the Copernican Revolution, and whatever that entails. I am sure that this will occupy generations of scholars for years to come. For the time being, however, we are fortunate to have a book that should stem the tide of nonsense that has been printed about Copernicus, destroy most of the myths that have been propagated about him in the secondary literature, and finally give a new impetus to the line of research that considers the works of Copernicus as a continuation of the works of the "Maragha School" astronomers. This reviewer can only add his voice to those at the committee on prizes in the History of Science Society who awarded this book the Pfizer Prize at their last meeting in the Fall of 1985.